Operational Research in Engineering Sciences: Theory and Applications Vol. 4, Issue 1, 2021, pp. 115-135 ISSN: 2620-1607 eISSN: 2620-1747 DOI: https://doi.org/10.31181/oresta20401156j * Corresponding author. antras1209@gmail.com (Ž. Jokić), dbozanic@yahoo.com (D. Božanić), dpamucar@gmail.com (D. Pamučar) SELECTION OF FIRE POSITION OF MORTAR UNITS USING LBWA AND FUZZY MABAC MODEL Željko Jokić*, Darko Božanić, Dragan Pamučar Military Academy, University of Defence in Belgrade, Serbia Received: 18 February 2021 Accepted: 26 March 2021 First online: 28 March 2021 Original scientific paper Abstract: The paper presents a hybrid model based on the LBWA method and the fuzzy MABAC method, applied when selecting firing positions' locations of the Serbian Army's mortar units. Using a questionnaire, the experts determined the criteria for choosing the firing position. The LBWA method is used to determine the weighting coefficients of the criteria, while the fuzzy MABAC method is used to determine the most favorable location of the firing position by choosing between six specific options - alternatives. By changing the value of the elasticity coefficients, the sensitivity analysis of the developed model was performed, and by applying the Spearman coefficient, it was determined that there is an ideal positive correlation of ranks. Key words: LBWA, MABAC, fuzzy numbers, mortar units, firing position. 1. Introduction The entire twentieth and the beginning of the 21st century were marked by dizzying technological developments that could not but include the military industry. The impact of technological development on armaments and military equipment also conditioned a change in the armed conflicts' physiognomy. Modern combat conflicts are characterized by: sudden and rapid actions of forces from a distance, with mass use of armored and mechanized units and special forces on land, frequent use of helicopter landings, strong air support, and constant possibility and the threat of using weapons of mass destruction. However, there are means of military equipment which, despite the stated technological development, have not undergone significant changes during all this time, and even without them, no major armed conflict can be imagined. From its appearance in 1904, in the Russo-Japanese War, to the present day, mortars have undergone small changes. They are produced in various calibers, of which the most common are 60 mm, 81-82 mm, and 120 mm, as traction or self-propelled. With the mailto:antras1209@gmail.com mailto:dbozanic@yahoo.com Jokić et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (1) (2021) 115-135 116 possibility of shooting in a vertical path, they are suitable for shooting from bays, ravines, and from the back slope. Mortar units are the basis of the infantry battalion's fire support in performing all types of combat operations. With their firepower and the possibility of quick maneuver, they can bring an advantage on the field, provided they are used correctly, which, above all, depends on the correct choice of the location of the combat schedule elements. The article discusses the choice of the location of the battalion fire group fire position (BFG) formed by the company-platoon of 120 mm mortars. The ultimate goal of the article is to apply a model that will support the decision-maker in choosing the location of the firing position, which would significantly reduce the response time and the possibility of making an inadequate decision. With the relatively newer LBWA method, the weight coefficients of the criteria for the selection of the location of the firing position (FP) will be determined. Experts in the subject area identified eight criteria, based on the applicable rules and instructions, on which the choice of the location of the firing position directly depends. The choice of the specific location of the firing position, between the six options, will be solved by applying the fuzzy MABAC method. 2. The place and role of mortars in contemporary combat actions The 120 mm mortar is an accompanying infantry weapon, intended for neutralizing and destroying manpower and firepower, creating smoke curtains, blinding observation posts and firing points, illuminating battlefields, opening passages through wire barriers and minefields, and demolishing light fortification barriers at distances of about 6500 m (Military Encyclopedia, 1973). According to the formation of MB 120 mm, mortar companies or platoons are formed, depending on whether it is an infantry or mechanized battalion. During combat operations, when operating within a battalion, a company-platoon of 120 mm mortars forms a battalion fire group of temporary composition. At the decision of the Commander, the battalion fire group may be attached to another unit or perform tasks for the needs of a higher unit. The tasks of the battalion fire group, during the execution of combat operations, derive from the purpose of the 120 mm mortar (Military Encyclopedia, 1973): - neutralization and destruction of the enemy's manpower, firepower and fire support, - fight against enemy landings, - neutralization of enemy observation posts and observation posts, - neutralization of enemy command posts and communication centers, - demolition and destruction of field-type fortifications and opening of passages in obstacles, - smoking and lighting of certain areas and rooms. When conducting combat operations, BFG possesses elements of the combat schedule, which is part of the combat schedule of the battalion in which it operates, ie the unit it supports. The combat schedule of the BFG consists of an observation post, a firing position and a place of means of transport. The command part has an Selection of fire position of mortar units using LBWA and Fuzzy MABAC model 117 observation post, and the fire part has a fire position. Before possessing the stated elements of the combat schedule, the selection of optimal regions-locations for the execution of the obtained task is performed. As the topic of the paper is the choice of the location of the firing position in the future, the paper will not deal with the observation post. The fire position (FP) is a region on the land where people, tools, ammunition and traction equipment are deployed in order to perform a fire task (Kurtov et al., 2014). According to the purposes, FP can be: basic, reserve, temporary, next and false, while according to the degree of shelter: sheltered, semi-sheltered and discovered (unprotected). There are no works in the domestic and foreign literature that deal exclusively with the problem of choosing the firing position for mortar units. In addition to the rules and manuals that deal with mortars from the point of view of construction, some authors in works such as Department of the Army (2017), Jenzen-Jones (2015), consider mortars from the aspect of their application. The choice of the location of the basic VP for mortar units belongs to the group of location problems, which are considered in the literature in different ways, both by the type of location and by the applied methods. The problem of the location of military facilities was discussed by Karatas et al. (2019). Božanić & Pamučar (2010) select the location of the bridge crossing using Fuzzy logic. Also, Pamučar et al. (2019) select the optimal location for water barriers using the Interval-Valued Fuzzy-Rough Numbers and MAIRCA methods. Sennaroglu & Celebi (2018) use the AHP, PROMETHEE and VIKOR methods to select the location of the military airport. Pamučar et al. (2016) selected the firing position of the brigade artillery group in the defensive operation using a hybrid model fuzzy AHP - TOPSIS and a fabricated Satie scale. Hamurcu & Eren (2019) using multi-criteria decision-making using the AHP and TOPSIS methods select the best motorcycle route in Ankara. Stoilova (2020) using the AHP and SIMUS optimal railway route in case of an emergency. Liang et al. (2020) address the problem of route selection for perishable goods vehicles. Xu et al. (2020) solve a similar problem by multi-criteria analysis. Darbari et al. (2016) using multi-criteria analysis determine the optimal locations for the collection and disposal of recycled electrical equipment. Ortiz-Astorquiza et al. (2018) conduct a comprehensive overview of problems with the location of accommodation facilities. A similar problem is addressed by Küçükaydın & Aras (2020) using the Fuzzy C-means cluster. Contreras & O’Kelly (2019) address the problem of hub location when designing networks in transportation and telecommunications systems. The hybrid model AHP and PROMETHEE, Abdel-Basset et al. (2021) are used to select the location of coastal wind farms. Pan et al. (2021) are conducting a case study on the selection of the most suitable pedestrian overhead bridge location for the installation of elevators in Singapore using an adaptive Bayesian network. 3. Description of the method The hybrid model, applied when solving the problem of choosing the location of FP mortars, consists of LBWA and fuzzy MABAC method. The LBWA method is used to determine the weight coefficients of the criteria, while the fuzzy MABAC method is used to determine the most favorable location of the mortars position. Figure 1 shows the scheme of the model. Jokić et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (1) (2021) 115-135 118 Phase 1 - Defining criteria Phase 2 – Calculation of weight coefficients of criteria Phase 4 - Sensitivity analysis Phase 3 - Choosing the best alternative Expert evaluation Expert evaluation, LBWA method Fuzzy MABAC Change weighting coefficients, Spearman coefficient Figure 1. Model scheme 3.1. Level Based Weight Assessment (LBWA) model The model of weight assessment based on levels (LBWA) was presented for the first time in their work by Žižović & Pamučar in 2019. Although a relatively new method, LBWA has so far been applied in several papers in solving various problems. After the first presentation of the method, Božanić et al. (2020) use a hybrid LBWA - IR-MAIRCA model of multi-criteria decision-making for weapon selection. Fuzzy LBWA – MACBETH – RAFSI was used to develop a multi-criteria model for the sustainable reorganization of the health system in the emergency situation caused by the COVID-19 pandemic (Pamučar et al., 2020b). The choice of the way passengers arrive at the airport in Istanbul was made, also by applying the fuzzy LBWA-WASPAS- H model (Pamučar et al., 2020a). LBWA is a subjective model for determining weighting coefficients. Advantage of LBWA metod over other is in next keys (Žižović & Pamučar, 2019): (1) Calculation of weighting coefficients can be realized with a small number of comparison criteria; (2) A simple algorithm of the LBWA method; (3) А simple mathematical apparatus is used to obtain the weighting coefficients; (4) After realized comparisons of criteria, the coefficient of elasticity enables additional corrections of the values of weight coefficient. Criteria must be defined before applying the LBWA method. If the number of criteria is denoted by n, then a set of criteria is available  1 2, , , nS C C C= . After that, the LBWA method is approached through the following steps (Žižović & Pamučar, 2019): Step 1. Determining the most significant criterion. The most important criterion is the one that, in the opinion of experts, has the greatest influence. Selection of fire position of mortar units using LBWA and Fuzzy MABAC model 119 Step 2. Grouping the criteria by levels of significance in relation to the most significant criterion, according to the following: Level 1 S : From the set S, at the level of S1, criteria are grouped that are of equal importance as C1 or are up to twice less significant than C1; Level 2 S : From the set S, at the level of S2, the criteria are grouped exactly two times less significant than C1 or are up to three times less significant than C1; Level S3: ... By applying the previously, the decision-maker is grouping criteria according levels of significance. If the significance of the criteria j C marks with ( ) j s C , wherein  1, 2, ,j n , then we have 1 2 kS S S S=    , where for each level  1, 2, ,i k , it is true that it is     1 2 , , , : ( ) 1 si i i i j j S C C C C S i s C i= =    + (1) Also, for everyone  , 1, 2, ,p q k such that it is p q the intersection of the sets is p q S S =  . Step 3. Within the formed subsets, criteria according to significance are compared. Each criterion pi i C S from the set   1 2 , , , si i i i S C C C= is assigned an integer  0,1, , pi I r , that the most important criterion 1 C is assigned a number 1 0I = . If it is pi C more significant than qi C than it is p q I I , and if it is pi C of the same importance as qi C , than it is p q I I= . Expression (2) gives the maximum value of the scale for comparing the criteria r.  1 2max , , ,= kr S S S (2) Step 4. Based on the defined maximum value of the scale for comparing the criteria, coefficient of elasticity is determined 0 r N which should satisfy the condition that 0 r r . Step 5. The calculation of the criterion influence function is realized in the following way: 0 0 ( ) p p i i r f C i r I =  + (3) where i is the number of levels / subsets into which the criterion is classified, r0 represents the coefficient of elasticity, while  0,1, , pi I r represents the value assigned to the criterion pi C within the observed level. Step 6. Calculation of optimal values of weight coefficients of criteria: Jokić et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (1) (2021) 115-135 120 1 2 1 1 ( ) ( ) = + + + n w f C f C (4) The values of the weighting coefficients of the other criteria: 1 ( ) j j w f C w=  (5) where in 2,3, ,j n= , and n the total number of criteria. If expert decision-making is performed, as previously stated, after each expert determines the values of weighting coefficients, the aggregation of individual judgments according to the considered criterion (Aij) is started (Blagojević et al., 2017). In the Aij method, to obtain the group coefficient, the weighted geometric mean method is used, which is calculated as the nth root of the product of all elements of the data set using expression (6): 1 1 2 1 * *...* = = =  n n n jC j j jn j i w w w w w (6) where in 1jC w combined weighting factor for the criterion C1, 1jw expert weighting factor (E1) and n number of weighting coefficients according to the given criterion. 3.2 Fuzzy sets In classical set theory, the membership of elements in a set is estimated in a binary sense according to the bivalent condition - the element either belongs or does not belong to the set (Chatterjee & Stević, 2019). However, it is not always possible to make a clear division, especially of complex phenomena, which cannot be easily described by traditional mathematical methods, especially when the goal is to find an approximately good solution (Bojadziev & Bojadziev, 1996). Modeling using fuzzy sets has been shown to be an effective way of formulating decision-making problems, where available input information is subjective or imprecise (Zimmermann, 1998). Fuzzy sets are sets whose elements have membership degrees. The theory of obscure sets was first introduced by Zadeh (1965), whose application enables decision-makers to deal effectively with uncertainties. Since then, fuzzy sets have been used by many researchers in solving various problems alone or in combination with other methods of multi-criteria decision- making. Thus, Kushwaha et al. (2020) and Panchal et al. (2019a, 2019b) use Fuzzy FMEA to assess risk and improve safety in various engineering systems. Also Pamučar et al. (2016) use Fuzzy Logic System of Type 2 to assess the risk of natural and other disasters in the Republic of Serbia while Božanć et al. (2015) in risk assessment when overcoming water obstacles in a defense operation. Similar to the previous one, Gopal & Panchal (2021) use the Fuzzy Lambda-Tau (λ-τ) approach in the dairy processing industry. The Lambda-Tau fuzzy method was also applied when determining the time interval of regular maintenance of a coal-fired thermal power plan (Panchal et al., 2020) as well as when analyzing the performance problems of a chemical process plan (Panchal & Srivastava, 2019). Fuzzy sets are used mainly with triangular (TFN), trapezoidal and Gaussian fuzzy numbers. A fuzzy set à is a set of ordered pairs consisting of elements x of the Selection of fire position of mortar units using LBWA and Fuzzy MABAC model 121 universal set H and a certain degree of affiliation μÃ(h), shape Ã={ (x, μÃ(x))︱xX, μÃ(x)[0,1](Zadeh, 1965). The membership function of the μà fuzzy set à is the mapping μÃ: X → [0,1], where à is a subset of the universal set H. Due to its fairly simple membership function, the triangular fuzzy number is one of the most commonly used fuzzy numbers, is defined by the following form: 1 , 1 , 0, otherwise  − − −    − = −   +     A m x m l x m l x m m x m u u (7) Fuzzy number is denoted as ( , , )=A l m u . The value of m mark the basic value of the fuzzy number, a l deviation from the left, that is, u to the right of the modal value. A very important concept associated with the application of fuzzy numbers is the dephasing process, which converts a fuzzy number into a real number. Several methods for performing dephasification can be found in the literature. The most widely used dephasification procedure is the centroid method, which is also known as the center of gravity or the Kwong method (Kwong & Bai, 2003). The triangular fuzzy of the number à = (l, m, u) is translated into a real number using the following expression: ( 4 * ) 6 + + = l m u M (8) 3.3. Fuzzy MABAC method Multi-Attributive Border Approximation area Comparasion (MABAC) The MABAC method is a reliable tool for rational decision-making (Pamučar & Ćirović, 2015). So far, it has been used in a large number of works independently or in one of the modifications. Alinezhad & Khalili (2019) in their book, among others, deal with the MABAC method. Sun et al. (2017) use the MABAC method to determine the priority of patient care. Using a modified rough method, AHP-MABAC, Sharma et al. (2018) determined the priority stations in the Indian Railways. Some authors combine the basic MABAC motor with fuzzy sets q-ROFS (Wang, 2020). Wei et al. (2019) apply the MABAC meter in ranking medical equipment suppliers. Božanić et al. (2016) applied the MABAC method in support of decision-making on the use of force in a defensive operation. Liang et al. (2019) use the MABAC method when assessing risk. Also, using this method, some authors selected the most suitable route of new lines in road and railway traffic (Luo et al., 2019). When defining the new interval-valued fuzzy-rough numbers (IVFRN) method, Pamučar et al. (2018) modified the BWM (Best – Worst method) and MABAC methods. Mishra et al. (2020) select a programming language using the MABAC method. Due to its consistency mentioned earlier, the MABAC method can be found in many more papers. Тhe fuzzy MABAC method solves the problem in three steps (Bobar et al., 2020): Jokić et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (1) (2021) 115-135 122 Step 1. Forming the initial decision matrix ( X ). The first thing is to do assessment m alternatives according to n criteria. Alternatives are shown in vector form ( )1 2, ...,=i i i inA x x x . 1 2 1 11 12 1 2 11 22 2 1 2 ... ... ... ... ... ... ... ...      =       n n n m m m mn C C C A x x x A x x x X A x x x (9) Step 2. Normalization the initial matrix ( X ) 1 2 1 11 12 1 2 11 22 2 1 2 ... ... ... ... ... ... ... ...      =       n n n m m m mn C C C A n n n A n n n X A n n n (10) Elements of a normalized matrix ( N ) are determined by the equation: a) For benefit (max) type criteria − + − − = − ij i ij i i x x n x x (11) b) For Cost (min) type criteria ij i ij i i x x n x x + − + − = − (12) Step 3. Calculate the elements from the weighted matrix ( V ). * ij i ij i v w n w= + (13) ij n is the elements of a normalized matrix ( N ), and i w is the weighting coefficients of the criteria. Using equation (19) we receive a weighted matrix 11 12 1 21 22 2 1 2 ... ... ... ... ... ... ... n n m m mn v v v v v v V v v v      =       (14) Step 4. Determining the matrix of the approximate boundary area ( G ). Selection of fire position of mortar units using LBWA and Fuzzy MABAC model 123 Boundary approximation area (BAA) is determined based on the expression: 1/ 1 m m i ij j g v =   =      (15) After calculating the value i g for each criterion, a matrix of boundary approximate domains is formed G .   1 2 1 2 ... ... n n C C C G g g g= (16) Step 5. The calculation of the distance of the alternatives from the boundary approximate domain is obtained as follows: 11 12 1 21 22 2 1 2 ... ... ... ... ... ... n n m m mn q q q q q q Q q q q      =       (17) Q V G= − (18) Alternative i A may belong to the boundary approximate domain ( G ), upper approximate area ( G + ) or the lower approximate domain ( G − ). Belonging to an alternative i A area of approximation ( G , G + or G − ) is determined on the basis of the equation (19). 0 0 0 ij i ij ij G if q A G if q G if q + −    =   (19) Step 6. Ranking alternatives. Through the sum of the distances of the alternatives from the boundary approach area ( i q ) the calculation of the values of the criterion functions for alternatives was received. The final value of the criterion functions of the alternatives was received by calculating the sum of the elements of the matrix Q by rows. 1 , 1, 2,..., , 1, 2,..., n i ij j S q j n i m = = = = (20) 4. Application of the hybrid model of multi-criteria decision making The LBWA – Fuzzy MABAC hybrid model consists of four phases. In the first phase of the model, based on expert assessment, the criteria are defined. In the second phase, the calculation of the weight coefficients of the criteria is realized using expert Jokić et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (1) (2021) 115-135 124 assessment and the LBWA method. In the third phase, the best alternative is selected using the fuzzy MABAC method. The last phase includes the sensitivity analysis of the developed model and the correlation of ranks. 4.1. Criteria for choosing the firing position In the first phase, the criteria are defined that in the further work directly affect the choice of the best alternative, ie. optimal locations for the firing position. Defining criteria and their weighting coefficients represents an important phase for decision- making models (Pamučar et al., 2016). Due to the complexity of the problem in defining the selection criteria, experts were hired. Experts identified eight criteria for the considered problem, which are listed from C1 to C8. The criteria for selecting the location of the fire position (FP) of the battalion fire group (BFG), which is formed by a company of 120 mm mortars, were defined on the basis of expert opinion, and the data from the rules served as the basis for the survey. The selection of the most favorable location of BFG is made on the basis of eight criteria: C1 - distance to the target, expressed in meters (ideal location is generally defined at 1/3 of the range of the weapon from the front end of its own forces when the unit is in attack, or 2/3 when in defense). C2 - the ability to observe the firing position by the enemy. In the professional literature, the stated criterion is defined as the shelter of the firing position, and on that basis, the division into sheltered, semi-sheltered and discovered (un sheltered) firing position was made. The detected firing position allows direct aiming at the target. On it the enemy can spot people and tools. The semi-sheltered firing position makes it impossible for the enemy to visually spot people, but it can detect it by smoke and flash when firing a mine. The sheltered firing position prevents the enemy from observing from the ground or detecting the firing position by the smoke and flash of a fired mine. C3 - masking conditions (terrain characteristics that enable successful masking of BFG and movement of parts or the whole BFG). C4 - soil bearing capacity - terrain characteristics on which the accuracy of shooting depends. When shooting from too hard ground, the ground bounces off the ground, while on soft ground it collapses, which requires additional soil reinforcement. C5 - the size of the parallax expressed in thousands of parts of the angle. The parallax of the target is the angle between the line of sight and the line of fire. If it is in the range from 0-00 to 3-00 it is small, from 3-00 to 5-00 it is medium and over 5-00 it is large. C6 - distance of the observation post from the firing position. The distance of the observation post from the firing position directly affects the duration and accuracy of the correction. The smaller the distance, the more precise the correction will be, and thus the faster it will be completed. Based on that, there is a division into near and far observatories. The observation post is close if it is within 10% of the shooting distance. Selection of fire position of mortar units using LBWA and Fuzzy MABAC model 125 C7 - access conditions. The approach conditions directly affect the speed of the firing position, and thus the time of preparation of the unit for opening fire. Having in mind the mass of individual parts of the 120 mm mortar, it is not at all negligible whether the tools can be brought by motor vehicle to the firing position or the handlers have to carry them by hand. C8 - distance to own units. The duration of the correction also depends on the distance of the firing position to one's own units. The closer the units are to the firing position, the easier it is to make a correction, and thus in a shorter time. The set of criteria Cj consists of two subsets, a subset of the benefit type criteria, which means that a higher value of the criterion is more desirable, ie. better, denoted by C + and a subset of cost-type criteria, which means that a smaller value is more desirable, ie. better marked with C -. In this particular case, the subset of criteria C+ includes criteria C2, C3, C4 and C7, while the subset of criteria C- includes criteria C1, C5, C6 and C8. The values of criteria C1 and C8 are shown as numerical values while the values of criteria C2 are shown through a linguistic scale from 1 to 5 as seen in Table 1. Table 1. Linguistic descriptors for criterion C2 Linguistic descriptor discovered (Ds) semi-sheltered (SS) sheltered (S) Assigned numeric value 1 3 5 The values of criteria C3, C4, C5, C6 and C7 are presented as fuzzy linguistic descriptors (table 2, table 3 and table 4). Table 2.Fuzzy linguistic descriptors for criteria C3, C4, C7 Linguistic condition fuzzy number Bad (B) (0, 1, 3) Good (G) (2, 3, 5) Excellent (E) (4, 5, 5) Table 3. Fuzzy linguistic descriptor for criterion C5 Linguistic condition Fuzzy number Small (S) (0, 2, 3.5) Medium (M) (2.5, 4, 5.5) Large (L) (4.5, 6, 7.5) Table 4: Fuzzy linguistic descriptor for criterion C6 Linguistic condition fuzzy number Close (C) (0, 450, 600) Remote (R) (480, 640, 1000) 4.2. Calculation of weight coefficients of criteria using LBWA method In the second phase, the calculation of the weight coefficients of the criteria is performed using the LBWA method, in the previously described manner. After the selection of the most important criteria by the experts, the determination of the weighting criteria is presented in this text. Determination of weighting coefficients is shown for one expert (E1). As 11 experts participated in the research, in the end the Jokić et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (1) (2021) 115-135 126 aggregation of weight coefficients from all of them was performed and the weight coefficients were obtained, which were further used when choosing the firing position of the 120 mm mortar using the MABAC method. Step 1: For the most important, the E1 expert chose criterion C2. Step 2: The criteria are classified into three levels: S1 = {C2,C8, S2 = {C1,C6,C4, S3 = {C7,C5,C3. Step 3: Based on expression (2), the maximum value of the scale for comparing the criteria is defined r = max {|S1|, |S2|, |S3| = 3 Based on the comparison of criteria according to their significance, C2 gets the value I2 = 0 as the most significant criterion, while other criteria according to their importance in their sub-levels, get the following values: S1: I8=1; S2: I1=1, I6=2, I4=4; S3: I7=1, I5=3, I3=3. Step 4: Based on the defined maximum value of the scale for comparing the criteria r = 3, the coefficient of elasticity r0 = 4 is defined. Step 5: Using expression (3), the influence functions of the criteria were calculated: 2 8 1 6 4 7 5 3 4 4 ( ) 1, ( ) 0.8, 1* 4 0 1* 4 1 4 4 4 ( ) 0.444, ( ) 0.4, ( ) 0.333, 2 * 4 1 2 * 4 2 2 * 4 4 4 4 4 ( ) 0.308, ( ) 0.267, ( ) 0.267 3* 4 1 3* 4 3 3* 4 3 f C f C f C f C f C f C f C f C = = = = + + = = = = = = + + + = = = = = = + + + Step 6: Using expression (4), the weight coefficient of the most influential criterion was obtained 2 1 0.262 1 0.8 0.444 0.4 0.333 0.308 0.267 0.267 w = = + + + + + + + while the values of weight coefficients of the remaining criteria were obtained by applying the expression (5): 1 3 8 0.262 * 0.444 0.116, 0.262 * 0.267 0.209, ... 0.262 * 0.8 0.209. w w w = = = = = = Based on the previous, the vector of expert weight coefficients (E1) was obtained: Selection of fire position of mortar units using LBWA and Fuzzy MABAC model 127 wj1=(0.116, 0.262, 0.070, 0.087, 0.070, 0.105, 0.081, 0.209). Using the expression (6), the aggregation of weight coefficients obtained from the experts was performed, on the basis of which a vector of weight coefficients was formed: wj =(0.187, 0.234, 0.072, 0.104, 0.094, 0.085, 0.067, 0.156). After determining the weight coefficients of the criteria, it is possible to move on to the next phase of the model. 4.3. Choosing the best alternative using the fuzzy MABAC method The third phase of the model involves selecting the best alternative for the firing position using the fuzzy MABAC method as described previously. The paper discusses six potential locations-alternatives to the firing position of the mortar unit. The characteristics of the considered locations were obtained by the intelligence-reconnaissance work of the superior command. Step 1. The first step is to form the initial matrix according to expression (9), which is shown in Tables 5 and 6. Table 5 shows the initial decision matrix. Numerical and linguistic values are given for the considered alternatives according to the stated criteria. Table 5. Initial decision matrix Alt. Criteria C1 (min) C2 (max) C3 (max) C4 (max) C5 (min) C6 (min) C7 (max) C8 (min) A1 5850 S E G S C G 1000 A2 4925 SS G E M R E 950 A3 3762 Ds G E L R E 1250 A4 4558 SS B G M C G 1187 A5 5321 S E B S C B 1530 A6 4789 S G G L R G 1987 wi 0.187 0.234 0.072 0.104 0.094 0.085 0.067 0.156 Linguistic values, in Table 6, are quantified into numerical ones. Criterion C2 is shown as a real number after quantification while criteria C3 to C7 are shown as triangular fuzzy numbers. Table 6. Quantification of the initial decision matrix Alt Criteria C1 (min) C2 (max) C3 (max) C4 (max) C5 (min) C6 (min) C7 (max) C8 (min) A1 5850 5 (4, 5, 5) (2, 3, 5) (0, 2, 3.5) (0, 450, 600) (2, 3, 5) 1000 A2 4925 3 (2, 3, 5) (4, 5, 5) (2.5, 4, 5.5) (480, 640, 1000) (4, 5, 5) 950 A3 3762 1 (2, 3, 5) (4, 5, 5) (4.5, 6, 7.5) (480, 640, 1000) (4, 5, 5) 1250 A4 4558 3 (0, 1, 3) (2, 3, 5) (2.5, 4, 5.5) (0, 450, 600) (2, 3, 5) 1187 A5 5321 5 (4, 5, 5) (0, 1, 3) (0, 2, 3.5) (0, 450, 600) (0, 1, 3) 1530 A6 4789 5 (2, 3, 5) (2, 3, 5) (4.5, 6, 7.5) (480, 640, 1000) (2, 3, 5) 1987 Jokić et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (1) (2021) 115-135 128 Step 2. Normalization of initial matrix elements (X). Normalization of elements from the confirmed initial decision matrix was performed using expressions (11) and (12), and the results are shown in Table 7. Table 7. Normalized matrix (N) Alt. Criteria C1 (min) C2 (max) C3 (max) … C7 (max) C8 (min) A1 0 1 (0.8, 1, 1) … (0.4, 0.6, 1) 0.952 A2 0.443 0.500 (0.4, 0.6, 1) … (0.8, 1, 1) 1 A3 1.000 0.000 (0.4, 0.6, 1) … (0.8, 1, 1) 0.771 A4 0.619 0.500 (0, 0.2, 0.6) … (0.4, 0.6, 1) 0.771 A5 0.253 1.000 (0.8, 1, 1) … (0, 0.2, 0.6) 0.441 A6 0.508 1.000 (0.4, 0.6, 1) … (0.4, 0.6, 1) 0 Step 3. Calculation of elements from a weighted matrix (V). The elements from the weighted matrix (V) are calculated on the basis of expression (13) which is shown in Table 8. Table 8. Difficult normalized matrix (V) Alt. Criteria C1 (min) C2 (max) C3 (max) … C7 (max) C8 (min) A1 0.187 0.468 (0.130, 0.144, 0.144) … (0.094, 0.107, 0.134) 0.305 A2 0.270 0.351 (0.101, 0.115, 0.144) … (0.120, 0.134, 0.134) 0.312 A3 0.374 0.234 (0.101, 0.115, 0.144) … (0.120, 0.134, 0.134) 0.267 A4 0.303 0.351 (0.072, 0.087, 0.115) … (0.094, 0.107, 0.134) 0.277 A5 0.234 0.469 (0.130, 0.144, 0.144) … (0.067, 0.080, 0.107) 0.225 A6 0.282 0.469 (0.101, 0.115, 0.144) … (0.094, 0.107, 0.134) 0.156 Step 4. Determination of the boundary approximate domain matrix (G) The boundary approximate area was obtained by applying expression (15), which is shown in Table 9. Table 9. Boundary approximate domain matrix BAA Criteria C1 (min) C2 (max) C3 (max) … C7 (max) C8 (min) gi 0.269 0.379 (0.104, 0.118, 0.139) … (0.096, 0.110, 0.129) 0.251 Step 5. Calculating the distance of the alternative from the area of the approximate boundary for the matrix elements ( Q ) The distance of alternatives from BAA was obtained by applying expressions (18) and (19), Table 10. Selection of fire position of mortar units using LBWA and Fuzzy MABAC model 129 Table 10. Matrix distance alternatives from BAA Alt. Criteria C1 (min) C2 (max) C3 (max) … C7 (max) C8 (min) A1 -0.082 0.089 (-0.009, 0.026, 0.040) … (-0.035,-0.003, 0.038) 0.054 A2 0.001 -0.028 (-0.038,-0.003, 0.040) … (-0.009, 0.024, 0.038) 0.062 A3 0.105 -0.145 (-0.038,-0.003, 0.040) … (-0.009, 0.024, 0.038) 0.017 A4 0.034 -0.028 (-0.067,-0.032, 0.012) … (-0.035,-0.003, 0.038) 0.026 A5 -0.035 0.089 (-0.009, 0.026, 0.040) … (-0.062,-0.030, 0.011) -0.026 A6 0.013 0.089 (-0.038,-0.003, 0.040) … (-0.035,-0.003, 0.038) -0.094 Step 6. Ranking alternatives. To make it easier to represent the final rank of the alternatives using expression (8) the triangular fuzzy number is translated into a real number. According to expression (20), by calculating the sum of the elements of the matrix Q by rows, the final values of the criterion functions of the alternatives were obtained, which is shown in Table 11. Table 11. Rank of alternatives by MABAC method Alt. Criteria Qj Ran k C1 (min) C2 (max) C3 (max) C4 (max) C5 (min) C6 (min) C7 (max) C8 (min) A1 -0.082 0.089 0.022 -0.002 0.027 0.012 -0.002 0.054 0.120 1 A2 0.001 -0.028 -0.002 0.032 0.001 -0.011 0.021 0.062 0.077 2 A3 0.105 -0.145 -0.002 0.032 -0.024 -0.011 0.021 0.017 -0.006 5 A4 0.034 -0.028 -0.030 -0.002 0.001 0.012 -0.002 0.026 0.011 4 A5 -0.035 0.089 0.022 -0.044 0.027 0.012 -0.028 -0.026 0.019 3 A6 0.013 0.089 -0.002 -0.002 -0.024 -0.011 -0.002 -0.094 -0.032 6 Based on the obtained results, it is concluded that alternative A1 is ranked first, ie that the ranking of alternatives is as follows: A1 > A2 > A5 > A4 > A3 > A6. 4.4. Sensitivity analysis The fourth phase includes testing the sensitivity of the applied model, in order for the decision-maker to receive confirmation of the quality of the obtained solution, ie to determine how changes in the weight of criteria lead to changes in alternative ranks (Tešić & Božanić, 2018; Durmić et al., 2020). Checking the stability of the used methods of multi-criteria decision-making is an indispensable step in the process of developing a model to support decision-making (Pamučar et al., 2017). Stability was examined by changing the weight coefficients wi, ie by changing the value of the coefficient of elasticity r0, whose value in the work is r0 = 4. Table 12 shows the influence of the value of r0 on the change in the rank of the alternative: Jokić et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (1) (2021) 115-135 130 Table 12: Rank alternative depending on r0 Alt. r0 = 4 r0 = 5 r0 = 6-9 r0 = 10-20 A1 1 1 1 1 A2 2 2 2 2 A3 5 5 5 4 A4 4 4 3 3 A5 3 3 4 5 A6 6 6 6 6 Based on Table 12, it can be noticed that with the change of the weight coefficient, ie the coefficient of elasticity, the model shows stability. Three alternatives (A1, A2 and A6) retain the rank regardless of the value of r0 while the other alternatives suffer changes of rank for r0 = 5, r0 = 6 and r0 = 10. For the values r0 = 7, r0 = 8 and r0 = 9, the rank is identical as for r0 = 6. Also, for r0 = 11-20 the rank is identical as for r0 = 10. In order to establish the correlation of the ranks obtained by changing r0, the Spearman coefficient was used as in expression (20): n 2 i i 1 2 6 D S 1 n(n 1) == − −  (21) where Di represents the difference of rank according to the given r0 and rank in the corresponding r0, and n the number of ranked alternatives. The Spiraman coefficient belongs to the value interval [-1,1] (Radovanović et al., 2020). When the ranks of the alternatives completely match the Spearman coefficient is 1 (“ideal positive correlation”), when the ranks are completely opposite the Spearman coefficient is -1 (“ideal negative correlation”), ie when S = 0 the ranks are uncorrelated. The values of the Spearman coefficient for the considered problem are shown in Table 13. Table 13: Spearman coefficient values r0 = 4 r0 = 5 r0 = 6-9 r0 = 10-20 r0 = 4 1 1 0.94 0.83 r0 = 5 1 0.94 0.83 r0 = 6-9 1 0.94 r0 = 10-20 1 From the results shown in Table 13, it can be concluded that the values of the Spearman coefficient for all values of r0 are extremely high, ie that there is an ideal positive correlation of ranks. There is no deviation from the ideal positive correlation as well as the negative correlation. Based on the above, it is possible to conclude that the model has sufficient sensitivity. 5. Conclusion On a hybrid model based on the LBWA and fuzzy MABAC method, the paper explains the process of creating a multi-criteria decision model. Through a multi- criteria model, the paper solves the problem of choosing the location of the firing Selection of fire position of mortar units using LBWA and Fuzzy MABAC model 131 position for mortar units of company size MB 120 mm, which has not been considered in this way in the existing literature so far. The paper describes in detail the steps of the LBWA and fuzzy MABAC methods. The experts determined eight criteria for influencing the choice of firing position. Further, the experts identified the most significant criterion, defined the levels of significance and determined the values of the criteria by levels. Part of the criteria, of the linguistic type, obtained numerical values using fuzzy linguistic and linguistic descriptors. As the best choice-alternative, the MABAC method suggests the A1 alternative. Alternative A1 in relation to the others, has the largest battle of criteria belonging to the above approximate domain. As the last phase, the sensitivity analysis of the presented model was performed in the paper, by changes in the weight coefficients of the criteria (by changing the coefficient of elasticity) from r0 = 4 to r0 = 20. The results of the analysis indicate sufficient stability of the model. The first-ranked alternative A1 retains the first position regardless of the growth of the coefficient of elasticity r0. Also, the Spearman coefficient has a great value, which shows that there is an ideal positive correlation of ranks. Based on the existing literature, the LBWA method has not been combined with the MABAC method so far. Based on the results, presented in the model, it is concluded that the combination of these two methods gives consistent results. Further research should focus on testing the applied model through new problems, as well as the application of other methods to the existing problem. The presented model of mortar firing position can have great application in military management. The application of the methods used significantly shortens the time in the decision-making process, which is very important in an uncertain environment. References Abdel-Basset, M., Gamal, A., Chakrabortty, R. K., & Ryan, M. (2021). 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SELECTION OF FIRE POSITION OF MORTAR UNITS USING LBWA AND FUZZY MABAC MODEL Željko Jokić*, Darko Božanić, Dragan Pamučar 1. Introduction 2. The place and role of mortars in contemporary combat actions 3. Description of the method 3.1. Level Based Weight Assessment (LBWA) model 3.2 Fuzzy sets 3.3. Fuzzy MABAC method Multi-Attributive Border Approximation area Comparasion (MABAC) 4. Application of the hybrid model of multi-criteria decision making 4.1. Criteria for choosing the firing position 4.2. Calculation of weight coefficients of criteria using LBWA method 4.3. Choosing the best alternative using the fuzzy MABAC method 4.4. Sensitivity analysis 5. Conclusion References