Plane Thermoelastic Waves in Infinite Half-Space Caused Decision Making: Applications in Management and Engineering Vol. 3, Issue 2, 2020, pp. 131-148. ISSN: 2560-6018 eISSN: 2620-0104 DOI: https://doi.org/10.31181/dmame2003131r * Corresponding author. E-mail addresses: markoradovanovicgdb@yahoo.com (M. Radovanović), aca.r.0860.ar@gmail.com (A. Ranđelović), antras1209@gmail.com (Ž. Jokić) APPLICATION OF HYBRID MODEL FUZZY AHP - VIKOR IN SELECTION OF THE MOST EFFICIENT PROCEDURE FOR RECTIFICATION OF THE OPTICAL SIGHT OF THE LONG- RANGE RIFLE Marko Radovanović1*, Aca Ranđelović1 and Željko Jokić1 1 University of Defence, Military academy, Belgrade, Serbia Received: 12 July 2020; Accepted: 25 September 2020; Available online: 10 October 2020. Original scientific paper Abstract: The paper presents a decision support model when choosing the most efficient rectification procedure of the optical sight of the long - range rifle. The model is based on the fuzzy AHP method and the VIKOR method. Using the fuzzy AHP method, coefficient values of the criteria were defined. Fuzzification of the AHP method was performed by combining data obtained from experts - comparison of criteria in pairs and the degree of confidence in the comparison. Using the VIKOR method, the best alternative was selected. Through the paper, the criteria that condition this choice are elaborated and the application of the method in a specific situation is presented. Also, the paper presents the sensitivity analysis of the developed model. Key words: Fuzzy AHP, VIKOR, multi-criteria decision-making, rectification, long-range rifle. 1. Introduction The Serbian Army is a complex organizational system, where the decision-making process is a very important element. Therefore, the application of multi-criteria decision-making methods is an indispensable segment in this process. This paper presents a model for selecting the most efficient rectification method of a 12.7 mm M93 long - range rifle optical sight. A long-range rifle is a weapon to support infantry platoons in attack and defense. It is a type of small arms that is specially designed for fire action on people, non- combat and lightly armored combat vehicles, at distances up to 1800 m (Randjelovic et al. 2019a). It is a weapon of high accuracy and precision and achieves its firepower on targets by direct shooting. Successful rectification of sights achieves the accuracy and precision of a long- range rifle. Based on accuracy and precision, the probability of hitting the target is mailto:markoradovanovicgdb@yahoo.com mailto:aca.r.0860.ar@gmail.com mailto:antras1209@gmail.com Radovanović et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 131-148 132 determined, which affects the efficiency of long-range rifle 12.7 mm M93 solving fire tasks in operations. Having in mind the importance of rectification of the optical sight of a long - range rifle for performing combat actions, the most efficient rectification procedure was selected by applying the method of multi - criteria decision - making. 2. Problem description Through this paper, a model is presented which determines the most efficient and most economical procedure of rectification of the optical sight of a long - range rifle. Procedures for rectification of the optical sight of the 12.7 mm M93 long-range rifle are defined on the basis of the provisions of the technical and temporary instructions for the optical sight of the long-range rifle and the instructions for use for the optical sight of the long-range rifle (Long-range rifle 12.7 mm M93 (description, handling and maintenance), 2010; Purpose, description and handling of the 12.7 mm long- range rifle, 1999; The long-rifle Optical sight ON M93 for the long-range rifle "Zastava" 12.7 mm M93, 1998). In addition to the above, as one alternative, a modeled rectification procedure was taken, which was reached on the basis of the results of previous research in this area, presented in detail in Radovanović (2016), Radovanović et al. (2016) and Randjelovic et al. (2019a). The aim of this paper is to select the most efficient rectification procedure using the method of multi-criteria decision-making in order to indirectly increase the efficiency of realization of fire tasks with a long-range rifle. The results used for the analysis were obtained on the basis of realized shootings at the training field "Pasuljanske livade". Most units of the Serbian Army for the process of rectification of the optical sight of the long-rifle 12.7 mm M93, use the model shown in the temporary instructions for long-range rifle (Purpose, description and handling of long-range rifle 12.7 mm, 1999). To a lesser extent, other methods of rectification are used in the units. According to the above, it can be concluded that there is no universality regarding the rectification of the optical sight of a long-range rifle. Comparisons regarding quality, but also other parameters of rectification have not been performed so far. In other words, there are several satisfactory ways of rectification, but so far no detailed analysis has been made as to which way (model) would be the most acceptable from several aspects (quality, price, required resources, etc.). Accordingly, it is clear that the presented problem is an ideal field for the application of multi-criteria decision- making methods. In the literature available to the authors, it was found that there is not a large number of papers dealing with this issue. Radovanović (2016) models a new rectification procedure and the software program Correction of sights. In the paper Radovanović et al. (2016) performed a numerical analysis of different ways of rectification in relation to certain criteria such as ammunition consumption, time and price of rectification. Randjelovic et al. (2019a) show the dependence of the rectification procedure on the execution of fire tasks in a counter-terrorist operation. The available literature describes only a part of the criteria on the basis of which the most efficient rectification procedure is selected. 3. Description of applied methods The hybrid model, applied when solving the problem of choosing the most efficient rectification method of the long - range rifle optical sight, was defined by a Application of hybrid model fuzzy AHP - VIKOR in selection of the most efficient procedure ... 133 combination of the fuzzy AHP and VIKOR methods. This part of the paper describes the methods used in the paper. The fuzzy AHP method was used to define the coefficient values, while the VIKOR method was used to select the best alternative. Figure 1 shows the phases through which this model was realized. Figure 1. Appearance of the model for rectification of the optical sight of a long-range rifle 3.1. Fuzzy AHP method The AHP method was developed by Thomas Saaty (1980). To date, this method has undergone a large number of modifications (Božanić et al., 2013; Stević et al., 2017; Petrović et al., 2018; Chatterjee et al., 2019; Afriliansyah et al., 2019; Osintsev et al., 2020; Zhu et al., 2020;), but in some cases it is still used in its original form (Radovanović et al., 2019; Radovanović and Stevanović, 2020; Ranđelović et al., 2019b) both in the individual (Badi and Abdulshahed, 2019) and in group decision making (Srđević and Zoranović, 2003). Analytical hierarchical process is a method based on the decomposition of a complex problem into a hierarchy, with the goal at the top, criteria, sub-criteria and alternatives at the levels and sublevels of the hierarchy (Saaty, 1980). For comparisons in pairs, which is the basis of the AHP method, the Saaty’s scale is usually used, Table 1. Table 1. Saaty’s pair-wise comparison scale Standard values Definition Inverse values 1 Same meaning 1 3 Weak dominance 1/3 5 Strong dominance 1/5 7 Very strong dominance 1/7 9 Absolute dominance 1/9 2, 4, 6, 8 Intermediate values 1/2, 1/4, 1/6, 1/8 The comparison in pairs leads to the initial decision matrices. The Saaty’s scale is most commonly used to determine the coefficient values of the criteria, but can also be used to rank alternatives. Radovanović et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 131-148 134 Very often when taking values from the Saaty’s scale in the pair-wise comparison process, decision makers hesitate between the values they will assign to a particular comparison. In other words, it happens that they are not sure of the comparison they are making. Due to the above, various modifications of the Saaty’s scale are often made. One of them is the application of fuzzy numbers. There are different approaches in the fuzzification of the Saaty scale, and in principle they can be divided into two groups: sharp (hard) and soft fuzzification (Božanić et al., 2015b). Fasification can be done with different types of fuzzy numbers, and is most often done using a triangular fuzzy number Figure 2. t1 t2 t3 1  T x    2 1, T x x t     1 1 2 2 1 , T x x t t x t t t         3 2 3 3 2 , T x t x t x t t t         1 0, T x x t     3 0, T x x t    0 αT1 αT2 Figure 2. Triangular phase number T (Pamučar et al., 2016b) By "sharp" fuzzification is meant that a fuzzy number  1 2 3, ,T t t t is a predetermined confidence interval, that is, it is predetermined that the value of the fuzzy number will not be greater than 3 t or less than 1 t (Božanić et al., 2015b). Based on the predefined fuzzy Saaty’s scale, a comparison is made in pairs. In soft fuzzification, the confidence interval of the values in the Saaty’s scale is not predetermined, but is defined during the decision-making process, based on additional parameters. The definition of the coefficient values of the criteria in this paper was performed by applying the phased Saaty’s scale presented in the works of Božanić et al. (2016), Pamučar et al. (2016a), Božanić (2017), Božanić et al. (2018), Bojanic et al. (2018) and Bobar et al. (2020). The starting elements of this fuzzification are (Bobar et al., 2020): 1) introducing the fuzzy numbers instead of classic numbers of the Saaty scale, 2) introducing the degree of confidence of decision makers/analysts/experts (DM/A/E) in the statements they make when comparing in pairs -  . The degree of confidence () is defined at the level of each comparison in pairs. The value of the degree of confidence belongs to the interval 0,1, where =1 describes the absolute confidence of DM/A/E in the defined comparison. The decrease in the confidence of DM/A/E in the performed comparison is accompanied by a decrease in the degree of confidence ji. Forms for calculating fuzzy numbers are given in Table 2. Application of hybrid model fuzzy AHP - VIKOR in selection of the most efficient procedure ... 135 Table 2. Fuzzification of the Saaty's scale using the degree of confidence (Bobar et al., 2020) Definition Standard values Fuzzy number Inverse values of fuzzy number Same meaning 1 (1, 1, 1) (1, 1, 1) Weak dominance 3   3 , 3, 2 3 ji ji   1 2 3,1/ 3,1 3  ji ji Strong dominance 5   5 , 5, 2 5 ji ji   1 2 5,1/ 5,1 5  ji ji Very strong dominance 7   7 , 7, 2 7 ji ji   1 2 7 ,1/ 7,1 7  ji ji Absolute dominance 9   9 , 9, 2 9 ji ji  1 (2 )9 ,1 / 9,1 9  ji ji Intermediate values 2, 4, 6, 8   , , 2 , ji jix x x 2, 4, 6, 8x    1 2 ,1/ ,1  ji jix x x 2, 4, 6, 8x  An example of the appearance of a fuzzy number with different degrees of confidence is given in Figure 3. For example, the value of low dominance from the Saaty’s scale and degrees of confidence =1, =0.7 and =0.3 are taken. 0 1 0.7  53.5 6.5 b) 0 1 0.3  51.5 8.5 c ) 0 1 1  5 a) Figure 3. Dependence of fuzzy number on degree of confidence By introducing different values of the degree of confidence, the left and right distributions of fuzzy comparisons change according to the expression (Bobar et al., 2020):           1 2 1 2 1 2 1 2 3 2 2 2 3 2 3 2 2 3 , , , 1 / 9, 9 , , , 1 / 9,9 2 , , , 1 / 9, 9                    t t t t t t T t t t t t t t t t t t t (1) where the value of t2 represents the value of the linguistic expression from the classical Saaty’s scale, which in the fuzzy number has the maximum affiliation t2=1. Fuzzy number     1 2 3, , , , 2   T t t t x x x ,  1, 9x  is defined by expressions (Božanić, 2017): 1 , 1 1, 1             x x x t x x (2)  2 , 1, 9t x x   (3) Radovanović et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 131-148 136    3 2 , 1, 9   jit x x (4) Inverse fuzzy number     1 1 2 31/ ,1/ ,1/ 1 2 ,1/ ,1     ji jiT t t t x x x ,  1, 9x  is defined as (Božanić, 2017):          3 1 2 , 1 2 1 1 / 1 2 , 1, 9 1, 1 2 1                 ji ji ji ji x x t x x x (5) 21/ 1/ , 1/ 1, 9  t x x (6)  31 / 1 , 1 / 1, 9  jit x x (7) Accordingly, the initial decision matrix has the following form (Božanić et al., 2015a): 1 2 1 11 11 12 12 1 1 2 21 21 22 22 2 2 1 1 2 2 ; ; ; ; ; ; ; ; ;                       n n n n n n n n n n nn nn C C C C a a a A C a a a C a a a (8) where ji=ij. Reaching the final results implies further application of the standard steps of the AHP method. At the end of the application, the fuzzy number is converted to a real number. Numerous methods are used for this procedure (Herrera and Martinez, 2000). Some of the known terms for defuzzification are (Liou and Wang, 1992; Seiford, 1996): 3 1 2 1 1 (( ) ( )) / 3    A t t t t t (9)  3 2 11 / 2      A t t t (10) where  represents the optimism index, which can be described as the belief/ratio DM/A/E in decision-making risk. Most often, the optimism index is 0, 0.5 or 1, which corresponds to the pessimistic, average or optimistic view of the decision maker (Milićević, 2014). 3.2. VIKOR method VIKOR (VIšekriterijumsko KOmpromisno Rangiranje) is a method of multi- criteria decision-making whose use is very common. It was developed by Serafim Opricović (1986). It is suitable for solving various decision-making problems. It is especially emphasized for situations where criteria of a quantitative nature prevail. The VIKOR method starts from the "boundary" forms of Lp - metrics, where the choice of the solution that is closest to the ideal is made. The presented metric represents the distance between the ideal point F* and the point F (x) in the space of criterion functions (Opricović, 1986). Minimizing this metric determines a compromise solution. As a measure of the distance from the ideal point, the following is used: Application of hybrid model fuzzy AHP - VIKOR in selection of the most efficient procedure ... 137  *pL (F , F)       1/p pn * j jj=1 f -f (x) ,1 p (11) The VIKOR method has been applied in a large number of papers in its original form (Nisel, 2014; Kuo and Liang, 2011; Opricović and Tzeng, 2004; Jokić et al., 2019, Radovanović et al. 2020), but also in fuzzy (Chatterjeea and Chakrabortyb, 2016; Ince, 2007; Shemshadi et al., 2011;) and a rough (Li and Song, 2016; Wang et al. 2018) environment. When applying the VIKOR method, the following terms are used:  n – number of criteria  m – number of alternatives for multicriteria ranking  fij – the values of the i criterion function for the j alternative,  wj – the value of the j criterion function,  v – the weight of the strategy, meeting most of the criteria,  i – ordinal number of the alternative, i = 1, ..., m.,  j – ordinal number of the criteria, j = 1, ..., n,  Qi – measure for multi-criteria ranking of the j alternative. For each alternative, there are Qi values, after which the alternative with the lowest Qi value is selected. The measure for multi-criteria ranking of the i action (Qi) is calculated according to the expression (Opricović, 1998): 1 i i Q v* QS ( v )* QR   (12) where: * i i * S S QS S S     (13) * i i * R R QR R R     (14) By calculating the QSi, QRi, and Qi values for each alternative, it is possible to form three independent rankings. The QSi value, is a measure of deviation that displays the requirement for maximum group benefit (first ranking list). QRi value is a measure of deviation that shows the requirement to minimize the maximum distance of an alternative from the "ideal" alternative (second ranking list). Qi value represents the establishment of a compromise ranking list that combines QSi and QRi values (third ranking list). By choosing a smaller or larger value for v (the weight of strategies to meet most criteria), the decision maker can favor the influence of QSi value or QRi value in the compromise ranking list. For example, higher values for v (v > 0.5) indicate that the decision maker gives greater relative importance to the strategy of satisfying most of the criteria (Nikolić et al., 2010). Modeling the preferential dependence of criteria usually includes the weights of individual criteria. If the given values are weights w1,w2,…..,wn, the multi-criteria ranking by the VIKOR method is realized by using the measure Si and Ri. In the previous terms, the labels used have the following meanings:    * * 1 1 / n n i i i ij i i j ij j j S w f f f f w d         (15)    * *max / maxi i i ij i i j ij j j R w f f f f w d        (16) Radovanović et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 131-148 138 i = 1,2, ..., m, j=1,2,...,n, and where: * * * min max min max max min i i i i i i i i ij i ij i S S S S R R R R f f f f          Alternative ai is better than alternative ak according to j criterion, if:  ij kj f f (for max fj, that is when the criterion has a maximum requirement),  ij kj f f (for min fj, that is when the criterion has a minimum requirement). In multi-criteria ranking by the VIKOR method, alternative ai is better than alternative ak (in total, according to all criteria), if: Qi