International Journal of Analysis and Applications Volume 18, Number 6 (2020), 989-997 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-989 EXTENSION OF VIKOR METHOD FOR MCDM UNDER BIPOLAR FUZZY SET BASHAYER ALSOLAME∗ AND NOURA OMAIR ALSHEHRI Department of Mathematics, Faculty of Science, University of Jeddah ∗Corresponding author: baalsolame@uj.edu.sa Abstract. In this paper, we extend the VIKOR method (Serbian name: VlseKriterijumska Optimizacija I Kompromisno Resenje, means multi-criteria optimization and compromise solution) to solve multi-criteria decision making (MCDM) with bipolar fuzzy environment. Firstly, the bipolar fuzzy set concept is described. Then, proposed the VIKOR strategy in bipolar set environment to handle MCDM problems. Finally, two numerical examples illustrate an application of BF-VIKOR method, and analysis the results of different values of the decision making weights of criteria on ranking order of the alternatives. 1. Introduction Multi-Criteria Decision Making (MCDM) has seen an incredible amount of use over last several decades. Its role in different applications areas has increased significantly, especially as new methods develop and as old methods improve. In classical MCDM methods, the goal is to designate a preferred alternative, classify alternatives in a small number of categories and rank alternatives in a subjective preference order. However, most of the real world decisions are made in an environment in which goals and constraints cannot be precisely expressed due to their complexity. The fuzzy sets theory was first proposed by Zadeh in 1965 ( [1], [2]). Afterward, Bellman and Zadeh described a decision-making method in a fuzzy environment [3]. Multi-criteria optimization and compromise solution (VIKOR in Serbian) method, developed by Opricovic (1998) [4], Opricovic and Tzeng (2004) [5], is one of the most outranking MCDM method. The first paper that proposes to use fuzzy inputs with VIKOR method was published in 2002 [6]. Later, Opricovic [7] proposes a Received August 15th, 2020; accepted September 7th, 2020; published September 29th, 2020. 2010 Mathematics Subject Classification. 03E72. Key words and phrases. bipolar fuzzy set; VIKOR method. ©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 989 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-989 Int. J. Anal. Appl. 18 (6) (2020) 990 fuzzy extension of VIKOR to find a fuzzy compromise solution.Wang and Chang [8] introduced fuzzy VIKOR for solving multi-criteria group decision-making problem. Since that, many researchers have been dealing with decision-making problems by applying VIKOR method in fuzzy environment ( [9], [10], [11], [12], [13]), interval-valued intuitionistic fuzzy environment [14], intuitionistic fuzzy VIKOR [15] and hesitant fuzzy VIKOR [16]. In 1994, Zhang [17] proposed the concept of bipolar fuzzy sets, which is a generalization of fuzzy set allowing the membership degree having two values. One locates in the interval [0,1] represents the satisfaction degree of a certain property associated to the fuzzy set and the other locates in the interval [-1,0] and represents the satisfaction degree of a counter-property to the concerned fuzzy set. Bipolarity is important to distinguish between positive information, which represents what is guaranteed to be possible, for example what has already been observed or experienced and negative information, which represents what is impossible or forbidden, or surely false. Thus providing an efficient approach to solving a widely larger number of complex decision making problems. In recent years, many decision-making problems have been applied in bipolar fuzzy environment. In this paper, we present bipolar fuzzy VIKOR (BF-VIKOR) method solving MCDM problems that are equipped with bipolar fuzzy information. We illustrate our proposed method with numerical examples. 2. Preliminaries In this section, some preliminaries from the fuzzy set theory and bipolar fuzzy sets are induced. Definition 2.1. A fuzzy set of the universe X is a mathematical object A described by its characteristic function (membership function) µA : X → [0, 1] defined as [1]: (2.1) A = {(x,µA(x)) | x ∈ X} Definition 2.2. A bipolar fuzzy set A on the universe X is defined as [17]: (2.2) A = {(x,µA (x) ,νA (x)) | x ∈ X} where µA : X → [0, 1] is a positive membership degree denotes the satisfaction degree of an element to the property corresponding to A, and νA : X → [−1, 0] is the negative membership degree denotes the satisfaction degree of an element to some implicit counter-property corresponding to A. 3. VIKOR Method The VIKOR method focuses on ranking and selecting from a set of alternatives with conflicting crite- ria, and determines compromise solution based on particular measure. Assuming that each alternative is Int. J. Anal. Appl. 18 (6) (2020) 991 evaluated according to each criterion function, the compromise ranking can be presented by comparing the degree of closeness to the ideal alternative. The compromise solution, whose foundation was established by Yu [18] and Zeleny [19], is a feasible solution, which is the closest to the ideal, and here “compromise” means an agreement established by mutual concessions. The multi-criteria measure for compromise ranking is developed from the Lp− metric used as an aggregating function in a compromise programming method defined as: (3.1) Lp,i =   n∑ j=1 [ wj ( f∗j −fij )( f∗j −f − j )]p   1 p , 1 ≤ p ≤∞ where L1,i is defined as the maximum group utility, and L∞,i is defined as the minimum individual regret of the opponent. For a MCDM problem involving m alternatives and n criteria, whereby the performances of the alternatives are expressed by using bipolar fuzzy set, Let ψ = {ψ1,ψ2, . . . ,ψm} = {ψi : i = 1, 2, . . . ,m} and C = {C1,C2, . . . ,Cn} = {Cj : j = 1, 2, . . . ,n} be the sets of alternatives and criteria that are determined by a decision maker, respectively. Formally, a bipolar fuzzy MCDM problem can be express in matrices format as follows: (3.2) D =   (µ11,ν11) (µ12,ν12) . . . (µ1n,ν1n) (µ21,ν21) (µ22,ν22) . . . (µ2n,ν2n) . . . . (µm1,νm1) (µm2,νm2) . . . (µmn,νmn)   m×n The ratings of any alternative ψi ∈ ψ for each criterion Cj are given by a bipolar fuzzy set Fi = {(ψi,fij) | j = 1, 2, . . . ,n} where fij = (µij,νij) represent respectively, the satisfaction degree (µij ∈ [0, 1]) and the dissatisfaction de- gree (νij ∈ [−1, 0]) that are determined for ψi with respect to Cj. Now, determine the best f∗j and the worst f-j value of all criterion functions, j = 1, 2, . . . ,n as: (3.3) f∗j = { (max µij, min νij) , for the benefit criterion (min µij, max νij) , for the cost criterion } , i = 1, 2, . . . ,m. (3.4) f−j = { (min µij, max νij) , for the benefit criterion (max µij, min νij) , for the cost criterion } , i = 1, 2, . . . ,m In the VIKOR development methodology Si (as L1,i in Equation 3.1) and Ri (as L∞,i in Equation 3.1) are used to formulate ranking measure [4], [5]. In light of the distance measure for each alternatives functions Int. J. Anal. Appl. 18 (6) (2020) 992 fij = (µij,νij) and the best f ∗ J and the worst f - j value of all criterion functions. (3.5) Si = n∑ j=1 wj d ( f∗j ,fij ) d ( f∗j ,f − j ), (3.6) Ri = max j [ wj d ( f∗j ,fij ) d ( f∗j ,f − j )] where wj are the weights of criteria, expressing the decision-makers preference of the criteria, such that wj ∈ [0, 1] and n∑ j=1 wj = 1. Furthermore, the relative closeness degree of each alternative ψi (i = 1, 2, . . . ,m) with respect to Si and Ri is given as: (3.7) Qi = v (Si −S∗) (S− −S∗) + (1 −v) (Ri −R∗) (R− −R∗) where, S∗ = min i Si , S − = max i Si R∗ = min i Ri , R − = max i Ri v and (1 −v) are the weights for the strategy of maximum group utility and individual regret, respectively. Usually, the value of v can be assumed to be ( v = 0.5 ).The Qirepresent the distance of ψi alternative from the best solution ”compromise solution”. So the alternative, which has the minimum value in Qi would be the compromise solution if the following two conditions are satisfied: C1) Acceptable advantage: (3.8) Q(ψ(2)) −Q(ψ(1)) ≥ 1 m− 1 where, ψ(1) and ψ(2) are the top two alternatives in Qi [such that Qi sort in ascending order], m is the number of the alternatives. C2) Acceptable stability: the compromise solution should be the best rank by Si and/or Ri. If one of the conditions is not satisfied, then a set of compromise solutions is proposed, which consists of: • ψ(1) and ψ(2) if only the condition C2 is not satisfied, or • ψ(1),ψ(2), . . . ,ψ(M) if the condition C1 is not satisfied (Equation 3.8); ψ(M) is determined by the relation: Q(ψ(M)) −Q(ψ(1)) < 1 m− 1 ; maximum M. Int. J. Anal. Appl. 18 (6) (2020) 993 4. VIKOR Algorithm According to the above discussion, the procedures of BF-VIKOR method can be summarized in Figure 1. Figure 1. The Flow Chart of BF-VIKOR Algorithm 5. Numerical Examples 5.1. Example 1. Assume that we have a decision maker who is confused in choosing the a new location to start his sales business. After the initial consideration of the available alternatives, four alternatives has been identified as suitable. Suppose that he is evaluating suitable alternatives based on the following criteria: C1 :Rental space quality; C2 :Rental space adequacy; C3 :Location quality; C4 :Location distance from the city center, and C5 :Rental price. So, these locations represent the alternatives {ψ1,ψ2,ψ3,ψ4} and the mentioned features represent the criteria {C1,C2,C3,C4,C5} in our MCDM problem. Ratings of the alternatives, in Table 1, and weights of the criteria are given by the decision maker in matrices format with bipolar fuzzy and fuzzy values, respectively, as follows: W = [ 0.18 0.24 0.21 0.16 0.21 ]T Clearly, the given weights satisfy the normalized condition. From Equation 3.3, Equation 3.4 the best and worst values of all criterion ratings are determined as follows in Table 2. Further, the Equations 3.5 - 3.7 calculated the values of Si, Ri and Qi in Table 3 were (v = 0.5) and rank the alternatives by ranking Si,Ri and Qi in decreasing order as shown in Table 4. Thus, the compromise solution is ψ4 and ψ1. Int. J. Anal. Appl. 18 (6) (2020) 994 Table 1. Ratings of the alternatives C1 C2 C3 C4 C5 ψ1 (0.67,−0.16) (0.79,−0.32) (1, 0) (1, 0) (0.81,−0.10) ψ2 (0.47,−0.18) (0.43,−0.42) (1,−.26) (1, 0) (0.77,−0.10) ψ3 (0.64,−0.10) (0.61,−0.16) (0.49, 0) (0.41,−0.10) (0.70,−0.13) ψ4 (0.81,−0.10) (1,−0.15) (0.53, 0) (0.53,−0.10) (0.60,−0.10) Table 2. The best and worst value of all criterion functions f∗j f − j C1 (0.81,−0.18) (0.47,−0.10) C2 (1,−0.42) (0.43,−0.15) C3 (1,−0.26) (0.49, 0) C4 (1,−0.10) (0.41, 0) C5 (0.60,−0.10) (0.80,−0.13) Table 3. The values of Si, Ri and Qi for all alternatives ψ1 ψ2 ψ3 ψ4 Si 0.491 0.595 0.751 0.467 Ri 0.208 0.217 0.21 0.197 Qi 0.311 0.726 0.826 0 Table 4. Final ranking of alternatives ψ1 ψ2 ψ3 ψ4 Ranking Si 0.491 0.595 0.751 0.467 ψ4,ψ1,ψ2,ψ3 Ri 0.208 0.217 0.21 0.197 ψ4,ψ1,ψ3,ψ2 Qi 0.311 0.726 0.826 0 ψ4,ψ1,ψ2,ψ3 5.2. Example 2. Assume that we have a decision maker who is confused in choosing the best airline. After the initial consideration of the available alternatives, five alternatives has been identified as suitable. Suppose that he is evaluating suitable alternatives based on the following criteria: C1 :Empathy, represents how airline deal with the customer complaints and provide thoughtful services; C2 :Assurance, represents the certainty that airline provides for customers; C3 :Tangibility, means the physical service presentation such as quality of the food, and C4 :Cost, represents the tickets and services prices. Int. J. Anal. Appl. 18 (6) (2020) 995 So, these airlines represent the alternatives {ψ1,ψ2,ψ3,ψ4,ψ5} and the mentioned features represent the criteria {C1,C2,C3,C4} in our MCDM problem. Ratings of the alternatives, in Table 5, and weights of the criteria are given by the decision maker in matrices format with bipolar fuzzy and fuzzy values, respectively, as follows: W = [ 0.25 0.3 0.25 0.2 ]T Clearly, the given weights satisfy the normalized condition. Using the Equations 3.5 - 3.7 the values of Si, Ri and Qi are calculated as in Table 6. Table 5. Rating of the alternatives C1 C2 C3 C4 ψ1 (0.5,−0.25) (0.8,−0.7) (0.3,−0.1) (0.6,−0.6) ψ2 (0.2,−0.8) (0.9,−0.4) (0.6,−0.3) (0.55,−0.5) ψ3 (0.33,−0.25) (0.75,−0.4) (0.25,−0.7) (0.3,−0.1) ψ4 (0.65,−0.6) (0.3,−0.75) (0.8,−0.35) (0.65,−0.7) ψ5 (1,−0.5) (0.4,−0.35) (0.2,−0.6) (0.25,−0.65) Table 6. The values ofSi,Ri and Qi for all alternatives ψ1 ψ2 ψ3 ψ4 ψ5 Ranking Compromise solutions Si 0.637 0.622 0.558 0.657 0.675 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3 Ri 0.23 0.206 0.223 0.25 0.266 ψ2,ψ3,ψ1,ψ4,ψ5 ψ2 Qi (v = 0.5) 0.538 0.274 0.143 0.781 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3,ψ2 After rank the alternatives by sorting the values Si, Ri and Qi in decreasing order. The results are three ranking lists, which is depicted in Table 6. So, the compromise solution are ψ3 and ψ2. The parameter v in VIKOR technique normally considered as (v = 0.5). Therefore, by changing value of v in the interval [0, 1] is performed for the obtained results. The ranking for five alternatives under different v values are illustrated in Table 7. As can be seen, when v is changed, there are some deviations in ranking of alternatives. ψ3 is the best ranked alternative for v ≥ 0.7; also, ψ2 has the best rank for v = 0. Moreover, ψ5 is the worst ranked alternative for different values of v. 6. Conclusion In MCDM methods the alternatives are compared against each other based on how they perform relative to each criterion. One the other hand, VIKOR method require comparison of the criteria to determine the relative importance of each criterion. The alternative with the highest rank is selected as the best compromise Int. J. Anal. Appl. 18 (6) (2020) 996 Table 7. Ranking orders of alternatives under different v values. ψ1 ψ2 ψ3 ψ4 ψ5 Ranking Compromise solutions Si 0.637 0.622 0.558 0.657 0.675 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3 Ri 0.23 0.206 0.223 0.25 0.266 ψ2,ψ3,ψ1,ψ4,ψ5 ψ2 Qi(v) 0 0.399 0 0.285 0.722 1 ψ2,ψ3,ψ1,ψ4,ψ5 ψ2 0.1 0.427 0.055 0.257 0.734 1 ψ2,ψ3,ψ1,ψ4,ψ5 ψ2,ψ3 0.2 0.455 0.109 0.228 0.746 1 ψ2,ψ3,ψ1,ψ4,ψ5 ψ2,ψ3 0.3 0.482 0.164 0.2 0.758 1 ψ2,ψ3,ψ1,ψ4,ψ5 ψ2,ψ3 0.4 0.51 0.219 0.171 0.769 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3,ψ2 0.5 0.538 0.274 0.143 0.781 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3,ψ2 0.6 0.566 0.328 0.114 0.793 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3,ψ2 0.7 0.594 0.383 0.086 0.805 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3 0.8 0.621 0.438 0.057 0.816 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3 0.9 0.649 0.493 0.029 0.828 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3 1 0.677 0.547 0 0.84 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3 solution. In this paper, we proposed BF-VIKOR method Since the VIKOR method is an effective MCDM method to reach a compromise solution, particularly in a situation where the decision maker is not able to express his preference at the beginning. 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Zhang, G.W Wei, Extension of VIKOR method for decision making problem based on hesitant fuzzy set, Appl. Math. Model. 37 (7) (2013), 4938-4947. [17] W. R. Zhang, Yin-Yang bipolar fuzzy sets, In 1998 IEEE International Conference on Fuzzy Systems Proceedings. 1998, 835-840. [18] P.L. Yu, A class of solutions for group decision problems. Manage. Sci. 19 (8) (1973), 936-946. [19] M. Zeleny, Multiple Criteria Decision Making, McGraw-Hill, New York, 1982. 1. Introduction 2. Preliminaries 3. VIKOR Method 4. VIKOR Algorithm 5. Numerical Examples 5.1. Example 1 5.2. Example 2 6. Conclusion References