Extension of the TOPSIS method for decision-making problems with fuzzy data

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1 Applied Mathematics and Computation 181 (2006) Extension of the TOPSIS method for decision-making problems with fuzzy data G.R. Jahanshahloo a, F. Hosseinzadeh Lotfi a, M. Izadikhah b, * a Department of Math, Science and Research Branch, Islamic Azad University, Tehran , Iran b Department of Math, Islamic Azad University, P.O. Box 38135/567, Arak, Iran Abstract Decision making problem is the process of finding the best option from all of the feasible alternatives. In this paper, from among multicriteria models in making complex decisions and multiple attribute models for the most preferable choice, technique for order preference by similarity to ideal solution (TOPSIS) approach has been dealt with. In real-word situation, because of incomplete or non-obtainable information, the data (attributes) are often not so deterministic, there for they usually are fuzzy/imprecise. Therefore, the aim of this paper is to extend the TOPSIS method to decision-making problems with fuzzy data. In this paper, the rating of each alternative and the weight of each criterion are expressed in triangular fuzzy numbers. The normalized fuzzy numbers is calculated by using the concept of a-cuts. Finally, a numerical experiment is used to illustrate the procedure of the proposed approach at the end of this paper. Ó 2006 Published by Elsevier Inc. Keywords: MCDM; TOPSIS; Fuzzy numbers; Fuzzy positive ideal solution; Fuzzy negative ideal solution 1. Introduction Multi-criteria decision making has been one of the fastest growing areas during the last decades depending on the changings in the business sector. Decision maker(s) need a decision aid to decide between the alternatives and mainly excel less preferrable alternatives fast. With the help of computers the decision making methods have found great acceptance in all areas of the decision making processes. Since multicriteria decision making (MCDM) has found acceptance in areas of operation research and management science, the discipline has created several methodologies. Especially in the last years, where computer usage has increased significantly, the application of MCDM methods has considerably become easier for the users the decision makers as the application of most of the methods are corresponded with complex mathematics. In discrete alternative multicriteria decision problems, the primary concern for the decision aid is the following: * Corresponding author. address: m_izadikhah@yahoo.com (M. Izadikhah) /$ - see front matter Ó 2006 Published by Elsevier Inc. doi: /j.amc

2 G.R. Jahanshahloo et al. / Applied Mathematics and Computation 181 (2006) (1) choosing the most preferred alternative to the decision maker (DM), (2) ranking alternatives in order of importance for selection problems, or (3) screening alternatives for the final decision. The general concepts of domination structures and non-dominated solutions play an important role in describing the decision problems and the decision maker s revealed preferences described above [10]. So far, various approaches have been developed as the decision aid (see, for example [9]). That is, for many such problems, the decision maker wants to solve a multiple criteria decision making (MCDM) problem. In MCDM problems, there does not necessarily exist the solution that optimizes all objectives functions, and then the concept which is called Pareto optimal solution (or efficient solution) is introduced. Usually, there exist a number of Pareto optimal solutions, which are considered as candidates of final decision making solution. It is an issue how decision makers decide one from the set of Pareto optimal solutions as the final solution (see, for more details [6]). A MCDM problem can be concisely expressed in matrix format as W ¼½w 1 ; w 2 ;...; w n Š; where A 1,A 2,...,A m are possible alternatives among which decision makers have to choose, C 1,C 2,...,C n are criteria with which alternative performance are measured, x ij is the rating of alternative A i with respect to criterion C j, w j is the weight of criterion C j. The main steps of multiple criteria decision making are the following: (a) establishing system evaluation criteria that relate system capabilities to goals; (b) developing alternative systems for attaining the goals (generating alternatives); (c) evaluating alternatives in terms of criteria (the values of the criterion functions); (d) applying a normative multicriteria analysis method; (e) accepting one alternative as optimal (preferred); (f) if the final solution is not accepted, gather new information and go into the next iteration of multicriteria optimization. Steps (a) and (e) are performed at the upper level, where decision makers have the central role, and the other steps are mostly engineering tasks. For step (d), a decision maker should express his/her preferences in terms of the relative importance of criteria, and one approach is to introduce criteria weights. This weights in MCDM do not have a clear economic significance, but their use provides the opportunity to model the actual aspects of decision making (the preference structure). Technique for order performance by similarity to ideal solution (TOPSIS) [7], one of known classical MCDM method, was first developed by Hwang and Yoon [4] for solving a MCDM problem. TOPSIS, known as one of the most classical MCDM methods, is based on the idea, that the chosen alternative should have the shortest distance from the positive ideal solution and on the other side the farthest distance of the negative ideal solution. The TOPSIS-method will be applied to a case study, which is described in detail. In classical MCDM methods, the ratings and the weights of the criteria are known precisely [3,4]. A survey of the methods has been presented in Hwang and Yoon [4]. In the process of TOPSIS, the performance ratings and the weights of the criteria are given as exact values. Recently, Abo-sinna and Amer [1] extend TOPSIS approach to solve multi-objective nonlinear programming problems. Jahanshahloo et al. [5] extend the concept of TOPSIS to develop a methodology for solving multi-criteria decision-making problems with interval data. In real-word situation, because of incomplete or non-obtainable information, for example, human judgements including preferences are often vague and cannot estimate his preference with an exact numerical data, the data (attributes) are often not so deterministic, there for they usually are fuzzy/imprecise [2,11], so, we try to extend TOPSIS for fuzzy data.

3 1546 G.R. Jahanshahloo et al. / Applied Mathematics and Computation 181 (2006) The rest of the paper is organized as follows: in the following section, first, we discus preliminary definitions of fuzzy data, then, we briefly introduce the original TOPSIS method. In Section 3, we introduce MCDM problems with fuzzy data, then, we present an algorithm to extend TOPSIS to deal with fuzzy data. In Section 4, we illustrate our proposed algorithmic method with an example. The final section concludes. 2. Background 2.1. Preliminary definitions of fuzzy data Let X be a classical set of objects, called the universe, whose generic elements are denoted by x. The membership in a crisp subset of X is often viewed as characteristic function l A from X to {0,1} such that: 1 if and only if x 2 A; l A ðxþ ¼ 0 otherwise; where {0,1} is called a valuation set. If the valuation set is allowed to be the real interval [0, 1], A is called a fuzzy set and denoted by A ~ and l AðxÞ ~ is the degree of membership of x in A. ~ Definition 1. If ~A be a fuzzy set, then ~A is completely characterized by the set of ordered pairs [8]: ~A ¼fðx; l AðxÞÞjx ~ 2 X g: Definition 2 (a-level set or a-cut). The a-cut of a fuzzy set ~A is a crisp subset of X and is denoted by [8]: ½~AŠ a ¼fxjl A ~ ðxþ P ag; where l A ~ ðxþ is the membership function of ~A and a 2 [0, 1]. The lower and upper points of any a-cut, ½ AŠ ~ a, are represented by inf½ AŠ ~ a and sup½ AŠ ~ a, respectively, and we suppose that both are finite. For convenient, we show inf½~aš a with ½~AŠ L a and sup½ ~AŠ a with ½~AŠ U a (see Fig. 1). Definition 3 (Normality). A fuzzy set A ~ is normal if and only if sup x l A ~ ðxþ ¼1: Definition 4 (Convexity). A fuzzy set ~A in X is convex if and only if for every pair of point x 1 and x 2 in X, the membership function of ~A satisfies the inequality l Aðdx 1 ~ þð1 dþx 2 Þ P minðl Aðx 1 ~ Þ; l Aðx 2 ~ ÞÞ; where d 2 [0, 1]. Alternatively, a fuzzy set is convex if all a-level sets are convex. Definition 5 (Fuzzy number). A fuzzy number ~A is a convex normalized fuzzy set ~A of the real line R with continuous membership function. Fig. 1. An example of a-cut.

4 G.R. Jahanshahloo et al. / Applied Mathematics and Computation 181 (2006) Fig. 2. A triangular fuzzy number ~A. Definition 6 (Triangular fuzzy numbers). The triangular fuzzy numbers can be denoted as ~A ¼ða; m; nþ, where a is the central value ðl ~A ðaþ ¼1Þ, m is the left spread and n is the right spread (see Fig. 2). Definition 7 (Multiplication of triangular fuzzy numbers). Suppose that we have two triangular fuzzy numbers ~A and ~B such that A ~ ¼ða; m; nþ and ~B ¼ðb; s; rþ, then, the multiplication of the fuzzy numbers A ~ and ~B is defined as follows [8]: 8 >< ðab; as þ bm ms; ar þ bn þ nrþ if ~A > 0; ~B > 0; ~AðÞ~B ¼ ðab; ar þ bm þ mr; as bn þ snþ if ~A < 0; ~B > 0; >: ðab; ar bn nr; as bm þ nrþ if ~A < 0; ~B < 0: Definition 8. A fuzzy number ~ A is called positive fuzzy number if l ~ AðxÞ ¼0 for all x <0; 6 1 for a 2 [0, 1], then ~A is called a nor- Definition 9. If ~A is a triangular fuzzy number and ½~AŠ L a > 0 and ½ ~AŠ U a malized positive triangular fuzzy number. Definition 10. Let ~A ¼ða; m; nþ, ~B ¼ðb; s; rþ be two triangular fuzzy numbers, then the distance between them using vertex method is defined as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dð~a; ~BÞ ¼ 3 ½ða bþ2 þða b mþsþ 2 þða bþn rþ 2 Š: Remark 1. If ~A ¼½½~AŠ L a ; ½ ~AŠ U a Š, then by choosing a = 1 we can identify the central value of ~A, and by a =0we can identify the left and right spreads of ~A TOPSIS method TOPSIS (technique for order preference by similarity to an ideal solution) method is presented in Chen and Hwang [2], with reference to Hwang and Yoon [4]. TOPSIS is a multiple criteria method to identify solutions from a finite set of alternatives. The basic principle is that the chosen alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. The procedure of TOPSIS can be expressed in a series of steps: (1) Calculate the normalized decision matrix. The normalized value n ij is calculated as s ffiffiffiffiffiffiffiffiffiffiffiffi X m n ij ¼ x ij ; i ¼ 1;...; m; j ¼ 1;...; n: i¼1 x 2 ij (2) Calculate the weighted normalized decision matrix. The weighted normalized value v ij is calculated as v ij ¼ w j n ij ; i ¼ 1;...; m; j ¼ 1;...; n; where w j is the weight of the ith attribute or criterion, and P n j¼1 w j ¼ 1.

5 1548 G.R. Jahanshahloo et al. / Applied Mathematics and Computation 181 (2006) (3) Determine the positive ideal and negative ideal solution A þ ¼fv þ 1 ;...; vþ n g¼fðmax j A ¼fv 1 ;...; v n g¼fðmin j v ij ji 2 IÞ; ðmin v ij ji 2 JÞg; j v ij ji 2 IÞ; ðmax v ij ji 2 JÞg; j where I is associated with benefit criteria, and J is associated with cost criteria. (4) Calculate the separation measures, using the n-dimensional Euclidean distance. The separation of each alternative from the ideal solution is given as ( )1 d þ i ¼ Xn 2 ðv ij v þ j Þ2 ; i ¼ 1;...; m; j¼1 Similarly, the separation from the negative ideal solution is given as ( )1 d i ¼ Xn 2 ðv ij v j Þ2 ; i ¼ 1;...; m; j¼1 (5) Calculate the relative closeness to the ideal solution. The relative closeness of the alternative A i with respect to A + is defined as R i ¼ d i =ðd þ i þ d i Þ; i ¼ 1;...; m: Since d i P 0andd þ i P 0, then, clearly, R i 2 [0,1]. (6) Rank the preference order. For ranking alternatives using this index, we can rank alternatives in decreasing order. The basic principle of the TOPSIS method is that the chosen alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. The TOPSIS method introduces two reference points, but it does not consider the relative importance of the distances from these points. 3. TOPSIS method with fuzzy data In real-word situation, because of incomplete or non-obtainable information, the data (attributes) are often not so deterministic, there for they usually are fuzzy/impresice, so, we try to extend TOPSIS for fuzzy data. Suppose A 1,A 2,...,A m are m possible alternatives among which decision makers have to choose, C 1,C 2,...,C n are criteria with which alternative performance are measured, ~x ij is the rating of alternative A i with respect to criterion C j and is a fuzzy number. A MCDM problem with fuzzy data can be concisely expressed in matrix format (namely, fuzzy decision matrix) as ~W ¼½~w 1 ; ~w 2 ;...; ~w n Š; where ~w j is the weight of criterion C j and is a normalized fuzzy number. The approach to extend the TOPSIS method to the fuzzy data is proposed in this section. First step is, identification the evaluation criteria. Step 2 is, generating alternatives. Step 3 is, evaluating alternatives in terms of criteria (the values of the criterion functions which are fuzzy). Step 4 is, identifying the weight of criteria.

6 Step 5. Construct the fuzzy decision matrix. In fuzzy decision matrix, we suppose that, each ~x ij is triangular fuzzy number, i.e., ~x ij ¼ðx ij ; a ij ; b ij Þ. Step 6. We calculate the normalized fuzzy decision matrix as follows: First, for each fuzzy number ~x ij ¼ðx ij ; a ij ; b ij Þ, we calculate the set of a-cut as ~x ij ¼ ½~x ij Š L a ; ½~x ijš U a ; a 2½0; 1Š: Therefore, each fuzzy number ~x ij is transform to an interval, now by an approach proposed in Jahanshahloo et al. [5] we can transform this interval in to normalized interval as follows: s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½~n ij Š L a ¼½~x X ijš L m a ð½~x ij Š L a Þ2 þð½~x ij Š U a Þ2 ; i ¼ 1;...; m; j ¼ 1;...; n; i¼1 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½~n ij Š U a ¼½~x X ijš U m a ð½~x ij Š L a Þ2 þð½~x ij Š U a Þ2 ; i ¼ 1;...; m; j ¼ 1;...; n i¼1 now, interval ½½~n ij Š L a ; ½~n ijš U a Š is normalized of interval ½½~x ijš L a ; ½~x ijš U a Š. According to Remark 1 we can transform this normalized interval in to a fuzzy number such as ~N ij ¼ðn ij ; a ij ; b ij Þ such that, n ij is obtained when a = 1 i.e., n ij ¼½~n ij Š L a¼1 ¼½~n ijš U a¼1, also by setting a = 0 we have ½~n ijš L a¼0 ¼ n ij a ij and ½~n ij Š U a¼0 ¼ n ij þ b ij then a ij ¼ n ij ½~n ij Š L a¼0 ; b ij ¼½~n ij Š U a¼0 n ij and ~N ij is a normalized positive triangular fuzzy number i.e., ~N ij is normalized of fuzzy number ~x ij. Now, we can work with these normalized fuzzy numbers. Step 7. By considering the different importance of each criterion, we can construct the weighted normalized fuzzy decision matrix as: ~v ij ¼ ~N ij :~w j ; where ~w j is the weight of jth attribute or criterion. Step 8. Now, each ~v ij is normalized fuzzy numbers and their ranges is belong to [0,1]. So, we can identify fuzzy positive ideal solution and fuzzy negative ideal solution as: ~A þ ¼ð~v þ 1 ;...; ~vþ n Þ; ~A ¼ð~v 1 ;...; ~v n Þ; where ~v þ i ¼ð1; 0; 0Þ and ~v i ¼ð0; 0; 0Þ; i ¼ 1;...; n for each criteria (benefit or cost criteria). Step 9. The separation of each alternative from the fuzzy positive ideal solution, using the distance measurement between two fuzzy number (see Definition 8) can be currently calculated as: ~d þ i ¼ Xn j¼1 dð~v ij ; ~v þ ij Þ; i ¼ 1;...; m: Similarly, the separation from the fuzzy negative ideal solution can be calculated as: ~d i ¼ Xn j¼1 G.R. Jahanshahloo et al. / Applied Mathematics and Computation 181 (2006) dð~v ij ; ~v ij Þ; i ¼ 1;...; m: Step 10. A closeness coefficient is defined to determine the ranking order of all alternatives once the d ~ þ i and ~d i of each alternative A i has been calculated. The relative closeness of the alternative A i with respect to A ~ þ is defined as: ~R i ¼ d ~ i =ð d ~ þ i þ d ~ i Þ; i ¼ 1;...; m:

7 1550 G.R. Jahanshahloo et al. / Applied Mathematics and Computation 181 (2006) Obviously, an alternative A i is closer to the ~A þ and farther from ~A as ~R i approaches to 1. Therefore, according to the closeness coefficient, we can determine the ranking order of all alternatives and select the best one from among a set of feasible alternatives. 4. Numerical example In this section, we work out a numerical example to illustrate the TOPSIS method for decision-making problems with fuzzy data. Suppose that we have three alternatives A 1, A 2 and A 3 among which decision makers have to choose and, also, five benefit criteria C 1,...,C 5, are identified as the evaluation criteria for these alternatives. These data and also the vector of corresponding weight of each criteria are given in Table 1. The Table 1 The fuzzy decision matrix and fuzzy weights of three alternatives C 1 C 2 C 3 C 4 C 5 A 1 (7.7, 2, 3.6) (7, 2, 2) (7.7, 2, 1.3) (9.67, 1.34, 0.33) (5, 2, 2) A 2 (8.3, 2, 1.4) (10, 1, 0) (9.7, 1.4, 0.3) (10, 1, 0) (9, 2, 1) A 3 (8, 1.7, 1) (9, 2, 1) (9, 2, 1) (9, 2, 1) (8.3, 2, 1.4) Weight (0.9, 0.2, 0.1) (1, 0.1, 0) (0.93, 0.16, 0.07) (1, 0.1, 0) (0.63, 0.2, 0.2) Table 2 The normalized fuzzy decision matrix C 1 C 2 C 3 C 4 C 5 A 1 (0.39, 0.09, 0.1) (0.33, 0.09, 0.1) (0.36, 0.09, 0.07) (0.41, 0.04, 0.04) (0.27, 0.11, 0.11) A 2 (0.42, 0.14, 0.09) (0.47, 0.04, 0) (0.45, 0.05, 0.03) (0.43, 0.03, 0) (0.48, 0.1, 0.06) A 3 (0.41, 0.08, 0.06) (0.42, 0.08, 0.06) (0.42, 0.08, 0.06) (0.38, 0.07, 0.07) (0.44, 0.1, 0.09) Table 3 The weighted normalized fuzzy decision matrix C 1 C 2 C 3 C 4 C 5 A 1 (0.35, 0.14, 0.14) (0.33, 0.11, 0.1) (0.33, 0.12, 0.1) (0.41, 0.08, 0.04) (0.17, 0.1, 0.15) A 2 (0.38, 0.18, 0.13) (0.48, 0.08, 0) (0.42, 0.11, 0.06) (0.45, 0.09, 0) (0.3, 0.14, 0.15) A 3 (0.37, 0.14, 0.1) (0.42, 0.11, 0.06) (0.39, 0.13, 0.09) (0.38, 0.1, 0.07) (0.28, 0.13, 0.16) Table 4 Closeness coefficients ~d j ~d þ j A A A Table 5 Ranking ~R i Rank A A A

8 normalized fuzzy decision matrix and weighted normalized fuzzy decision matrix are given in Tables 2 and 3, respectively. The closeness coefficients, which are defined to determine the ranking order of all alternatives by calculating the distance to both the fuzzy positive-ideal solution and the fuzzy negative-ideal solution simultaneously, are given in Table 4. Now a preference order can be ranked according to the order of ~R i. Therefore, the best alternative is the one with the shortest distance to the fuzzy positive ideal solution and with the longest distance to the fuzzy negative ideal solution. According to the closeness coefficient, ranking the preference order of these alternatives is as Table 5. The proposed approach presented in this paper can be applied to many areas of management decision problems. 5. Conclusion Considering the fact that, in some cases, determining precisely the exact value of the attributes is difficult and that, their values are considered as fuzzy data, therefore, in this paper TOPSIS for fuzzy data has been extended and an algorithm to determine the most preferable choice among all possible choices, when data is fuzzy, is presented. The normalized fuzzy decision matrix is calculated by using the concept of a-cuts. In this approach, as well as considering the distance of an alternative from the fuzzy positive ideal solution, its distance from the fuzzy negative ideal solution is also considered. That is to say, the less the distance of the alternative under evaluation from the fuzzy positive ideal solution and the more its distance from the fuzzy negative ideal solution, the better its ranking. References G.R. Jahanshahloo et al. / Applied Mathematics and Computation 181 (2006) [1] M.A. Abo-Sinna, A.H. Amer, Extensions of TOPSIS for multi-objective large-scale nonlinear programming problems, Applied Mathematics and Computation 162 (2005) [2] S.J. Chen, C.L. Hwang, Fuzzy Multiple Attribute Decision Making: Methods and Applications, Springer, Berlin, [3] J.S. Dyer, P.C. Fishburn, R.E. Steuer, J. Wallenius, S. Zionts, Multiple criteria decision making, multiattribute utility theory: the next ten years, Management Science 38 (5) (1992) [4] C.L. Hwang, K. Yoon, Multiple Attribute Decision Making Methods and Applications, Springer, Berlin Heidelberg, [5] G.R. Jahanshahloo, F. Hosseinzadeh Lotfi, M. Izadikhah, An algorithmic method to extend TOPSIS for decision-making problems with interval data, Applied Mathematics and Computation (2005). [6] P. Korhonen, J. Wallenius, S. Zionts, Solving the discrete multiple criteria problem using convex cones, Management Science 30 (1981) [7] Y.J. Lai, T.Y. Liu, C.L. Hwang, TOPSIS for MODM, European Journal of Operational Research 76 (3) (1994) [8] Y.J. Lai, T.Y. Liu, C.L. Hwang, Fuzzy mathematical programming, methods and applications, Springer, Berlin/Heidelberg, [9] D.L. Olson, Decision Aids for Selection Problems, Springer, New York, [10] P.L. Yu, Multiple-Criteria Decision Making, Concepts, Techniques, and Extensions, Plenum Press, New York, [11] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965)