Journal of Computer System Sciences 70 (2005) 73 85 www.elsevier.com/locate/css Multiattribute decision making models methods using intuitionistic fuzzy sets Deng-Feng Li Department Two, Dalian Naval Academy, Dalian 116018, Liaoning, China Received 27 September 2003; received in revised form 28 June 2004 Available online 18 August 2004 Abstract The concept of intuitionistic fuzzy sets is the generalization of the concept of fuzzy sets. The theory of intuitionistic fuzzy sets is well suited to dealing with vagueness. Recently, intuitionistic fuzzy sets have been used to build soft decision making models that can accommodate imprecise information, two solution concepts about the intuitionistic fuzzy core the consensus winner for group decision-making have also been developed by other researchers using intuitionistic fuzzy sets. However, it seems that there is little investigation on multicriteria /or group decision making using intuitionistic fuzzy sets with multiple criteria being explicitly taken into account. In this paper, multiattribute decision making using intuitionistic fuzzy sets is investigated, in which multiple criteria are explicitly considered, several linear programming models are constructed to generate optimal weights for attributes, the corresponding decision-making methods have also been proposed. Feasibility effectiveness of the proposed method are illustrated using a numerical example. 2004 Elsevier Inc. All rights reserved. Keywords: Fuzzy set; Intuitionistic fuzzy set; Multiattribute decision making; Linear programming model 1. Introduction The theory of fuzzy sets proposed by Zadeh [23] has attracted wide spread attentions in various fields, especially where conventional mathematical techniques are of limited effectiveness, including biological social sciences, linguistic, psychology, economics, more generally soft sciences. In such fields, variables are difficult to quantify dependencies among variables are so ill-defined E-mail addresses: lidengfeng65@hotmail.com, dengfengli@sina.com. 0022-0000/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/.css.2004.06.002
74 D.-F. Li / Journal of Computer System Sciences 70 (2005) 73 85 that precise characterization in terms of algebraic, difference or differential equations becomes almost impossible. Even in fields where dependencies between variables are well defined, it might be necessary or advantageous to employ fuzzy rather than crisp algorithms to arrive at a solution [18]. Out of several higher-order fuzzy sets, intuitionistic fuzzy sets introduced by Atanassov [1 3] have been found to be well suited to dealing with vagueness. The concept of an intuitionistic fuzzy set can be viewed as an alternative approach to define a fuzzy set in cases where available information is not sufficient for the definition of an imprecise concept by means of a conventional fuzzy set. In general, the theory of intuitionistic fuzzy sets is the generalization of fuzzy sets. Therefore, it is expected that intuitionistic fuzzy sets could be used to simulate human decision-making processes any activities requiring human expertise knowledge [12,13], which are inevitably imprecise or not totally reliable. Gau Buehrer [10] presented the concept of vague sets. Burillo Bustince [4] showed that the notion of vague sets coincides with that of intuitionistic fuzzy sets. Szmidt Kacprzyk [20] proposed a non-probabilistic type of entropy measure for intuitionistic fuzzy sets. De et al. [8] studied Sanchez s approach for medical diagnosis extended this concept with the notion of intuitionistic fuzzy set theory. Turanli Coker [22] introduced several types of fuzzy connectedness in intuitionistic fuzzy topological spaces. De et al. [7] defined some operations on intuitionistic fuzzy sets. Szmidt Kacprzyk [21] discussed distances between intuitionistic fuzzy sets. Bustince [5] presented different theorems for building intuitionistic fuzzy relations on a set with predetermined properties. Ciftcibasi Altunay [6] discussed different forms of fuzzy propositional expressions their relations. Li Cheng [15] studied similarity measures of intuitionistic fuzzy sets their application to pattern recognitions. Szmidt Kacprzyk [16 19] considered the use of intuitionistic fuzzy sets for building soft decision-making models with imprecise information, proposed two solution concepts about the intuitionistic fuzzy core the consensus winner for group decision making using intuitionistic fuzzy sets. A novel effective approach to deal with decision making in medical diagnosis using the composition of intuitionistic fuzzy relations was proposed in [9]. However, it seems that so far there has been little research on multicriteria or multiattribute in discrete decision situations /or group decision making using intuitionistic fuzzy sets, which is indeed one of the most important areas in decision analysis as most real world decision problems involve multiple criteria a group of decision makers [12,13]. In this paper, multiattribute decision making using intuitionistic fuzzy sets is investigated, in which attributes are explicitly considered, several corresponding linear programming models are constructed to generate optimal weights of attributes, the corresponding decision-making methods are also proposed. This paper is organized as follows. The definitions properties of intuitionistic fuzzy sets are briefly introduced in Section 2. Multiattribute decision-making models with intuitionistic fuzzy values are then proposed, the corresponding linear programming models methods are established in Section 3. A numerical example a short conclusion are given in Section 4 5, respectively. 2. Definitions properties of intuitionistic fuzzy sets Definition 1. (Atanassov [1 3]) Let X ={x 1,x 2,...,x n } be a finite universal set. An intuitionistic fuzzy set A in X is an obect having the following form: A ={<x, μ A (x ), υ A (x )> x X},
where the functions μ A : X [0, 1], x X μ A (x ) [0, 1] υ A : X [0, 1], x X υ A (x ) [0, 1] D.-F. Li / Journal of Computer System Sciences 70 (2005) 73 85 75 define the degree of membership degree of non-membership of the element x X to the set A X, respectively, for every x X,0 μ A (x ) + υ A (x ) 1.We call π A (x ) = 1 μ A (x ) υ A (x ), the intuitionistic index of the element x in the set A. It is the degree of indeterminacy membership of the element x to the set A. It is obvious that for every x X 0 π A (x ) 1. Distance between intuitionistic fuzzy sets was first introduced by Atanassov [2]. A deeper discussion of the distance is given by Szmidt Kacprzyk [21]. Here, we introduce a normalized Hamming distance [14], which will be employed in Section 3. Let A B be two intuitionistic fuzzy sets in the set X. Namely, A ={<x, μ A (x ), υ A (x )> x X} B ={<x, μ B (x ), υ B (x )> x X}. The normalized Hamming distance between A B is defined as follows where D(A, B) = 1 2n n ( μ A (x ) μ B (x ) + υ A (x ) υ B (x ) + π A (x ) π B (x 10 ) ), (1) =1 π A (x ) = 1 μ A (x ) υ A (x ) π B (x ) = 1 μ B (x ) υ B (x ). Theorem 1. D defined by Eq. (1) is a metric. Proof. Evidently, D is symmetric D(A, A) = 0. Conversely, if D(A, B) = 0, according to Eq. (1), we must have μ A (x ) = μ B (x ), υ A (x ) = υ B (x ) π A (x ) = π B (x ) for all x X. Hence, it follows that A = B according to Definition 1. Thus D is positive definite.
76 D.-F. Li / Journal of Computer System Sciences 70 (2005) 73 85 For any intuitionistic fuzzy sets A, B C, where C ={<x, μ C (x ), υ C (x )> x X}. Using Eq. (1), we have i.e., D(A, B)= 1 2n 1 2n n ( μ A (x ) μ B (x ) + υ A (x ) υ B (x ) + π A (x ) π B (x ) ) =1 n ( μ A (x ) μ C (x ) + υ A (x ) υ C (x ) + π A (x ) π C (x ) ) =1 + 1 2n n ( μ C (x ) μ B (x ) + υ C (x ) υ B (x ) + π C (x ) π B (x ) ) =1 = D(A, C) + D(C, B), D(A, B) D(A, C) + D(C, B). So D is triangular. Hence, we have completed the proof of Theorem 1. If A B are conventional fuzzy sets, i.e., A ={<x, μ A (x ), 1 μ A (x )> x X} B = {<x, μ B (x ), 1 μ B (x )> x X}, D(A, B) defined by Eq. (1) becomes D(A, B) = 1 n n μ A (x ) μ B (x ). =1 If A B are crisp sets, i.e., A ={<x, μ A (x ), 1 μ A (x )> x X} B ={<x, μ B (x ), 1 μ B (x )> x X}, where { 1 if x X, μ A (x ) = 0 otherwise { 1 if x X μ B (x ) = 0 otherwise. D(A, B) is the cardinality of the symmetric difference of A B, i.e., the set-theoretic difference between their union intersection. 3. Models methods for multiattribute decision makingusingintuitionistic fuzzy values 3.1. Presentation of multiattribute decision-making problems under intuitionistic fuzzy environment Suppose there exists an alternative set X ={x 1,x 2,...,x n } which consists of n non-inferior decisionmaking alternatives from which a most preferred alternative is to be selected. Each alternative is assessed on m attributes. Denote the set of all attributes A ={a 1,a 2,...,a m }. Assume that μ i υ i are the degree of membership the degree of non-membership of the alternative x X with respect to the
D.-F. Li / Journal of Computer System Sciences 70 (2005) 73 85 77 attribute a i A to the fuzzy concept excellence, respectively, where 0 μ i 1, 0 υ i 1 0 μ i + υ i 1. In other words, the evaluation of the alternative x X with respect to the attribute a i A is an intuitionistic fuzzy set. Denote X i ={<x, μ i, υ i >}. The intuitionistic indices π i = 1 μ i υ i are such that the larger π i the higher a hesitation margin of the decision maker as to the excellence of the alternative x X with respect to the attribute a i A whose intensity is given by μ i. Intuitionistic indices allow us to calculate the best final result ( the worst one) we can expect in a process leading to a final optimal decision. During the process the decision maker can change his evaluations in the following way. He can increase his evaluation by adding the value of the intuitionistic index. So in fact his evaluation lies in the closed interval [μ l i, μu i ]=[μ i, μ i + π i ], where μ l i = μ i μ u i = μ i + π i = 1 υ i. Obviously, 0 μ l i μu i 1 for all x X a i A. Similarly, assume that ρ i τ i are the degree of membership the degree of non-membership of the attribute a i A to the fuzzy concept importance, respectively, where 0 ρ i 1, 0 τ i 1 0 ρ i + τ i 1. The intuitionistic indices η i = 1 ρ i τ i are such that the larger η i the higher a hesitation margin of decision maker as to the importance of the attribute a i A whose intensity is given by ρ i. Intuitionistic indices allow us to calculate the biggest weight ( the smallest one) we can expect in a process leading to a final decision. During the process the decision maker can change his evaluating weights in the following way. He can increase his evaluating weights by adding the value of the intuitionistic index. So in fact his weight lies in the closed interval [ω l i, ωu i ]=[ρ i, ρ i + η i ], where ω l i = ρ i ω u i = ρ i + η i = 1 τ i. Obviously, 0 ω l i ωu i 1 for each attribute a i A. In addition, in this paper assume that ω l i 1 m ω u i 1 in order to find weights ω i [0, 1] (i = 1, 2,...,m) satisfying ω l i ω i ω u i m ω i = 1. 3.2. Optimization model of multiattribute decision making under intuitionistic fuzzy environment For each alternative x X, its optimal comprehensive value can be computed via the following programming { max z = m } β i ω i μ l i β i μ u i (i = 1, 2,...,m; = 1, 2,...,n), ω s.t l i ω i ω u (2) i (i = 1, 2,...,m), ω i = 1 for each = 1, 2,...,n. To solve Eq. (2), we can solve the following two linear programmings min{z l = m μ l i ω i} ω l i ω i ω u i (i = 1, 2,...,m) ω i = 1 (3)
78 D.-F. Li / Journal of Computer System Sciences 70 (2005) 73 85 m max{z u = μ u i ω i} ω l i ω i ω u i ω i = 1 (i = 1, 2,...,m), for each = 1, 2,...,n. Solving Eqs. (3) (4) by Simplex method, we can obtain their optimal solutions ω = ( ω 1, ω 2,..., ω m) T ω =( ω 1, ω 2,..., ω m )T ( =1, 2,...,n), respectively. In total, 2n linear programmings need to be solved since there are n alternatives in the set X. After generating the corresponding optimal weight vectors, the optimal comprehensive value of alternative x X can be computed as an interval [ z l, zu ], where m z l = m μ l i ω i = μ i ω i m z u = μ u i ω m i = 1 υ i ω i for each = 1, 2,...,n. That is, the optimal comprehensive value of the alternative x X is an intuitionistic fuzzy set Ā ={<x, z l, 1 zu {<x >}=, μ i ω m } i, υ i ω i >. (7) However, optimal solutions of Eqs. (3) (4) are different in general, i.e., the weight vectors ω = ω for all alternatives x X, or ω i = ω i for all i = 1, 2,...,m = 1, 2,...,n. Therefore, the comprehensive values of all n alternatives x X can not be compared. Since X is a non-inferior alternative set, there exists no evident preference on some alternatives. Hence, for each alternative x X, its obective function z l in Eq. (3) should be assigned a equal weight 1/n. Eq. (3) is then aggregated into the following linear programming: n μ l i ω i min z0 l = =1 n ω l i ω i ω u i (i = 1, 2,...,m), ω i = 1. (8) (4) (5) (6)
D.-F. Li / Journal of Computer System Sciences 70 (2005) 73 85 79 In a similar way, Eq. (4) is aggregated into the following linear programming n μ u i ω i max z0 u = =1 n ω l i ω i ω u i (i = 1, 2,,m) ω i = 1 Solving Eqs. (8) (9) by Simplex method, we can obtain their optimal solutions (9) ω 0 = ( ω 0 1, ω0 2,..., ω0 m )T ω 0 = ( ω 0 1, ω 0 2,..., ω 0 m )T, respectively. After generating the corresponding optimal weight vectors, the optimal comprehensive value of the alternative x X can be computed as an interval [ z l, z u ], where z l m = m μ l i ω0 i = μ i ω 0 i z u m = μ u i ω 0 m i = 1 υ i ω 0 i (11) for each = 1, 2,...,n. That is, the optimal comprehensive value of the alternative x X is an intuitionistic fuzzy set given by { } Ā = {<x, z l, 1 z u m } > = <x, μ i ω 0 i, υ i ω 0 i >. (12) In generating the above-intuitionistic fuzzy set only two linear programmings (i.e. Eqs. (8) (9)) need to be solved. However, the optimal solutions of Eqs. (8) (9) are normally different, so ω 0 = ω 0 in general, or ω 0 i = ω 0 i for all i = 1, 2,...,m. Therefore, it is possible that z l > z u. If this is the case, it follows that the intuitionistic index is negative, or π = 1 z l (1 z u ) = z u z l < 0. However, this is not permitted by Definition 1. (10)
80 D.-F. Li / Journal of Computer System Sciences 70 (2005) 73 85 Note that Eq. (8) is equivalent to the following linear programming n μ l i ω i max z 0 l = =1 n ω l i ω i ω u i (i = 1, 2,...,m), ω i = 1. Since Eqs. (9) (13) have the same constraints, they can be combined to formulate the following linear programming n (μ u i μl i )ω i =1 max z = n ω l i ω i ω u i (i = 1, 2,, m), (14) ω i = 1. Normally, Eqs. (9) (13) are not equivalent to Eq. (14). However, Some of solutions of Eqs. (9) (13) can be generated by solving Eq. (14). Eq. (14) can be rewritten as follows n π i ω i =1 max z = n ω l i ω i ω u i (i = 1, 2,, m), (15) ω i = 1. The optimal solution ω 0 = (ω 0 1, ω0 2,...,ω0 m )T can be obtained solving Eq. (14) or Eq. (15) by Simplex method. Then, the optimal comprehensive value of the alternative x X can be computed as an interval ], where [z 0l,z0u (13) z 0l m = m μ l i ω0 i = μ i ω 0 i z 0u = m μ u i ω0 i = 1 υ i ω 0 i (17) (16)
D.-F. Li / Journal of Computer System Sciences 70 (2005) 73 85 81 for each = 1, 2,...,n. That is, the optimal comprehensive value of the alternative x X is an intuitionistic fuzzy set given by { m } A 0 ={<x,z 0l, 1 z0u >}= <x, μ i ω 0 i, υ i ω 0 i >. (18) Theorem 2. Suppose intuitionistic fuzzy sets Ā A 0 Then for each alternative x X [ z l, zu ] [z0l,z0u ]. are defined by Eqs. (7) (18), respectively. (19) Proof. Since ω 0 = (ω 0 1, ω0 2,...,ω0 m )T is an optimal solution of Eq. (14) or Eq. (15), it is a feasible solution of Eq. (3). Note that ω = ( ω 1, ω 2,..., ω m) T is an optimal solution of Eq. (3). Then, according to Eqs. (3) (5), we have m z l = m μ l i ω i μ l i ω0 i = z0l (20) for each alternative x X. In a similar way, ω = ( ω 1, ω 2,, ω m )T is an optimal solution of Eq. (4). Then, according to Eqs. (4) (6), we have m z u = μ u i ω m i μ u i ω0 i = z0u (21) for each alternative x X. Obviously, we have z 0u = m μ u i ω0 i μ l i ω0 i = z0l (22) for each alternative x X. Then, combining Eqs. (20) (21) with Eq. (22), we have z u z0u z0l zl. Therefore, it follows that for each alternative x X. [ z l, zu ] [z0l,z0u ]. 3.3. Multiattribute decision-making method under an intuitionistic fuzzy environment Using the above Eq. (14) or Eq. (15), n optimal comprehensive values A 0 of all alternatives x X( = 1, 2,...,n) can be obtained. Now, we are interested in how a final best compromise alternative or the final ranking order of the alternative set X can be generated.
82 D.-F. Li / Journal of Computer System Sciences 70 (2005) 73 85 In a similar way to the TOPSIS method proposed by Hwang Yoon [11], we define the following index for each alternative x X D(A 0 ξ =,B) D(A 0,B)+ D(A0,G), (23) where A 0 ={<x,z 0l, 1 z0u >}={<x, m μ i ω 0 i, m υ i ω 0 i >} given by Eq. (18) is an intuitionistic fuzzy set corresponding to the optimal comprehensive value of the alternative x X. G ={<g, 1, 0 >} is an intuitionistic fuzzy set corresponding to the evaluation of the ideal alternative g. B ={<b,0, 1 >} is an intuitionistic fuzzy set corresponding to the evaluation of the negative ideal alternative b. Obviously, normally g/ X b/ X. D(A 0,B)is a distance measure between the intuitionistic fuzzy sets A0 B. D(A 0,G)is a distance measure between the intuitionistic fuzzy sets A0 G. There are several distance formulae between intuitionistic fuzzy sets [3]. In this paper, we choose the distance formula given by Eq. (1) in Section 2. Obviously, for each alternative x X,wehave 0 ξ 1 Furthermore, ξ = 0ifA 0 = B (or x is the negative ideal alternative b); ξ = 1if(A 0 = Gor x is the ideal alternative g). It is easy to see that the higher ξ the better the alternative x. According to Eq. (1), D(A 0,B) D(A0,G)are reduced into the following formulae D(A 0,B)= z0l 0 + 1 z0u = z0l + z0u + (z 0u z 0l) 2 = z 0u 1 + 1 z 0l 2 (1 z0u ) 0 (24) D(A 0,G)= z0l = 1 + 1 z0u 0 + 1 z 0l 2 (1 z0l ) + (1 z0u ) + (z0 z0l ) 2 = 1 z 0l. (1 z0u ) (25) Hence, Eq. (23) can be simply written as follows ξ = z 0u 1 + z 0u z 0l. (26)
D.-F. Li / Journal of Computer System Sciences 70 (2005) 73 85 83 From Eqs. (16) (17), Eq. (26) can also be written as follows μ u i ω0 i ξ = 1 + m (μ u i μl i )ω0 i 1 m υ i ω 0 i = 1 + (1 m υ i ω 0 i ) m μ i ω 0 i 1 m υ i ω 0 i = 2 m. (μ i + υ i )ω 0 i Thus, the best alternative x X can be generated so that ξ = max{ξ x X} (27) (28) the alternatives are ranked according to the increasing order of ξ for all x X. 4. An numerical example Consider an air-condition system selection problem. Suppose there exist three air-condition systems x 1, x 2 x 3 Denote the alternative set by X ={x 1,x 2,x 3 }. Suppose three attributes a 1 (economical), a 2 (function) a 3 (being operative) are taken into consideration in the selection problem. Denote the set of all attributes by A ={a 1,a 2,a 3 }. Using statistical methods, the degrees μ i of membership the degrees υ i of non-membership for the alternative x X with respect to the attribute a i A to the fuzzy concept excellence can be obtained, respectively. Namely, x 1 x 2 x 3 ((μ i, υ i )) 3 3 = a 1 a 2 a 3 ((μ i, υ i )) 3 3 = a 1 a 2 a 3 (0.75, 0.10) (0.80, 0.15) (0.40, 0.45), (0.60, 0.25) (0.68, 0.20) (0.75, 0.05) (0.80, 0.20) (0.45, 0.50) (0.60, 0.30) x 1 x 2 x 3 [0.75, 0.90] [0.80, 0.85] [0.40, 0.55], [0.60, 0.75] [0.68, 0.80] [0.75, 0.95] [0.80, 0.80] [0.45, 0.50] [0.60, 0.70] In a similar way, the degrees ρ i of membership the degrees τ i of non-membership for the three attributes a i A to the fuzzy concept importance can be obtained, respectively. Namely, a 1 a 2 a 3 ((ρ i, τ i )) 1 3 =((0.25, 0.25) (0.35, 0.40) (0.30, 0.65)).
84 D.-F. Li / Journal of Computer System Sciences 70 (2005) 73 85 Therefore, attribute weights lie in the closed interval as follows, a 1 a 2 a 3 ([ω l i, ωu i ]) 1 3 = ([0.25, 0.75] [0.35, 0.60] [0.30, 0.35]). According to Eq. (14) or Eq. (15), the following linear programming can be obtained { max z = 0.35ω } 1 + 0.47ω 2 + 0.15ω 3, 3 0.25 ω 1 0.75, 0.35 ω 2 0.60, 0.30 ω 3 0.35, ω 1 + ω 2 + ω 3 = 1. Solving the above linear programming, its optimal solution can be obtained as follows ω 0 = (ω 0 1, ω0 2, ω0 3 )T = (0.25, 0.40, 0.35) T. Using Eqs. (16) (17), the optimal comprehensive value of the alternative x X can be computed as follows: z 0l 1 z 0u 1 = 0.7075, z0l 2 = 0.8050, z0u 2 = 0.6295, z0l 3 = 0.610 = 0.7075, z0u 3 = 0.7625. Thus, the optimal comprehensive value of the alternative x X can be expressed as an intuitionistic fuzzy set A 0 1 ={<x 1, 0.7075, 0.1950 >},A 0 2 ={<x 2, 0.6295, 0.2925 >}, A 0 3 ={<x 3, 0.610, 0.2375 >}, respectively. For alternatives x 1, x 2 x 3, the following index for each alternative can be generated using Eq. (26): ξ 1 = ξ 2 = ξ 3 = z1 0u 0.8050 1 + z1 0u = z0l 1 + 0.8050 0.7075 = 0.7335, 1 z2 0u 0.7075 1 + z2 0u = z0l 1 + 0.7075 0.6295 = 0.6563, 2 z3 0u 0.7625 1 + z3 0u = z0l 1 + 0.7625 0.610 = 0.6616. 3 Then, the best alternative is x 1. The optimal ranking order of the alternatives is given by x 1 x 3 x 2. (29) 5. Conclusions In the above analysis, we have proposed several linear programming models methods for multiattribute decision making under intuitionistic fuzziness. In such decision situations, attributes are
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