Multicriteria Decision-Making Approach Based on ... - Semantic Scholar
510
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 3, JUNE 2013
Multicriteria Decision-Making Approach Based on Atanassov’s Intuitionistic Fuzzy Sets With Incomplete Certain Information on Weights Jian-Qiang Wang and Hong-Yu Zhang
Abstract—To handle multicriteria fuzzy decision-making problems, a new multicriteria decision-making method is proposed in which the information about criteria’s weights is not completely certain, and the criteria values of alternatives are Atanassov’s intuitionistic fuzzy sets (A-IFSs). Using evidential reasoning algorithms, the criteria values are aggregated; receiving the overall A-IFS for alternatives and the distances between each alternative and the ideal, as well as anti-ideal alternative, are computed. Combining the incomplete certain information of weights, a nonlinear programming model is developed and resolved by particle swarm optimization algorithms to obtain the optimal criteria’s weights. The corresponding decision-making procedure is given in detail. Finally, two examples are given to show the feasibility and availability of the proposed method. Index Terms—Evidential reasoning, incomplete certain information, intuitionistic fuzzy sets (IFSs), multicriteria decision making, particle swarm optimization (PSO) algorithms.
I. INTRODUCTION INCE the fuzzy set was proposed by Zadeh in 1965 [1], it has been widely studied, developed, and successfully applied in various fields, such as multicriteria decision making, logic programming, pattern recognition, and so on. In real multicriteria decision-making cases, due to the fuzziness and uncertainty of decision-making problems, the criteria’s weights and criteria values of alternatives can be inaccurate, uncertain, or incomplete. For problems such as these, fuzzy sets, especially fuzzy numbers, can provide good solutions. In fuzzy sets, the membership degree of the element in a universe is a single value between zero and one. However, those single values tell us nothing about the lack of knowledge. In practice, however, the information of alternatives corresponding to a fuzzy concept may be incomplete, i.e., the sum of the membership degree and the nonmembership degree of element in the universe corresponding to the fuzzy concept may be less than one. The fuzzy sets theory is not capable of co-opting the lack of knowledge with membership degrees, while Atanassov’s intuitionistic fuzzy sets (A-IFSs), which is the extension of Zadeh’s fuzzy
S
Manuscript received November 25, 2011; revised April 10, 2012; accepted June 22, 2012. Date of publication July 26, 2012; date of current version May 29, 2013. This work was supported by the National Natural Science Foundation of China under Grant 70921001 and Grant 71271218 and the Two-oriented Society Research Center of Central South University 985 Project under Grant ZNLX1103. The authors are with the School of Business, Central South University, Changsha 410083, China (e-mail:[email protected]; [email protected]). Digital Object Identifier 10.1109/TFUZZ.2012.2210427
sets that was introduced by Atanassov in [2]–[4], can handle it by using an additional degree. Therefore, it is expected that the IFS could be used to simulate human decision-making process and activities requiring human expertise and knowledge, which are inevitably imprecise or not totally reliable. Gau and Buehrer [5] defined vague sets in 1993. Bustince and Burillo [6] pointed out that the notion of vague sets is the same with that of IFS. Chen [7], [8] proposed a set of methods to measure the degree of similarity between vague sets and between elements. Hong and Kim [9] showed by examples that the similarity measures proposed by Chen do not fit well in some cases and proposed a set of modified measures. Li and Cheng [10] proposed several similarity measures between IFSs. Liu [11] showed that Li and Cheng’s methods have the same disadvantages with Chen’s methods, and Hong and Kim’s modified similarity measures still do not fit well in the same cases. He also proposed some new methods to measure the degree of similarity between both IFSs and elements. Li et al. [12] analyzed and compared the existing similarity measures between A-IFS/vague sets, which could help us on selection and application of similarity measures between A-IFS/vague sets in practice. Szmidt and Kacprzyk [13] proposed a nonprobabilistic type of entropy measure for IFSs. Atanassov [14] defined some operations on A-IFS. Szmidt and Kacprzyk [15] discussed distances between A-IFS. De et al. [16] studied Sanchez’s method for medical diagnosis and extended the concept with IFS. Angelov [17] proposed an approach to solve A-IFS optimization problems. Beliakov et al. [18] defined the median aggregation operator for A-IFS and Atanassov’s interval-valued IFSs. Szmidt and Kacprzyk [19]–[22] considered the use of IFS to build soft decision-making models with imprecise information, and proposed two solution concepts about the A-IFS core and the consensus winner for group decision making with A-IFS. Deschrijver and Kerre [23] proposed a novel and effective method to deal with decision making in medical diagnosis using the composition of A-IFS relations. Chen and Tan [24] presented a new approach to handle multicriteria decision-making problems based on vague sets. Hong and Choi [25] proposed another method for the same problems, while, in this method, the new functions were provided to measure the degree of accuracy of alternatives with respect to criteria. In Hong and Choi’s method, the degree of importance to each criterion is constant. Szmidt and Kacprzyk [26] proposed a ranking method for A-IFS based on distance. Lin et al. [27] defined the degree of suitability for alternatives based on the core function and the accuracy function. They also constructed a linear programming on the degree
1063-6706/$31.00 © 2012 IEEE
WANG AND ZHANG: MULTICRITERIA DECISION-MAKING APPROACH BASED ON ATANASSOV’S INTUITIONISTIC FUZZY SETS
of suitability to alternatives to obtain weights of criteria. Liu and Wang [28] presented some new methods to solve multicriteria decision-making problem based on A-IFS. Zhi and Li [29] proposed a novel approach to multiattribute decision-making problems in intuitionistic fuzzy environment. Chen [30] proposed the optimistic operator and pessimistic operator to solve multiattribute decision-making problems in intuitionistic fuzzy environment. Zeng and Su [31] proposed an ordered weighting distance operator to solve the same kind of problems. Xu [32] used the Choquet integral to propose some intuitionistic fuzzy aggregation operators and apply them to a practical decisionmaking problem. Xu [33] established an interactive method for multicriteria decision-making problems, where the criteria values are A-IFS, and the weight information on criteria is not completely known. Among the aforementioned methods, the weights of criteria are constant or A-IFS. However, in real multicriteria decision-making problems, it is difficult to give the exact criteria’s weights, or compare the important degrees between them. That is why we cannot use analytical hierarchy process (AHP), analytical network process (ANP), cognitive network process (CNP), etc., to obtain criteria’s weights. Instead, decision maker can give the incomplete certain information [34], [35]. In A-IFS, the arithmetic operations were defined in [36] and [37]. However, the revised definition still possesses some shortcomings. Owing to the deficiency of arithmetic operations of A-IFS’s current definition, we proposed a new approach to handle the multicriteria decision-making problems, in which information on the criteria’s weights is not completely certain and the criteria values are A-IFS. The proposed method uses evidential reasoning algorithms to aggregate criteria values of alternatives and constructs a nonlinear programming model to generate the weights of criteria. The rest of this paper is organized as follows. The incomplete certain information on weights of criteria and the definition as well as properties of A-IFS are briefly introduced in Section II. Multicriteria decision-making approach that is based on IFS with incomplete certain information on criteria’s weights is proposed, together with corresponding nonlinear programming model established in Section III. Two examples and a short conclusion are given in Sections IV and V, respectively. II. INCOMPLETE CERTAIN INFORMATION ON WEIGHTS OF CRITERIA AND ATANASSOV’S INTUITIONISTIC FUZZY SETS A. Incomplete Certain Information of the Criteria’s Weights In real decision making, it is difficult for the decision makers to give the exact values of the criteria’s weights, or compare the importance of the criteria, making AHP, ANP, CHP, etc., not capable of getting the criteria’s weights. However, usually, the relation of weights is given in the form of incomplete or uncertain information, such as the weight of a criterion changes in an interval, a criterion is more important than another, and so on. The incomplete information on the weights can be divided into the following five forms [34]∼ [35]: Form 1: {ωi ≥ ωj } Form 2: {ωi − ωj ≥ αi } Form 3: {ωi ≥ αi ωj }
511
Form 4: {αi ≤ ωi ≤ αi + εi } Form 5: {ωi − ωj ≥ ωk − ωl |j = k = l} where αi and εi are nonnegative constants. Those five incomplete information forms are all linear inequalities. However, in real applications, relation between criteria’s weights may be linear equality. Here, we assume that the incomplete certain information of the criteria’s weights can be both linear inequality and linear equality and can be divided into the following three types: Type 1: {ω : A1 ω ≥ b, ω > 0, b ≥ 0} Type 2: {ω : A1 ω ≤ b, ω > 0, b ≥ 0} Type 3: {ω : A1 ω = b, ω > 0, b ≥ 0} where A1 is a l × t matrix, and ω = (ω1 , ω2 , ..., ωt )T . Those three types of incomplete certain information are expanded from incomplete information, uncertain information, partial certain information, etc. B. Atanassov’s Intuitionistic Fuzzy Sets Atanassov first proposed the concept of IFSs. It is enlargement and development of Zadeh’s fuzzy sets. The A-IFS adds a new parameter: the degree of nonmembership, which gives us the possibility to model unknown information. Atanassov gave the definition of the A-IFS as follows. Definition 1 [2]–[4]: Assume X = {x1 , x2 , ..., xn } is a finite universal set. An A-IFS A on X is an object having the following form: A = {< xj , μA (xj ), νA (xj ) > |xj ∈ X} where the functions μA : X → [0, 1] xj ∈ X → μA (xj ) ∈ [0, 1] and νA : X → [0, 1] xj ∈ X → νA (xj ) ∈ [0, 1] define the degrees of membership and nonmembership of the element xj ∈ X to the set A ⊆ X, respectively, and for every element, we have xj ∈ X, 0 ≤ μA (xj ) + νA (xj ) ≤ 1. Call πA (xj ) = 1 − μA (xj ) − νA (xj ) the intuitionistic index of the element xj in set A. It is the degree of indeterminacy membership of the element xj to set A. It can measure the hesitation degree of xj to A. It is obvious that for every xj ∈ X 0 ≤ πA (x) ≤ 1. Obviously, the A-IFS corresponding to Zadeh’s fuzzy set can be stated as A = {< x, μA (x), 1 − μA (x) > |x ∈ X} ∀x ∈ X, πA (x) = 1 − μA (x) − (1 − μA (x)) = 0. The A-IFS on the universal set X will be denoted as A-IFS (X). In addition, the operations of the A-IFS are given in [14]. Definition 2: Assume that X is a finite universal set. A = {< xj , μA (xj ), νA (xj ) > |xj ∈ X} and B = {< xj , μB (xj ),
512
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 3, JUNE 2013
νB (xj ) > |xj ∈ X} are two intuitionistic sets. Then, the Hamming distance between those two A-IFSs are as follows [15]: 1 (|μA (xj ) − μB (xj )| 2n j =1 n
D(A, B) =
+ |νA (xj ) − νB (xj )| + |πA (xj ) − πB (xj )|). (1) Definition 3: If A = < x, μA (x), νA (x) >, x, 1, 0 >, the ranking function of A is [26]
A+ =
, and the intuitionistic index is βH (al , ω). Assume that the degrees of membership and nonmembership of the ideal alternative G+ corresponding to fuzzy concept “excellence” are 1 and 0, respectively. Thus, we can denote the ideal alternative with < G+ , 1, 0 >, and for the anti-ideal alternative G− , its membership and nonmembership degrees to “excellence” are 0 and 1, respectively, denoted by < G− , 0, 1 >. If G+ ∈ / A or G− ∈ / A, then involve it with A. The distance between alternative al and the ideal alternative G+ is 1 Dl+ = D(al , G+ ) = [|β1 (al , ω) − 1|+β2 (al , ω)+βH (al , ω)]. 2
WANG AND ZHANG: MULTICRITERIA DECISION-MAKING APPROACH BASED ON ATANASSOV’S INTUITIONISTIC FUZZY SETS
The distance between alternative al and the anti-ideal alternative G− is 1 Dl− = D(al , G− ) = [β1 (al , ω) + |β2 (al , ω) − 1|+βH (al , ω)]. 2 The closer the ideal alternative G+ , and the farther the antiideal alternative G− , the better the alternative al . Therefore, for each alternative al , the optimal programming models are as follows: min Dl+ = D(al , G+ ) ⎧ ω∈Ω ⎪ ⎪ ⎪ t ⎨ ωj = 1 s.t. j =1 ⎪ ⎪ ⎪ ⎩ω ≥ 0 j
(3)
into an unconstrained one. A part of particles with infeasible solutions during each generation are kept, and then, the optimal solution can be found from the feasible and infeasible region simultaneously. The key steps are designed as follows. Step 1: Preprocess particle. Let (ω1 , ω2 , ..., ωt−1 ) be a particle, and the last weight ωt can be calculated by ωt = 1 − ω1 − ω2 − · · · − ωt−1 . Therefore, the sum of weights can remain 1 after the particle updating operation. Step 2: Initializing particle swarm. By solving the following linear programming, the optimal solution ω = (ω1 , ω2 , ..., ωt ) can be obtained: min
and
513
t
ωi (μi − νi )
i=1
max Dl− = D(al , G− ) ⎧ ω∈Ω ⎪ ⎪ t ⎪ ⎨ ωj = 1 s.t. j =1 ⎪ ⎪ ⎪ ⎩ ω ≥ 0. j
(4)
For each alternative, the distances between alternative and the ideal alternative as well as the anti-ideal alternative should come from the same criteria’s weights; therefore, (3) and (4) can be combined into the following programming problem: min Dl =
D(al , G+ ) D(al , G+ ) + D(al , G− )
⎧ ω∈Ω ⎪ ⎪ ⎪ t ⎨ ωj = 1 s.t. j =1 ⎪ ⎪ ⎪ ⎩ ω ≥ 0. j
(5)
m i=1
We set the first t−1 elements of the optimal solution ω = (ω1 , ω2 , ..., ωt ) to be the initial particle. If the optimal solution of (7) does not exist, then there are conflicts in the incomplete certain information of the criteria’s weights and an adjustment will be needed. Step 3: Updating the particle velocity and position. To improve the convergent speed and capability of algorithms, the particle velocity is updated by using the inertia weight particle optimization algorithms. Namely
+ c2 randk2 (gbestkd − xkid )
D(al , G+ ) D(al , G+ ) + D(al , G− )
⎧ ω∈Ω ⎪ ⎪ ⎪ t ⎨ ωj = 1 s.t. j =1 ⎪ ⎪ ⎪ ⎩ ω ≥ 0. j
(7)
k +1 k = wvid + c1 randk1 (pbestkid − xkid ) vid
There is no evident preference for alternatives in A; hence, for each alternative al ∈ A, its objective function Dl in (5) is assigned the equal weight and could be aggregated into the following programming model: min D =
⎧ ω∈Ω ⎪ ⎪ ⎪ t ⎨ ωi = 1 s.t. i=1 ⎪ ⎪ ⎪ ⎩ 0 ≤ ωi ≤ 1.
(6)
C. Solving the Programming Model Equation (6) is a nonlinear programming model. It is difficult to be solved by using traditional optimization approach. However, many new evolution algorithms have emerged for it today, such as genetic algorithms, particle swarm optimization (PSO) algorithms, ant algorithms, and so on. For (6), we chose the improved inertia weight PSO algorithms [39]. Utilizing penalty strategies and constructing penalty function to punish infeasible solutions, we can transform the constrained optimal problem
k +1 xkid+1 = xkid + vid
where w is the inertia weight, which controls the influence of the previous particle velocity on the updated one. c1 and c2 are the weights of particle’s acceleration. Step 4: Calculating the fitness value of particles. For each generation, the object function value is the fitness of the particle. Step 5: Convergent condition of algorithms. When the iterative number surpasses the maximum one we set, the algorithms end. In addition, the particle having the smallest fitness value is optimal. D. Rank Alternatives The optimal criteria’s weight ω ∗ can be obtained by using the aforementioned improved inertia weight PSO algorithms to solve (6). Then, we can compute the optimal A-IFS of the alternative al ∈ A, denoted by B = {< al , β1 (al , ω ∗ ), β2 (al , ω ∗ ) >, l = 1, 2, ..., n}. According to definition 3, the ranking of alternatives can be gained by comparing the elements in B.
514
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 3, JUNE 2013
IV. ILLUSTRATION OF EXAMPLES In this section, we will illustrate the model based on the following two examples. We first take the example shown in [36] so that we can compare the results of the two methods. Example 1: Consider an air-condition system selection problem in [40]. Suppose there are three air conditioners: a1 , a2 , and a3 . Let A = {a1 , a2 , a3 } denote the alternative set. Suppose that three criteria C1 (economical), C2 (function), and C3 (being operative) are taken into consideration during selection. Let C = {C1 , C2 , C3 } denote the set of all criteria. Using statistical methods, the degrees of membership μij and the degrees of nonmembership νij for the alternative ai ∈ A on criterion Cj ∈ C can be obtained, respectively, namely C1
((μij , νij )) =
a1
⎛ ⎝
C2
C3
⎞
a2
(0.75, 0.10) ⎜(0.80, 0.15)
(0.60, 0.25) (0.68, 0.20)
(0.80, 0.20) . (0.45, 0.50)⎟
a3
(0.40, 0.45)
(0.75, 0.05)
(0.60, 0.30)
⎠
In a similar way, the degrees of membership ρj and the degrees of nonmembership τj for the alternative Cj ∈ C to fuzzy concept “importance” can be obtained, respectively, where j = 1,2,3, namely
((ρj , τj )) =
C1
C2
C3
((0.25, 0.25)
(0.35, 0.40)
(0.30, 0.65))
Using definition 3, we obtain R1 = 0.3031, R2 = 0.3530, and R3 = 0.3998. Therefore, we go for the ranking a1 a2 a3 . Furthermore, the best alternative obtained by our method is the same to what is in [40]. However, the ranking of a2 and a3 is different, because the ranking method in Definition 3 implies that the hesitant part of an A-IFS tends to be divided into the nonmembership. Example 2: Consider a multicriteria selection or ranking problem. Suppose there are five alternatives a1 , a2 , ..., a5 , and five criteria C1 , C2 , ..., C5 . Decision makers give the degree of membership and nonmembership of each alternative to each criterion corresponding to “excellence” based on their knowledge, experience, and statistics, and the data are shown as follows: ⎛ C1 C2 ⎜ (0.75, 0.10) (0.80, 0.15) a1 ⎜ ⎜ a2 ⎜ ⎜ (0.60, 0.25) (0.68, 0.20) ((μij , νij )) = → ⎜ (0.80, 0.20) (0.45, 0.50) a3 ⎜ ⎜ ⎜ a4 ⎝ (0.70, 0.25) (0.78, 0.20) a5 (1.00, 0.00) (0.85, 0.10) C4 C5 C3
.
Therefore, criteria weights lie in the closed intervals as follows:
[ωjl , ωju ] =
C1
C2
C3
([0.25, 0.75]
[0.35, 0.60]
[0.30, 0.35])
.
According to (6), the programming model can be obtained min D =
1 − β1 (a1 , ω) + β2 (a1 , ω) + βH (a1 , ω) 2(1 + βH (a1 , ω)) + +
1 − β1 (a2 , ω) + β2 (a2 , ω) + βH (a2 , ω) 2(1 + βH (a2 , ω)) 1 − β1 (a3 , ω) + β2 (a3 , ω) + βH (a3 , ω) 2(1 + βH (a3 , ω))
⎧ 0.25 ≤ ω1 ≤ 0.75 ⎪ ⎪ ⎪ ⎨ s 0.35 ≤ ω2 ≤ 0.60 t. ⎪ ⎪ 0.30 ≤ ω3 ≤ 0.35 ⎪ ⎩ ω1 + ω2 + ω3 = 1.
Using PSO algorithm to solve the aforementioned programming model in MATLAB, the optimal weights of the criteria can be obtained as ω ∗ = (ω1 , ω2 , ω3 )T = (0.25, 0.45, 0.30)T . The A-IFS of alternatives are {< a1 , 0.7257, 0.1693 >, < a2 , 0.6736, 0.2449 > < a3 , 0.6546, 0.1879 >}.
(0.40, 0.45)
(0.60, 0.15)
(0.75, 0.05)
(0.40, 0.40)
(0.60, 0.30)
(0.60, 0.30)
(0.85, 0.05)
(0.60, 0.30)
(0.90, 0.05)
(0.70, 0.20)
⎞
(0.55, 0.45) ⎟ ⎟ ⎟ (0.70, 0.15) ⎟ ⎟ ⎟. (0.65, 0.20) ⎟ ⎟ (0.80, 0.15) ⎟ ⎠ (0.80, 0.15)
The incomplete certain information on the criteria’s weights is given as follows: ω1 > ω3 > ω2 > ω5 > ω4 , 0.2 ≤ ω1 ≤ 0.3, 0.15 ≤ ω2 ≤ 0.25, 0.1 ≤ ω3 ≤ 0.3, 0.1 ≤ ω4 ≤ 0.2, and 0.1 ≤ ω5 ≤ 0.25. According to (6), the programming model can be stated by min D =
5 1 − β1 (al , ω) + β2 (al , ω) + βH (al , ω) l=1
2(1 + βH (al , ω))
⎧ 0.2 ≤ ω1 ≤ 0.3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.15 ≤ ω2 ≤ 0.25 ⎪ ⎪ ⎪ ⎪ ⎪ 0.10 ≤ ω3 ≤ 0.30 s ⎨ 0.10 ≤ ω ≤ 0.20 4 t. ⎪ ⎪ ⎪ 0.10 ≤ ω ⎪ 5 ≤ 0.25 ⎪ ⎪ ⎪ ⎪ ω 1 > ω3 > ω2 > ω5 > ω4 ⎪ ⎪ ⎪ ⎪ ⎩ ω + ω + ω + ω + ω = 1. 1 2 3 4 5 By using PSO algorithm to solve the model in MATLAB, the optimal weights of criteria can be obtained as 0.3000, 0.1929, 0.2692, 0.1000, and 0.1380. In addition, the A-IFS of alternatives are {, , , , and }.
WANG AND ZHANG: MULTICRITERIA DECISION-MAKING APPROACH BASED ON ATANASSOV’S INTUITIONISTIC FUZZY SETS
The ranking values of alternatives can be then calculated: 0.3860, 0.3711, 0.3491, 0.2155, and 0.0903. Thus, the best alternative is a5 , and the ranking of alternatives is a5 > a4 > a3 > a2 a1 . V. CONCLUSION In this paper, we have proposed a new method to handle the multicriteria fuzzy decision-making problems, where the information on criteria’s weights is not completely certain or A-IFS, and the criteria values of alternatives are A-IFS. The proposed method applies evidential reasoning algorithm to aggregate those criteria values for each alternative, and constructs nonlinear programming model to obtain the optimal weights of criteria. The proposed method is suitable for fuzzy decision making and can satisfy the situation in which incomplete certain information or A-IFS is provided to criteria’s weights. In the end, two examples have been presented to illustrate the multicriteria fuzzy decision-making procedure showing that the proposed method in this paper can help decision makers to make efficient decisions. REFERENCES [1] L. A. Zadeh, “Fuzzy sets,” Inf. Control, vol. 8, pp. 338–356, 1965. [2] K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets Syst., vol. 20, pp. 87–96, 1986. [3] K. T. Atanassov, Intuitionistic Fuzzy Sets. Heidelberg, Germany: Springer, 1999. [4] K. T. Atanassov, “Two theorems for intuitionistic fuzzy sets,” Fuzzy Sets Syst., vol. 110, pp. 267–269, 2000. [5] W. L. Gau and D. J. Buehrer, “Vague sets,” IEEE Trans. Syst., Man, Cybern., vol. 23, no. 2, pp. 610–614, Mar./Apr. 1993. [6] H. Bustince and P. Burillo, “Vague sets are intuitionistic fuzzy sets,” Fuzzy Sets Syst., vol. 79, pp. 403–405, 1996. [7] S. M. Chen, “Measure of similarity between vague sets,” Fuzzy Sets Syst., vol. 74, pp. 217–223, 1995. [8] S. M. Chen, “Similarity measures between vague sets and between elements,” IEEE Trans. Syst., Man, Cybern., vol. 27, no. 1, pp. 153–158, Feb. 1997. [9] D. H. Hong and C. Kim, “A note on similarity measures between vague sets and between elements,” Inf. Sci., vol. 115, pp. 83–96, 1999. [10] D. F. Li and C. T. Cheng, “New similarity measures of intuitionistic fuzzy sets and application to pattern recognition,” Pattern Recognit. Lett., vol. 33, pp. 221–225, 2002. [11] H. W. Liu, “New similarity measures between intuitionistic fuzzy sets and between elements,” Math. Comput. Model., vol. 42, pp. 61–70, 2005. [12] Y. H. Li, D. L. Olson, and Z. Qin, “Similarity measures between intuitionistic fuzzy (vague) sets: A comparative analysis,” Pattern Recognit. Lett., vol. 28, pp. 278–285, 2007. [13] E. Szmidt and J. Kacprzyk, “Entropy for intuitionistic fuzzy sets,” Fuzzy Sets Syst., vol. 118, pp. 467–477, 2001. [14] K. T. Atanassov, “Intuitionistic fuzzy sets,” presented at the VII ITKR’s Session, Sofia, Bulgaria, Jun. 1983. [15] E. Szmidt and J. Kacprzyk, “Distances between intuitionistic fuzzy sets,” Fuzzy Sets Syst., vol. 114, pp. 505–518, 2001. [16] S. K. De, R. Biswas, and A. R. Roy, “An application of intuitionistic fuzzy sets in medical diagnosis,” Fuzzy Sets Syst., vol. 117, pp. 209–213, 2001. [17] P. Angelov, “Optimization in an intuitionistic fuzzy environment,” Fuzzy Sets Syst., vol. 86, pp. 299–306, 1997. [18] G. Beliakov, H. Bustince, S. James, T. Calvo, and J. Fernandez, “Aggregation for Atanassov’s intuitionistic and interval valued fuzzy sets: The median operator,” IEEE Trans. Fuzzy Syst., vol. 20, no. 3, pp. 487–498, Jun. 2012. [19] E. Szmidt and J. Kacprzyk, “Intuitionistic fuzzy sets in group decision making,” NIFS, vol. 2, pp. 15–32, 1996. [20] E. Szmidt and J. Kacprzyk, “Remark on some application of intuitionistic fuzzy sets in decision making,” NIFS, vol. 2, pp. 22–31, 1996. [21] E. Szmidt and J. Kacprzyk, “Group decision making via intuitionistic fuzzy sets,” in Proc. 2nd Workshop Fuzzy Based Expert Syst., Sofia, Bulgaria, Oct. 9–11, 1996, pp. 107–112.
515
[22] E. Szmidt and J. Kacprzyk, “Intuitionistic fuzzy sets for more realistic group decision making,” in Proc. Int. Conf. Transition Adv. Market Inst. Econ., Warsaw, Poland, Jun. 18-21,1997, pp. 430–433. [23] G. Deschrijver and E. E. Kerre, “On composition of intuitionistic fuzzy relations,” Fuzzy Sets Syst., vol. 136, pp. 333–361, 2003. [24] S. M. Chen and J. M. Tan, “Handling multicriteria fuzzy decision-making problems based on vague set theory,” Fuzzy Sets Syst., vol. 67, pp. 163– 172, 1994. [25] D. H. Hong and C. H. Choi, “Multicriteria fuzzy decision-making problems based on vague sets theory,” Fuzzy Sets Syst., vol. 114, pp. 103–113, 2000. [26] E. Szmidt and J. Kacprzyk, “Ranking of intuitionistic fuzzy alternatives in a multi-criteria decision making problem,” in Proc. 28th Annu. Conf. North Amer. Fuzzy Inf. Process. Soc., Cincinnati, OH, Jun. 14–17, 2009, pp. 1–6. [27] L. Lin, X. H. Yuan, and Z. Q. Xia, “Multicriteria fuzzy decision methods based on intuitionistic fuzzy sets,” J. Comput. Syst. Sci., vol. 73, pp. 84– 88, 2007. [28] H. W. Liu and G. J. Wang, “Multi-criteria methods based on intuitionistic fuzzy sets,” Eur. J. Oper. Res., vol. 179, pp. 220–233, 2007. [29] P. Zhi and Z. Li, “A novel approach to multi-attribute decision making based on intuitionistic fuzzy sets,” Expert Syst. Appl., vol. 39, pp. 2560– 2566, 2012. [30] Y. T. Chen, “A outcome-oriented approach to multicriteria decision analysis with intuitionistic fuzzy optimistic/pessimistic operators,” Expert Syst. Appl., vol. 37, pp. 7762–7774, 2010. [31] S. Z. Zeng and W. H. Su, “Intuitionistic fuzzy ordered weighted distance operator,” Knowledge-Based Syst., vol. 24, pp. 1224–1232, 2011. [32] Z. S. Xu, “Choquet integrals of weighted intuitionistic fuzzy information,” Inf. Sci., vol. 180, pp. 726–736, 2010. [33] Z. S. Xu, “Intuitionistic fuzzy multiattribute decision making: An interactive method,” IEEE Trans. Fuzzy Syst., vol. 20, no. 3, pp. 514–525, Jun. 2012. [34] H. K. Soung and H. H. Chang, “An interactive procedure for multi-attribute group decision making with incomplete information,” Comput. Oper. Res., vol. 26, pp. 755–772, 1999. [35] H. K. Soung, H. C. Sang, and K. K. Jae, “An interactive procedure for multi-attribute group decision making with incomplete information: Range-based approach,” Eur. J. Oper. Res., vol. 118, pp. 139–152, 1999. [36] Z. S. Xu, “Intuitionistic fuzzy aggregation operators,” IEEE Trans. Fuzzy Syst., vol. 15, no. 6, pp. 1179–1187, Dec. 2007. [37] Z. S. Xu and R. R. Yager, “Some geometric aggregation operators based on intuitionistic fuzzy sets,” Int. J. Gen. Syst., vol. 35, pp. 417–433, 2006. [38] J. B. Yang, “Rule and utility based evidential reasoning approach for multi-attribute decision analysis under uncertainty,” Eur. J. Oper. Res., vol. 131, pp. 31–61, 2001. [39] Y. Shi and R. C. Eberhart, “Empirical study of particle swarms optimization,” in Proc. Congr. Evol. Comput., Piscataway, NJ, 1999, pp. 1945– 1949. [40] D. F. Li, “Multiattribute decision making models and methods using intuitionistic fuzzy sets,” J. Comput. Syst. Sci., vol. 70, pp. 73–85, 2005. Jian-Qiang Wang received the Ph.D. degree in management science and engineering from Central South University, Changsha, China, in 2005. He is currently a Professor with the Business School, Central South University. His current research interests include decision-making theory and application, risk management and control, and information management.
Hong-Yu Zhang received the M.S. degree in computer software and theory from the School of Information Science and Engineering, Central South University, Changsha, China, in 2005 and the Ph.D. degree in management science and engineering from the Business School, Central South University, Changsha, China, in 2009. She is currently a Lecturer with the Business School, Central South University. Her research interests include the area of information management and its applications in production operations. Her current research focuses on remanufacturing production management and decisionmaking theory.
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 3, JUNE 2013
Multicriteria Decision-Making Approach Based on Atanassov’s Intuitionistic Fuzzy Sets With Incomplete Certain Information on Weights Jian-Qiang Wang and Hong-Yu Zhang
Abstract—To handle multicriteria fuzzy decision-making problems, a new multicriteria decision-making method is proposed in which the information about criteria’s weights is not completely certain, and the criteria values of alternatives are Atanassov’s intuitionistic fuzzy sets (A-IFSs). Using evidential reasoning algorithms, the criteria values are aggregated; receiving the overall A-IFS for alternatives and the distances between each alternative and the ideal, as well as anti-ideal alternative, are computed. Combining the incomplete certain information of weights, a nonlinear programming model is developed and resolved by particle swarm optimization algorithms to obtain the optimal criteria’s weights. The corresponding decision-making procedure is given in detail. Finally, two examples are given to show the feasibility and availability of the proposed method. Index Terms—Evidential reasoning, incomplete certain information, intuitionistic fuzzy sets (IFSs), multicriteria decision making, particle swarm optimization (PSO) algorithms.
I. INTRODUCTION INCE the fuzzy set was proposed by Zadeh in 1965 [1], it has been widely studied, developed, and successfully applied in various fields, such as multicriteria decision making, logic programming, pattern recognition, and so on. In real multicriteria decision-making cases, due to the fuzziness and uncertainty of decision-making problems, the criteria’s weights and criteria values of alternatives can be inaccurate, uncertain, or incomplete. For problems such as these, fuzzy sets, especially fuzzy numbers, can provide good solutions. In fuzzy sets, the membership degree of the element in a universe is a single value between zero and one. However, those single values tell us nothing about the lack of knowledge. In practice, however, the information of alternatives corresponding to a fuzzy concept may be incomplete, i.e., the sum of the membership degree and the nonmembership degree of element in the universe corresponding to the fuzzy concept may be less than one. The fuzzy sets theory is not capable of co-opting the lack of knowledge with membership degrees, while Atanassov’s intuitionistic fuzzy sets (A-IFSs), which is the extension of Zadeh’s fuzzy
S
Manuscript received November 25, 2011; revised April 10, 2012; accepted June 22, 2012. Date of publication July 26, 2012; date of current version May 29, 2013. This work was supported by the National Natural Science Foundation of China under Grant 70921001 and Grant 71271218 and the Two-oriented Society Research Center of Central South University 985 Project under Grant ZNLX1103. The authors are with the School of Business, Central South University, Changsha 410083, China (e-mail:[email protected]; [email protected]). Digital Object Identifier 10.1109/TFUZZ.2012.2210427
sets that was introduced by Atanassov in [2]–[4], can handle it by using an additional degree. Therefore, it is expected that the IFS could be used to simulate human decision-making process and activities requiring human expertise and knowledge, which are inevitably imprecise or not totally reliable. Gau and Buehrer [5] defined vague sets in 1993. Bustince and Burillo [6] pointed out that the notion of vague sets is the same with that of IFS. Chen [7], [8] proposed a set of methods to measure the degree of similarity between vague sets and between elements. Hong and Kim [9] showed by examples that the similarity measures proposed by Chen do not fit well in some cases and proposed a set of modified measures. Li and Cheng [10] proposed several similarity measures between IFSs. Liu [11] showed that Li and Cheng’s methods have the same disadvantages with Chen’s methods, and Hong and Kim’s modified similarity measures still do not fit well in the same cases. He also proposed some new methods to measure the degree of similarity between both IFSs and elements. Li et al. [12] analyzed and compared the existing similarity measures between A-IFS/vague sets, which could help us on selection and application of similarity measures between A-IFS/vague sets in practice. Szmidt and Kacprzyk [13] proposed a nonprobabilistic type of entropy measure for IFSs. Atanassov [14] defined some operations on A-IFS. Szmidt and Kacprzyk [15] discussed distances between A-IFS. De et al. [16] studied Sanchez’s method for medical diagnosis and extended the concept with IFS. Angelov [17] proposed an approach to solve A-IFS optimization problems. Beliakov et al. [18] defined the median aggregation operator for A-IFS and Atanassov’s interval-valued IFSs. Szmidt and Kacprzyk [19]–[22] considered the use of IFS to build soft decision-making models with imprecise information, and proposed two solution concepts about the A-IFS core and the consensus winner for group decision making with A-IFS. Deschrijver and Kerre [23] proposed a novel and effective method to deal with decision making in medical diagnosis using the composition of A-IFS relations. Chen and Tan [24] presented a new approach to handle multicriteria decision-making problems based on vague sets. Hong and Choi [25] proposed another method for the same problems, while, in this method, the new functions were provided to measure the degree of accuracy of alternatives with respect to criteria. In Hong and Choi’s method, the degree of importance to each criterion is constant. Szmidt and Kacprzyk [26] proposed a ranking method for A-IFS based on distance. Lin et al. [27] defined the degree of suitability for alternatives based on the core function and the accuracy function. They also constructed a linear programming on the degree
1063-6706/$31.00 © 2012 IEEE
WANG AND ZHANG: MULTICRITERIA DECISION-MAKING APPROACH BASED ON ATANASSOV’S INTUITIONISTIC FUZZY SETS
of suitability to alternatives to obtain weights of criteria. Liu and Wang [28] presented some new methods to solve multicriteria decision-making problem based on A-IFS. Zhi and Li [29] proposed a novel approach to multiattribute decision-making problems in intuitionistic fuzzy environment. Chen [30] proposed the optimistic operator and pessimistic operator to solve multiattribute decision-making problems in intuitionistic fuzzy environment. Zeng and Su [31] proposed an ordered weighting distance operator to solve the same kind of problems. Xu [32] used the Choquet integral to propose some intuitionistic fuzzy aggregation operators and apply them to a practical decisionmaking problem. Xu [33] established an interactive method for multicriteria decision-making problems, where the criteria values are A-IFS, and the weight information on criteria is not completely known. Among the aforementioned methods, the weights of criteria are constant or A-IFS. However, in real multicriteria decision-making problems, it is difficult to give the exact criteria’s weights, or compare the important degrees between them. That is why we cannot use analytical hierarchy process (AHP), analytical network process (ANP), cognitive network process (CNP), etc., to obtain criteria’s weights. Instead, decision maker can give the incomplete certain information [34], [35]. In A-IFS, the arithmetic operations were defined in [36] and [37]. However, the revised definition still possesses some shortcomings. Owing to the deficiency of arithmetic operations of A-IFS’s current definition, we proposed a new approach to handle the multicriteria decision-making problems, in which information on the criteria’s weights is not completely certain and the criteria values are A-IFS. The proposed method uses evidential reasoning algorithms to aggregate criteria values of alternatives and constructs a nonlinear programming model to generate the weights of criteria. The rest of this paper is organized as follows. The incomplete certain information on weights of criteria and the definition as well as properties of A-IFS are briefly introduced in Section II. Multicriteria decision-making approach that is based on IFS with incomplete certain information on criteria’s weights is proposed, together with corresponding nonlinear programming model established in Section III. Two examples and a short conclusion are given in Sections IV and V, respectively. II. INCOMPLETE CERTAIN INFORMATION ON WEIGHTS OF CRITERIA AND ATANASSOV’S INTUITIONISTIC FUZZY SETS A. Incomplete Certain Information of the Criteria’s Weights In real decision making, it is difficult for the decision makers to give the exact values of the criteria’s weights, or compare the importance of the criteria, making AHP, ANP, CHP, etc., not capable of getting the criteria’s weights. However, usually, the relation of weights is given in the form of incomplete or uncertain information, such as the weight of a criterion changes in an interval, a criterion is more important than another, and so on. The incomplete information on the weights can be divided into the following five forms [34]∼ [35]: Form 1: {ωi ≥ ωj } Form 2: {ωi − ωj ≥ αi } Form 3: {ωi ≥ αi ωj }
511
Form 4: {αi ≤ ωi ≤ αi + εi } Form 5: {ωi − ωj ≥ ωk − ωl |j = k = l} where αi and εi are nonnegative constants. Those five incomplete information forms are all linear inequalities. However, in real applications, relation between criteria’s weights may be linear equality. Here, we assume that the incomplete certain information of the criteria’s weights can be both linear inequality and linear equality and can be divided into the following three types: Type 1: {ω : A1 ω ≥ b, ω > 0, b ≥ 0} Type 2: {ω : A1 ω ≤ b, ω > 0, b ≥ 0} Type 3: {ω : A1 ω = b, ω > 0, b ≥ 0} where A1 is a l × t matrix, and ω = (ω1 , ω2 , ..., ωt )T . Those three types of incomplete certain information are expanded from incomplete information, uncertain information, partial certain information, etc. B. Atanassov’s Intuitionistic Fuzzy Sets Atanassov first proposed the concept of IFSs. It is enlargement and development of Zadeh’s fuzzy sets. The A-IFS adds a new parameter: the degree of nonmembership, which gives us the possibility to model unknown information. Atanassov gave the definition of the A-IFS as follows. Definition 1 [2]–[4]: Assume X = {x1 , x2 , ..., xn } is a finite universal set. An A-IFS A on X is an object having the following form: A = {< xj , μA (xj ), νA (xj ) > |xj ∈ X} where the functions μA : X → [0, 1] xj ∈ X → μA (xj ) ∈ [0, 1] and νA : X → [0, 1] xj ∈ X → νA (xj ) ∈ [0, 1] define the degrees of membership and nonmembership of the element xj ∈ X to the set A ⊆ X, respectively, and for every element, we have xj ∈ X, 0 ≤ μA (xj ) + νA (xj ) ≤ 1. Call πA (xj ) = 1 − μA (xj ) − νA (xj ) the intuitionistic index of the element xj in set A. It is the degree of indeterminacy membership of the element xj to set A. It can measure the hesitation degree of xj to A. It is obvious that for every xj ∈ X 0 ≤ πA (x) ≤ 1. Obviously, the A-IFS corresponding to Zadeh’s fuzzy set can be stated as A = {< x, μA (x), 1 − μA (x) > |x ∈ X} ∀x ∈ X, πA (x) = 1 − μA (x) − (1 − μA (x)) = 0. The A-IFS on the universal set X will be denoted as A-IFS (X). In addition, the operations of the A-IFS are given in [14]. Definition 2: Assume that X is a finite universal set. A = {< xj , μA (xj ), νA (xj ) > |xj ∈ X} and B = {< xj , μB (xj ),
512
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 3, JUNE 2013
νB (xj ) > |xj ∈ X} are two intuitionistic sets. Then, the Hamming distance between those two A-IFSs are as follows [15]: 1 (|μA (xj ) − μB (xj )| 2n j =1 n
D(A, B) =
+ |νA (xj ) − νB (xj )| + |πA (xj ) − πB (xj )|). (1) Definition 3: If A = < x, μA (x), νA (x) >, x, 1, 0 >, the ranking function of A is [26]
A+ =
, and the intuitionistic index is βH (al , ω). Assume that the degrees of membership and nonmembership of the ideal alternative G+ corresponding to fuzzy concept “excellence” are 1 and 0, respectively. Thus, we can denote the ideal alternative with < G+ , 1, 0 >, and for the anti-ideal alternative G− , its membership and nonmembership degrees to “excellence” are 0 and 1, respectively, denoted by < G− , 0, 1 >. If G+ ∈ / A or G− ∈ / A, then involve it with A. The distance between alternative al and the ideal alternative G+ is 1 Dl+ = D(al , G+ ) = [|β1 (al , ω) − 1|+β2 (al , ω)+βH (al , ω)]. 2
WANG AND ZHANG: MULTICRITERIA DECISION-MAKING APPROACH BASED ON ATANASSOV’S INTUITIONISTIC FUZZY SETS
The distance between alternative al and the anti-ideal alternative G− is 1 Dl− = D(al , G− ) = [β1 (al , ω) + |β2 (al , ω) − 1|+βH (al , ω)]. 2 The closer the ideal alternative G+ , and the farther the antiideal alternative G− , the better the alternative al . Therefore, for each alternative al , the optimal programming models are as follows: min Dl+ = D(al , G+ ) ⎧ ω∈Ω ⎪ ⎪ ⎪ t ⎨ ωj = 1 s.t. j =1 ⎪ ⎪ ⎪ ⎩ω ≥ 0 j
(3)
into an unconstrained one. A part of particles with infeasible solutions during each generation are kept, and then, the optimal solution can be found from the feasible and infeasible region simultaneously. The key steps are designed as follows. Step 1: Preprocess particle. Let (ω1 , ω2 , ..., ωt−1 ) be a particle, and the last weight ωt can be calculated by ωt = 1 − ω1 − ω2 − · · · − ωt−1 . Therefore, the sum of weights can remain 1 after the particle updating operation. Step 2: Initializing particle swarm. By solving the following linear programming, the optimal solution ω = (ω1 , ω2 , ..., ωt ) can be obtained: min
and
513
t
ωi (μi − νi )
i=1
max Dl− = D(al , G− ) ⎧ ω∈Ω ⎪ ⎪ t ⎪ ⎨ ωj = 1 s.t. j =1 ⎪ ⎪ ⎪ ⎩ ω ≥ 0. j
(4)
For each alternative, the distances between alternative and the ideal alternative as well as the anti-ideal alternative should come from the same criteria’s weights; therefore, (3) and (4) can be combined into the following programming problem: min Dl =
D(al , G+ ) D(al , G+ ) + D(al , G− )
⎧ ω∈Ω ⎪ ⎪ ⎪ t ⎨ ωj = 1 s.t. j =1 ⎪ ⎪ ⎪ ⎩ ω ≥ 0. j
(5)
m i=1
We set the first t−1 elements of the optimal solution ω = (ω1 , ω2 , ..., ωt ) to be the initial particle. If the optimal solution of (7) does not exist, then there are conflicts in the incomplete certain information of the criteria’s weights and an adjustment will be needed. Step 3: Updating the particle velocity and position. To improve the convergent speed and capability of algorithms, the particle velocity is updated by using the inertia weight particle optimization algorithms. Namely
+ c2 randk2 (gbestkd − xkid )
D(al , G+ ) D(al , G+ ) + D(al , G− )
⎧ ω∈Ω ⎪ ⎪ ⎪ t ⎨ ωj = 1 s.t. j =1 ⎪ ⎪ ⎪ ⎩ ω ≥ 0. j
(7)
k +1 k = wvid + c1 randk1 (pbestkid − xkid ) vid
There is no evident preference for alternatives in A; hence, for each alternative al ∈ A, its objective function Dl in (5) is assigned the equal weight and could be aggregated into the following programming model: min D =
⎧ ω∈Ω ⎪ ⎪ ⎪ t ⎨ ωi = 1 s.t. i=1 ⎪ ⎪ ⎪ ⎩ 0 ≤ ωi ≤ 1.
(6)
C. Solving the Programming Model Equation (6) is a nonlinear programming model. It is difficult to be solved by using traditional optimization approach. However, many new evolution algorithms have emerged for it today, such as genetic algorithms, particle swarm optimization (PSO) algorithms, ant algorithms, and so on. For (6), we chose the improved inertia weight PSO algorithms [39]. Utilizing penalty strategies and constructing penalty function to punish infeasible solutions, we can transform the constrained optimal problem
k +1 xkid+1 = xkid + vid
where w is the inertia weight, which controls the influence of the previous particle velocity on the updated one. c1 and c2 are the weights of particle’s acceleration. Step 4: Calculating the fitness value of particles. For each generation, the object function value is the fitness of the particle. Step 5: Convergent condition of algorithms. When the iterative number surpasses the maximum one we set, the algorithms end. In addition, the particle having the smallest fitness value is optimal. D. Rank Alternatives The optimal criteria’s weight ω ∗ can be obtained by using the aforementioned improved inertia weight PSO algorithms to solve (6). Then, we can compute the optimal A-IFS of the alternative al ∈ A, denoted by B = {< al , β1 (al , ω ∗ ), β2 (al , ω ∗ ) >, l = 1, 2, ..., n}. According to definition 3, the ranking of alternatives can be gained by comparing the elements in B.
514
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 3, JUNE 2013
IV. ILLUSTRATION OF EXAMPLES In this section, we will illustrate the model based on the following two examples. We first take the example shown in [36] so that we can compare the results of the two methods. Example 1: Consider an air-condition system selection problem in [40]. Suppose there are three air conditioners: a1 , a2 , and a3 . Let A = {a1 , a2 , a3 } denote the alternative set. Suppose that three criteria C1 (economical), C2 (function), and C3 (being operative) are taken into consideration during selection. Let C = {C1 , C2 , C3 } denote the set of all criteria. Using statistical methods, the degrees of membership μij and the degrees of nonmembership νij for the alternative ai ∈ A on criterion Cj ∈ C can be obtained, respectively, namely C1
((μij , νij )) =
a1
⎛ ⎝
C2
C3
⎞
a2
(0.75, 0.10) ⎜(0.80, 0.15)
(0.60, 0.25) (0.68, 0.20)
(0.80, 0.20) . (0.45, 0.50)⎟
a3
(0.40, 0.45)
(0.75, 0.05)
(0.60, 0.30)
⎠
In a similar way, the degrees of membership ρj and the degrees of nonmembership τj for the alternative Cj ∈ C to fuzzy concept “importance” can be obtained, respectively, where j = 1,2,3, namely
((ρj , τj )) =
C1
C2
C3
((0.25, 0.25)
(0.35, 0.40)
(0.30, 0.65))
Using definition 3, we obtain R1 = 0.3031, R2 = 0.3530, and R3 = 0.3998. Therefore, we go for the ranking a1 a2 a3 . Furthermore, the best alternative obtained by our method is the same to what is in [40]. However, the ranking of a2 and a3 is different, because the ranking method in Definition 3 implies that the hesitant part of an A-IFS tends to be divided into the nonmembership. Example 2: Consider a multicriteria selection or ranking problem. Suppose there are five alternatives a1 , a2 , ..., a5 , and five criteria C1 , C2 , ..., C5 . Decision makers give the degree of membership and nonmembership of each alternative to each criterion corresponding to “excellence” based on their knowledge, experience, and statistics, and the data are shown as follows: ⎛ C1 C2 ⎜ (0.75, 0.10) (0.80, 0.15) a1 ⎜ ⎜ a2 ⎜ ⎜ (0.60, 0.25) (0.68, 0.20) ((μij , νij )) = → ⎜ (0.80, 0.20) (0.45, 0.50) a3 ⎜ ⎜ ⎜ a4 ⎝ (0.70, 0.25) (0.78, 0.20) a5 (1.00, 0.00) (0.85, 0.10) C4 C5 C3
.
Therefore, criteria weights lie in the closed intervals as follows:
[ωjl , ωju ] =
C1
C2
C3
([0.25, 0.75]
[0.35, 0.60]
[0.30, 0.35])
.
According to (6), the programming model can be obtained min D =
1 − β1 (a1 , ω) + β2 (a1 , ω) + βH (a1 , ω) 2(1 + βH (a1 , ω)) + +
1 − β1 (a2 , ω) + β2 (a2 , ω) + βH (a2 , ω) 2(1 + βH (a2 , ω)) 1 − β1 (a3 , ω) + β2 (a3 , ω) + βH (a3 , ω) 2(1 + βH (a3 , ω))
⎧ 0.25 ≤ ω1 ≤ 0.75 ⎪ ⎪ ⎪ ⎨ s 0.35 ≤ ω2 ≤ 0.60 t. ⎪ ⎪ 0.30 ≤ ω3 ≤ 0.35 ⎪ ⎩ ω1 + ω2 + ω3 = 1.
Using PSO algorithm to solve the aforementioned programming model in MATLAB, the optimal weights of the criteria can be obtained as ω ∗ = (ω1 , ω2 , ω3 )T = (0.25, 0.45, 0.30)T . The A-IFS of alternatives are {< a1 , 0.7257, 0.1693 >, < a2 , 0.6736, 0.2449 > < a3 , 0.6546, 0.1879 >}.
(0.40, 0.45)
(0.60, 0.15)
(0.75, 0.05)
(0.40, 0.40)
(0.60, 0.30)
(0.60, 0.30)
(0.85, 0.05)
(0.60, 0.30)
(0.90, 0.05)
(0.70, 0.20)
⎞
(0.55, 0.45) ⎟ ⎟ ⎟ (0.70, 0.15) ⎟ ⎟ ⎟. (0.65, 0.20) ⎟ ⎟ (0.80, 0.15) ⎟ ⎠ (0.80, 0.15)
The incomplete certain information on the criteria’s weights is given as follows: ω1 > ω3 > ω2 > ω5 > ω4 , 0.2 ≤ ω1 ≤ 0.3, 0.15 ≤ ω2 ≤ 0.25, 0.1 ≤ ω3 ≤ 0.3, 0.1 ≤ ω4 ≤ 0.2, and 0.1 ≤ ω5 ≤ 0.25. According to (6), the programming model can be stated by min D =
5 1 − β1 (al , ω) + β2 (al , ω) + βH (al , ω) l=1
2(1 + βH (al , ω))
⎧ 0.2 ≤ ω1 ≤ 0.3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.15 ≤ ω2 ≤ 0.25 ⎪ ⎪ ⎪ ⎪ ⎪ 0.10 ≤ ω3 ≤ 0.30 s ⎨ 0.10 ≤ ω ≤ 0.20 4 t. ⎪ ⎪ ⎪ 0.10 ≤ ω ⎪ 5 ≤ 0.25 ⎪ ⎪ ⎪ ⎪ ω 1 > ω3 > ω2 > ω5 > ω4 ⎪ ⎪ ⎪ ⎪ ⎩ ω + ω + ω + ω + ω = 1. 1 2 3 4 5 By using PSO algorithm to solve the model in MATLAB, the optimal weights of criteria can be obtained as 0.3000, 0.1929, 0.2692, 0.1000, and 0.1380. In addition, the A-IFS of alternatives are {, , , , and }.
WANG AND ZHANG: MULTICRITERIA DECISION-MAKING APPROACH BASED ON ATANASSOV’S INTUITIONISTIC FUZZY SETS
The ranking values of alternatives can be then calculated: 0.3860, 0.3711, 0.3491, 0.2155, and 0.0903. Thus, the best alternative is a5 , and the ranking of alternatives is a5 > a4 > a3 > a2 a1 . V. CONCLUSION In this paper, we have proposed a new method to handle the multicriteria fuzzy decision-making problems, where the information on criteria’s weights is not completely certain or A-IFS, and the criteria values of alternatives are A-IFS. The proposed method applies evidential reasoning algorithm to aggregate those criteria values for each alternative, and constructs nonlinear programming model to obtain the optimal weights of criteria. The proposed method is suitable for fuzzy decision making and can satisfy the situation in which incomplete certain information or A-IFS is provided to criteria’s weights. In the end, two examples have been presented to illustrate the multicriteria fuzzy decision-making procedure showing that the proposed method in this paper can help decision makers to make efficient decisions. REFERENCES [1] L. A. Zadeh, “Fuzzy sets,” Inf. Control, vol. 8, pp. 338–356, 1965. [2] K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets Syst., vol. 20, pp. 87–96, 1986. [3] K. T. Atanassov, Intuitionistic Fuzzy Sets. Heidelberg, Germany: Springer, 1999. [4] K. T. Atanassov, “Two theorems for intuitionistic fuzzy sets,” Fuzzy Sets Syst., vol. 110, pp. 267–269, 2000. [5] W. L. Gau and D. J. Buehrer, “Vague sets,” IEEE Trans. Syst., Man, Cybern., vol. 23, no. 2, pp. 610–614, Mar./Apr. 1993. [6] H. Bustince and P. Burillo, “Vague sets are intuitionistic fuzzy sets,” Fuzzy Sets Syst., vol. 79, pp. 403–405, 1996. [7] S. M. Chen, “Measure of similarity between vague sets,” Fuzzy Sets Syst., vol. 74, pp. 217–223, 1995. [8] S. M. Chen, “Similarity measures between vague sets and between elements,” IEEE Trans. Syst., Man, Cybern., vol. 27, no. 1, pp. 153–158, Feb. 1997. [9] D. H. Hong and C. Kim, “A note on similarity measures between vague sets and between elements,” Inf. Sci., vol. 115, pp. 83–96, 1999. [10] D. F. Li and C. T. Cheng, “New similarity measures of intuitionistic fuzzy sets and application to pattern recognition,” Pattern Recognit. Lett., vol. 33, pp. 221–225, 2002. [11] H. W. Liu, “New similarity measures between intuitionistic fuzzy sets and between elements,” Math. Comput. Model., vol. 42, pp. 61–70, 2005. [12] Y. H. Li, D. L. Olson, and Z. Qin, “Similarity measures between intuitionistic fuzzy (vague) sets: A comparative analysis,” Pattern Recognit. Lett., vol. 28, pp. 278–285, 2007. [13] E. Szmidt and J. Kacprzyk, “Entropy for intuitionistic fuzzy sets,” Fuzzy Sets Syst., vol. 118, pp. 467–477, 2001. [14] K. T. Atanassov, “Intuitionistic fuzzy sets,” presented at the VII ITKR’s Session, Sofia, Bulgaria, Jun. 1983. [15] E. Szmidt and J. Kacprzyk, “Distances between intuitionistic fuzzy sets,” Fuzzy Sets Syst., vol. 114, pp. 505–518, 2001. [16] S. K. De, R. Biswas, and A. R. Roy, “An application of intuitionistic fuzzy sets in medical diagnosis,” Fuzzy Sets Syst., vol. 117, pp. 209–213, 2001. [17] P. Angelov, “Optimization in an intuitionistic fuzzy environment,” Fuzzy Sets Syst., vol. 86, pp. 299–306, 1997. [18] G. Beliakov, H. Bustince, S. James, T. Calvo, and J. Fernandez, “Aggregation for Atanassov’s intuitionistic and interval valued fuzzy sets: The median operator,” IEEE Trans. Fuzzy Syst., vol. 20, no. 3, pp. 487–498, Jun. 2012. [19] E. Szmidt and J. Kacprzyk, “Intuitionistic fuzzy sets in group decision making,” NIFS, vol. 2, pp. 15–32, 1996. [20] E. Szmidt and J. Kacprzyk, “Remark on some application of intuitionistic fuzzy sets in decision making,” NIFS, vol. 2, pp. 22–31, 1996. [21] E. Szmidt and J. Kacprzyk, “Group decision making via intuitionistic fuzzy sets,” in Proc. 2nd Workshop Fuzzy Based Expert Syst., Sofia, Bulgaria, Oct. 9–11, 1996, pp. 107–112.
515
[22] E. Szmidt and J. Kacprzyk, “Intuitionistic fuzzy sets for more realistic group decision making,” in Proc. Int. Conf. Transition Adv. Market Inst. Econ., Warsaw, Poland, Jun. 18-21,1997, pp. 430–433. [23] G. Deschrijver and E. E. Kerre, “On composition of intuitionistic fuzzy relations,” Fuzzy Sets Syst., vol. 136, pp. 333–361, 2003. [24] S. M. Chen and J. M. Tan, “Handling multicriteria fuzzy decision-making problems based on vague set theory,” Fuzzy Sets Syst., vol. 67, pp. 163– 172, 1994. [25] D. H. Hong and C. H. Choi, “Multicriteria fuzzy decision-making problems based on vague sets theory,” Fuzzy Sets Syst., vol. 114, pp. 103–113, 2000. [26] E. Szmidt and J. Kacprzyk, “Ranking of intuitionistic fuzzy alternatives in a multi-criteria decision making problem,” in Proc. 28th Annu. Conf. North Amer. Fuzzy Inf. Process. Soc., Cincinnati, OH, Jun. 14–17, 2009, pp. 1–6. [27] L. Lin, X. H. Yuan, and Z. Q. Xia, “Multicriteria fuzzy decision methods based on intuitionistic fuzzy sets,” J. Comput. Syst. Sci., vol. 73, pp. 84– 88, 2007. [28] H. W. Liu and G. J. Wang, “Multi-criteria methods based on intuitionistic fuzzy sets,” Eur. J. Oper. Res., vol. 179, pp. 220–233, 2007. [29] P. Zhi and Z. Li, “A novel approach to multi-attribute decision making based on intuitionistic fuzzy sets,” Expert Syst. Appl., vol. 39, pp. 2560– 2566, 2012. [30] Y. T. Chen, “A outcome-oriented approach to multicriteria decision analysis with intuitionistic fuzzy optimistic/pessimistic operators,” Expert Syst. Appl., vol. 37, pp. 7762–7774, 2010. [31] S. Z. Zeng and W. H. Su, “Intuitionistic fuzzy ordered weighted distance operator,” Knowledge-Based Syst., vol. 24, pp. 1224–1232, 2011. [32] Z. S. Xu, “Choquet integrals of weighted intuitionistic fuzzy information,” Inf. Sci., vol. 180, pp. 726–736, 2010. [33] Z. S. Xu, “Intuitionistic fuzzy multiattribute decision making: An interactive method,” IEEE Trans. Fuzzy Syst., vol. 20, no. 3, pp. 514–525, Jun. 2012. [34] H. K. Soung and H. H. Chang, “An interactive procedure for multi-attribute group decision making with incomplete information,” Comput. Oper. Res., vol. 26, pp. 755–772, 1999. [35] H. K. Soung, H. C. Sang, and K. K. Jae, “An interactive procedure for multi-attribute group decision making with incomplete information: Range-based approach,” Eur. J. Oper. Res., vol. 118, pp. 139–152, 1999. [36] Z. S. Xu, “Intuitionistic fuzzy aggregation operators,” IEEE Trans. Fuzzy Syst., vol. 15, no. 6, pp. 1179–1187, Dec. 2007. [37] Z. S. Xu and R. R. Yager, “Some geometric aggregation operators based on intuitionistic fuzzy sets,” Int. J. Gen. Syst., vol. 35, pp. 417–433, 2006. [38] J. B. Yang, “Rule and utility based evidential reasoning approach for multi-attribute decision analysis under uncertainty,” Eur. J. Oper. Res., vol. 131, pp. 31–61, 2001. [39] Y. Shi and R. C. Eberhart, “Empirical study of particle swarms optimization,” in Proc. Congr. Evol. Comput., Piscataway, NJ, 1999, pp. 1945– 1949. [40] D. F. Li, “Multiattribute decision making models and methods using intuitionistic fuzzy sets,” J. Comput. Syst. Sci., vol. 70, pp. 73–85, 2005. Jian-Qiang Wang received the Ph.D. degree in management science and engineering from Central South University, Changsha, China, in 2005. He is currently a Professor with the Business School, Central South University. His current research interests include decision-making theory and application, risk management and control, and information management.
Hong-Yu Zhang received the M.S. degree in computer software and theory from the School of Information Science and Engineering, Central South University, Changsha, China, in 2005 and the Ph.D. degree in management science and engineering from the Business School, Central South University, Changsha, China, in 2009. She is currently a Lecturer with the Business School, Central South University. Her research interests include the area of information management and its applications in production operations. Her current research focuses on remanufacturing production management and decisionmaking theory.
