Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making

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Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making

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Xiaolu Zhang a , Zeshui Xu b,∗ a b

School of Economics and Management, Southeast University, Nanjing 211189, China Business School, Sichuan University, Chengdu, Sichuan 610065, China

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a r t i c l e

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a b s t r a c t

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Article history: Received 25 May 2013 Received in revised form 7 July 2014 Accepted 30 August 2014 Available online xxx

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Keywords: Multi-attribute group decision making Interval-valued intuitionistic fuzzy number Consensus Experts’ weights Fuzzy TOPSIS Multi-choice goal programming

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1. Introduction

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Multi-attribute group decision making (MAGDM) is an important research topic in decision theory. In recent decades, many useful methods have been proposed to solve various MAGDM problems, but very few methods simultaneously take them into account from the perspectives of both the ranking and the magnitude of decision data, especially for the interval-valued intuitionistic fuzzy decision data. The purpose of this paper is to develop a soft computing technique based on maximizing consensus and fuzzy TOPSIS in order to solve interval-valued intuitionistic fuzzy MAGDM problems from such two aspects of decision data. To this end, we first define a consensus index from the perspective of the ranking of decision data, for measuring the degree of consensus between the individual and the group. Then, we establish an optimal model based on maximizing consensus to determine the weights of experts. Following the idea of TOPSIS, we calculate the closeness indices of the alternatives from the perspective of the magnitude of decision data. To identify the optimal alternatives and determine their optimum quantities, we further construct a multi-choice goal programming model based on the derived closeness indices. Finally, an example is given to verify the developed method and to make a comparative analysis. © 2014 Published by Elsevier B.V.

Multi-attribute decision making (MADM), which addresses the problem of choosing an optimum choice that has the highest degree of satisfaction from a set of alternatives that are characterized in terms of multiple attributes, is an important research topic in decision theory. Over the past few decades, a larger number of MADM methods have been developed for solving the real-life problems in the fields of management science, operational research and industrial engineering [1–4], etc. However, increasing complexity of the socio-economic environment makes it less and less possible for a single expert to consider all relevant aspects of a problem [5]. As a result, many real-life decision making processes usually take place in group settings. Multi-attribute group decision making (MAGDM) can be viewed as the decision making situations where a group of experts provide their assessments on multiple attributes of a problem to be solved and try to find a common solution. Generally, the process of MAGDM consists of four phases: setting of the values of attributes, determination of weights of the experts and the attributes, the aggregation procedure and the overall ranking. The main goal of this paper is to focus on the MAGDM problem limited to the following three phases: setting of the values of attributes, determination of the experts’ weights and the overall ranking of alternatives. The decision data (i.e., the attribute values) in the MAGDM problems, commonly given by the form of decision matrix, can be roughly classified into two styles. The first is certain decision data that includes integers, ordinals, utility values and so on. The second is uncertain decision data that mainly includes fuzzy decision data (such as interval numbers, triangular fuzzy numbers, intuitionistic fuzzy numbers (IFNs), interval-valued IFNs (IVIFNs) and linguistic variables) and stochastic decision data. With the increasing complexity of the socioeconomic environment, it is more and more difficult for experts in the MAGDM problems to provide the certain assessment information of the alternatives with each attribute. Instead, these experts often employ the IVIFNs [6] to express their quantitative preferences or linguistic

∗ Corresponding author. E-mail addresses: xiaolu [email protected] (X. Zhang), [email protected] (Z. Xu). http://dx.doi.org/10.1016/j.asoc.2014.08.073 1568-4946/© 2014 Published by Elsevier B.V.

Please cite this article in press as: X. Zhang, Z. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.08.073

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variables [7] to represent their qualitative preferences. Interval-valued intuitionistic fuzzy sets (IVIFSs) developed by Atanassov and Gargov [8], which can consider not only the membership but also the non-membership of an element to a given set, and simultaneously allow these membership and non-membership functions to be a ranger respectively, are very suitable for the depiction of the uncertainty and vagueness of decision making problems. In the real-life decision process, people usually use the IVIFNs instead of IVIFSs to express their assessments. IVIFNs have been broadly applied in real-life MAGDM problems, and studies of both methods and applications of MAGDM problems with IVIFNs have received extensive attentions [6,9–14]. Consequently, in this study we discuss the MAGDM problem in which the values of attributes are represented by IVIFNs. On the other hand, because the experts in practical MAGDM problems usually come from various research areas and may have many differences in knowledge structure, express abilities, evaluation levels, individual preference as well as practical experience, they have a variety of views for the same problem and different perceptions for the importance degrees of various factors therein. Thus in the process of MAGDM, each expert should be assigned a weight that reflects the corresponding importance in the group. Pérez et al. [15] proposed a consensus model, which takes into account the experts’ weights not only to aggregate the experts’ preferences but also when advising experts to change their preferences, to solve the group decision making problems. How to assign objectively the experts’ weights becomes a critical step of the MAGDM process. Many useful and valuable methods [16–21] have been proposed to determine objectively the weights of experts over the last decades. For example, Bodily [16] derived the experts’ weights as a result of designation of voting weights from an expert to a delegation subcommittee made up of other experts of the group. Brock [17] used a Nash bargaining-based method to estimate the experts’ weights intrinsically. Xu [18] utilized some deviation measures between additive linguistic preference relations to give some straightforward formulas for determining the experts’ weights. Recently, Yue [9,19,20] proposed a series of methods to derive the experts’ weights for the MAGDM problems under real number environments [19], interval fuzzy environments [20] and IVIFN environments [9], respectively. Xu and Cai [21] presented a novel method based on minimizing group discordance to determine the experts’ weights in the MAGDM problems. However, it is noted that all the aforementioned methods to determine the expert’s weights only consider the decision information from the perspective of the sizes of values, but fail to consider the decision information from the perspective of the ranking. That is to say, in the process of MAGDM, the ranking information provided by experts is not used, which may affect the final results of the MAGDM problems. Therefore, one of the main goals of this paper is to develop a novel approach to derive the experts’ weights under IVIFNs contexts from the perspective of the ranking of decision information. After completing these two phases, namely, setting of the values of attributes and determination of the experts’ weights; the most important phase in the MAGDM process is how to take advantage of decision information to sort the alternatives and to select the best one(s). Many classical ranking methods, such as the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method [22], the VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) method [23], the PROMETHEE (Preference Ranking Organization METHod for Enrichment Evaluations) method [24] and the ELECTRE (ELimination Et Choix Traduisant la REalité) method [25,26] have been proposed. Opricovic and Tzeng [27,28] presented a detailed comparison analysis among the VIKOR method, the TOPSIS method, the PROMETHEE method and the ELECTRE method. They pointed out that the TOPSIS method is based on the principle that the optimal point should have the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS), which is suitable for cautious (risk avoiding) experts; while the VIKOR method is based on an aggregating function representing “closeness to the ideal”, using linear normalization; and ranking by the PROMETHEE method, with a linear preference function, gives the same results as ranking by the VIKOR method, with measure S representing “group utility”; and the results by the ELECTRE method, with linear “surrogate” attribute functions, are relatively similar to the results by the VIKOR method [28]. In this study, we will extend the TOPSIS method to rank the alternatives because TOPSIS has a simple computation process, systematic procedure and a sound logic that represents the rationale of human choice. Additionally, for some situations where the manager of the MAGDM problem wants to not only identify the optimal alternatives but also determine their optimum quantities, the multi-choice goal programming (MCGP) model is further integrated into TOPSIS for handling this issue, which is another main goal of this paper. Based on the aforementioned analysis, this study will develop a soft computing technique based on maximizing consensus and fuzzy TOPSIS to solve interval-valued intuitionistic fuzzy MAGDM problems from the perspective of the ranking and the magnitude of decision data. To do so, the reminder of this paper is organized as follows: In Section 2, we review some basic concepts related to IFSs and IVIFSs; and introduce the MCGP model as well as the description of MAGDM problems with IVIFNs. In Section 3, we define the consensus index and develop a technique to derive the weights of experts. In Section 4, we propose a novel decision method based on the TOPSIS and the MCGP for solving the MAGDM problems with IVIFNs. Section 5 employs an example to demonstrate the implementation process of the proposed method. Section 6 presents our conclusions.

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2. Preliminaries

42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88

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In what follows, we introduce some basic concepts and terminologies, which will be used in the next sections. For the sake of simplicity, throughout this paper we denote M = {1, 2, . . ., m}, N = {1, 2, . . ., n} and P = {1, 2, . . ., p}.

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2.1. The basic concepts related to IFSs and IVIFSs

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In 1986, Atanassov generalized the fuzzy set [29] to introduce the concept of intuitionistic fuzzy set (IFS) [30]. Its definition is introduced as follows:

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Definition 1.

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Let a set X be a universe of discourse. An IFS is an object having the form:



A = { x, A (x), A (x) |x ∈ X}

(1)

where the function A : X → [0, 1] defines the degree of membership and A : X → [0, 1] defines the degree of non-membership of the element x ∈ X to A, respectively, and for every x ∈ X, it holds that 0 ≤ A (x) + A (x) ≤ 1

(2)

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For any IFS A and x ∈ X, A (x) = 1 − A (x) − A (x) is called the degree of indeterminacy of x to A. For simplicity, Xu [31] denoted ˛ = (˛ , ˛ ) as an intuitionistic fuzzy number (IFN), where ˛ and ˛ are the degree of membership and the degree of non-membership of the element ˛ ∈ X to A, respectively. In human cognitive and decision-making activities, it is not completely justifiable or technically sound to quantify grades of membership and non-membership in terms of a single numeric value [14]. To this end, Atanassov and Gargov [8] further presented the notion of IVIFS, which is characterized by a membership function and a non-membership function whose values are intervals rather than exact numbers.

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Definition 2.

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Let a set X be a universe of discourse. An IVIFS is an object having the form:



˜ = { x,  ˜ (x),  ˜ (x) |x ∈ X} A A A

(3)

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where A˜ (x) = [L˜ (x), R˜ (x)] ⊆ [0, 1] and vA˜ (x) = [vL˜ (x), vR˜ (x)] ⊆ [0, 1] are intervals, L˜ (x) = inf A˜ (x), R˜ (x) = sup A˜ (x), vL˜ (x) =

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inf vA˜ (x), vR˜ (x) = sup vA˜ (x) and R˜ (x) + vR˜ (x) ≤ 1.

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A

A

A

A

A

A

A

A

A

A

˜ is given as: For each element x ∈ X, its uncertain interval relative to A A˜ (x) = [L˜ (x), R˜ (x)] = [1 − R˜ (x) − vR˜ (x), 1 − uL˜ (x) − vL˜ (x)] ⊆ [0, 1] A

A

A

A

A

(4)

A

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˜ is reduced to an IFS. Analogously, Xu [6] called the Especially, for every x ∈ X, if A˜ (x) = L˜ (x) = R˜ (x) and vA˜ (x) = vL˜ (x) = vR˜ (x), then A A A A A ˜ = ([a, b], [c, d]), where [a, b] ⊆ [0, 1], [c, ˛ ˜ = (˛˜ , ˛˜ ) an interval-valued intuitionistic fuzzy number (IVIFN) and denoted the IVIFN by ˛ d] ⊆ [0, 1] and b + d ≤ 1. The basic operational laws of IVIFNs defined by Xu [6] are introduced as follows:

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Definition 3.

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Let ˛ ˜ = ([a, b], [c, d]), ˛ ˜ 1 = ([a1 , b1 ], [c1 , d1 ]) and ˛ ˜ 2 = ([a2 , b2 ], [c2 , d2 ]) be three IVIFNs, and  > 0, then

˜1 + ˛ ˜ 2 = ([a1 + a2 − a1 a2 , b1 + b2 − b1 b2 ], [c1 c2 , d1 d2 ]); (1) ˛  (2) ˛ ˜ = ([1 − (1 − a) , 1 − (1 − b) ], [c  , d ]);      (3) ˛ ˜ = ([a , b ], [1 − (1 − c) , 1 − (1 − d) ]). In order to compare the magnitudes of two IVIFNs, Xu [6] introduced the score and accuracy functions for IVIFNs and gave a simple comparison law as follows: Definition 4. as follows:

Let ˛ ˜ = ([a, b], [c, d]) be an IVIFN, the score function s(˛) ˜ and the accuracy function h(˛) ˜ of ˛ ˜ can be defined, respectively,

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s(˛) ˜ =

1 (a − c + b − d) 2

(5)

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h(˛) ˜ =

1 (a + b + c + d) 2

(6)

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Based on the score and accuracy functions, a comparison law for IVIFNs is introduced as below: Definition 5. Let ˛ ˜ 1 = ([a1 , b1 ], [c1 , d1 ]) and ˛ ˜ 2 = ([a2 , b2 ], [c2 , d2 ]) be two IVIFNs, s(˛ ˜ 1 ) and s(˛ ˜ 2 ) be the scores of ˛ ˜ 1 and ˛ ˜ 2 , respectively, h(˛ ˜ 1 ) and h(˛ ˜ 2 ) be the accuracy degrees of ˛ ˜ 1 and ˛ ˜ 2 , respectively, then ˜ 1 ) < s(˛ ˜ 2 ), then  ˛ ˜1 ≺ ˛ ˜ 2; (1) If s(˛ h(˛ ˜ 1 ) < h(˛ ˜ 2) ⇒ ˛ ˜1 ≺ ˛ ˜2 (2) If s(˛ ˜ 1 ) = s(˛ ˜ 2 ), then . h(˛ ˜ 1 ) = h(˛ ˜ 2) ⇒ ˛ ˜ 1 ∼˛ ˜2 The weighted averaging operators for IVIFNs developed by Xu [6] are presented as follows:

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Definition 6. Let ˛ ˜ j = ([aj , bj ], [cj , dj ])(j ∈ N) be a collection of IVIFNs, and w = (w1 , w2 , . . ., wn )T be the weight vector of ˛ ˜ j (j ∈ N), where wj

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indicates the importance degree of ˛ ˜ j , satisfying wj ≥ 0 (j ∈ N) and

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IVIFWA(˛ ˜ 1, ˛ ˜ 2 , ..., ˛ ˜ n) =

n 

⎛⎡

wj ˛ ˜ j = ⎝⎣1 −

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IVIFA(˛ ˜ 1, ˛ ˜ 2 , ..., ˛ ˜ n) =

n 1

n

⎤ ⎡ ⎤⎞ n n   w (1 − bj ) j ⎦ , ⎣ cjwj , djwj ⎦⎠

n 

j=1

j=1

(1 − aj )wj , 1 −

j=1

(7)

j=1

⎛⎡

˛ ˜ j = ⎝⎣1 −

⎤ ⎡ ⎤⎞ n n   1/n 1/n 1/n (1 − bj ) ⎦ , ⎣ c , d ⎦⎠

n 

n 

j=1

j=1

(1 − aj )1/n , 1 −

j

j=1

j

(8)

j=1

Additionally, Xu and Chen [32] also proposed the Hamming and Euclidean distances for IVIFNs: Let ˛ ˜ 1 = ([a1 , b1 ], [c1 , d1 ]) and ˛ ˜ 2 = ([a2 , b2 ], [c2 , d2 ]) be two IVIFNs, then

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Definition 7.

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• The IVIFN Hamming distance:

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wj = 1, and let IVIFWA: n →  if

then the function IVIFWA is called the interval-valued intuitionistic fuzzy weighted averaging (IVIFWA) operator, and if w = (1/n, 1/n,. . ., 1/n)T , then the IVIFWA operator is reduced to the interval-valued intuitionistic fuzzy averaging (IVIFA) operator:

j=1 139

j=1

n 

j=1 136

n

d1 (˛ ˜ 1, ˛ ˜ 2) =

1 (|a1 − a2 | + |b1 − b2 | + |c1 − c2 | + |d1 − d2 |) 4

(9)

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• The IVIFN Euclidean distance:

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d2 (˛ ˜ 1, ˛ ˜ 2) =

1 2 2 ((a1 − a2 )2 + (b1 − b2 ) + (c1 − c2 )2 + (d1 − d2 ) ) 4

(10)

2.2. Multi-choice goal programming The multi-choice goal programming (MCGP) problem was first introduced by Chang [33], which describes a decision making situation existing in real-life where the decision makers/experts would like to make a decision on the problem, with the goal that can be achieved from some specific aspiration levels (i.e., one goal mapping many aspiration levels). This MCGP problem can be expressed as follows:

Min 149

m    fi (x) − gi1 or gi2 or gi3 or. . .or gin 

(M-1)

i=1

s.t. x ∈ F 150 151 152

where F is a feasible set, x is unrestricted in sign, fi (x) is the linear function of the ith goal, gij (i = 1, 2, . . ., m, j = 1, 2, . . ., n) is the jth aspiration level of the ith goal, and gij−1 ≤ gij ≤ gij+1 . To handle effectively this type of problem, Chang [33] presented a MCGP model as follows:

Min

m 

wi (di+ + di− )

⎧i=1 n  ⎪ + − ⎪ f (x) − d + d = gij Sij (B), i ∈ M ⎪ i ⎪ i i ⎪ ⎪ ⎨ j=1

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s.t.

Sij (B) ∈ Ri (x),

(M-2)

i∈M

⎪ ⎪ ⎪ di+ , di− ≥0, i ∈ M ⎪ ⎪ ⎪ ⎩ x∈F

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where di+ and di− are the positive and negative deviations attached to the ith goal; the parameter wi represents the weight attached to the deviation of ith goal; Sij (B) represents a function of binary serial number; Ri (x) is the function of resources limitations; other variables are defined as in Model (M-1). According to the different real-life decision making situations, gij Sij (B) can be usually classified into three cases [33]: (1) something more/higher is better in the aspiration levels, i.e., maximization of gij Sij (B); (2) something less/lower is better in the aspiration levels, i.e., minimization of gij Sij (B); and (3) no direction is provided in the aspiration levels, i.e., gij Sij (B). To reduce the binary variables, Chang [34] further developed a revised MCGP method which does not involve multiplicative terms of binary variables to model the multiple aspiration levels as below:

Min

m  



˛i (di+ + di− ) + ˇi (ei+ + ei− )

⎧i=1 f i (x) − di+ + di− = yi , i ∈ M ⎪ ⎪ ⎪ ⎪ ⎪ y − e+ + ei− = gi,max or gi,min , i ∈ M ⎪ ⎨ i i

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s.t.

gi,min ≤ yi ≤ gi,max ,

(M-3)

i∈M

⎪ ⎪ ⎪ ⎪ d+ , d− , e+ , e− ≥0, i ∈ M ⎪ ⎪ ⎩ i i i i x∈F

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where ˛i is the weight attached to the deviations di+ and di− ; di+ and di− are the positive and negative deviations attached to the ith goal |fi (x) − yi |; yi is the continuous variable with a range of interval values gi,min ≤ yi ≤ gi,max ; gi,max and gi,min are the upper and lower bounds of yi ; ˇi is the weight attached to deviations ei+ and ei− ; ei+ and ei− are the positive and negative deviations attached to |yi − gi,max | or |yi − gi,min |; other variables are defined as in Model (M-1).

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2.3. Description of the MAGDM problems with IVIFNs

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For a given MAGDM problem under interval-valued intuitionistic fuzzy environment, let A = {A1 , A2 , . . ., Am } (m ≥ 2) be a discrete set of T m feasible alternatives, n X = {x1 , x2 , . . ., xn } be a finite set of attributes, and w = (w1 , w2 , . . ., wn ) be the weight vector of attributes, which satisfies 0 ≤ wj ≤ 1, w = 1. Let E = {e1 , e2 , . . ., ep } be a group of experts/decision makers, and  = {1 , 2 , . . ., p } be the weight vector j=1 j

p

= 1. Let Rk = (˛ ˜ kij )

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of experts, where 0 ≤ k ≤ 1 and

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expert ek (k ∈ P). Therefore, the MAGDM problem with IVIFNs can be represented as the following matrix form:

 k=1 k

m×n

be an interval-valued intuitionistic fuzzy decision matrix provided by the

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where each of elements ˛ ˜ kij = ([akij , bkij ], [cijk , dijk ]) is an IVIFN representing the performance rating of the alternative Ai with respect to the attribute xj provided by the expert ek , which means that the degree to which the alternative Ai satisfies the attribute xj is the interval value [akij , bkij ] and the degree to which the alternative Ai dissatisfies the attribute xj is the interval value [cijk , dijk ]. In the real-life MAGDM process, the weights of the experts and the attributes are usually completely unknown or partially known. In this study, without loss of generality we suppose that the information about attribute weights is completely known in advance, and the information involving the experts’ weights which needs to be determined is completely unknown or partially known. If the weights of attributes are completely unknown in advance; the AHP (Analytical Hierarchical Process) method is often used to objectively determine their weights [41]. The structure forms of incomplete weight information of experts can be roughly divided into the following five basic forms [5,14]: (1) a weak ranking:  1 = {i ≥ j }; (2) a strict ranking:  2 = {i − j ≥ ˇi } (ˇi > 0); (3) a ranking of differences: 3 = {wi − wj ≥wk − wl }(i = / j= / k= / l); (4) a ranking with multiples: 4 = {wi ≥ˇi wj } (0 ≤ ˇi ≤ 1); (5) an interval form: 5 = {ˇi ≤ wi ≤ ˇi + εi }(0 ≤ ˇi ≤ ˇi + εi ≤ 1). The structure forms of the weights of the experts usually consist of several sets of the above basic sets or may contain all the five basic sets, which depend on the characteristic and need of the real-life decision problems. Let  denote a set of the known weights of experts and  =  1 ∪  2 ∪  3 ∪  4 ∪  5 . In what follows, we develop a novel approach, an integrated idea of maximizing consensus and fuzzy TOPSIS, for solving such MAGDM problems with IVIFNs.

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3. Maximizing consensus method for deriving the experts’ weights in the MAGDM problems with IVIFNs

174 175 176 177 178 179 180 181 182 183 184 185 186 187

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In this section, the concept of consensus index is first proposed. Then, an optimal model based on maximizing consensus is constructed to determine the experts’ weights.

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3.1. Consensus index

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In the real-life MAGDM process, the experts usually come from various research areas and may have many differences in knowledge structures, express abilities, evaluation levels, individual preferences as well as practical experience, thus they have a variety of views for the same MAGDM problem and may provide different assessments. In other words, there usually exists the inconsistency among the individual experts’ opinions. In such cases, if we aggregate the different experts’ opinions directly, it may have an unreasonable impact on the final result. Consequently, many different consensus approaches have been developed to improve the consensus degree between the individual expert’s opinion and the group opinion [35–37]. For example, Cabrerizo et al. [35] developed a method based on an allocation of information granularity as an important asset to increase the consensus achieved within the group of experts in group decision making situations. Cabrerizo et al. [36] analyzed the different consensus approaches in fuzzy group decision making problems and discussed their advantages and drawbacks. Herrera-Viedma et al. [37] analyzed comprehensively consensus approaches based on soft consensus measures in which the consensus reaching process is guided by a moderator. However, the existing consensus approaches only consider the consensus degree from the perspective of the magnitude of decision data, but fail to consider the consensus situations among the individual experts’ opinions in ranking. Borrowing ideas from consensus of the individual overall preference values with respect to the collective overall preference values in linguistic context [38], thus we define the concept of consensus index from the perspective of the ranking of decision data. For the aforementioned MAGDM problem, we say that ∀Ag , Af ∈ A(1 ≤ g, f ≤ m, g = / f), the alternative Af for the expert ek and the attribute xj dominates the alternative Ag if ˛ ˜ kfj ≥˛ ˜ kgj , denoted by Af (Rxekj )≥ Ag . The dominance relation (Rxekj )≥ can be expressed as:

 209

210

≥

=



[Af ]exk

≥

j

212 213

214

Ag , Af





∈ A × A ˛ ˜ kfj ≥˛ ˜ kgj



(11)

Then, the dominance class of the alternative Af with respect to the attribute xj under the expert ek can be defined as follows:

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Rxekj

=





Ag ∈ A|˛ ˜ kfj ≥˛ ˜ kgj |

(12)

Thus, the family set of the dominance classes under the expert ek for the alternative set A with respect to the attribute xj can be obtained as below:



A e

Rxkj

≥ =



[A2 ]xekj

≥  ,

[A2 ]exkj

≥

 , . . .,

[Am ]exkj

≥ 

(13)

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In the following, we introduce the concept of consensus index in order to measure the degree of consensus between the experts ek and el for all alternatives with respect to the attribute xj , which is defined as:

217

(14)

215

218 219 220

   e ≥ e ≥ m ([A ] k ) ∩ ([A ] l )   i i x x j j 1   , k, l ∈ P, j ∈ N Ckl (xj ) =  ek ≥ el ≥  m ([A ] ) ∪ ([A ] )   i xj i xj i=1

where m is the total number of the alternatives. Eq. (14) expresses the consensus degree between two experts for all alternatives with respect to a single attribute from the perspective of the ranking of decision data. It is easy to prove that 1/m ≤ Ckl (xj ) ≤ 1. The closer the Ckl (xj ) approaches to 1/m, the poorer the consensus.

225

Conversely, the closer the Ckl (xj ) approaches to 1, the better the consensus. If Ckl (xj ) = 1, it means that the opinion of the expert ek is completely consistent with the opinion of the expert el . Moreover, it is easily seen that the above definition of consensus index is simple and reasonable because it avoids the use of distance or similarity functions to measure the consensus index in group decision making and thus reduces the effect of the application of some different distance functions for measuring consensus as Chiclana et al. [39] pointed out that different distance functions can produce significantly different results.

226

3.2. Computing the optimal weights of experts: the maximizing consensus model

221 222 223 224

227 228 229 230

Clearly, in the practical MAGDM process the opinion of the group should be consistent with the individual expert’s opinion to the greatest extent. Assume that the common will of the group is the ideal decision of group, for convenience of description, which is presumed to be provided by the ideal expert e*. Similar to Ref. [11], it is reasonable to assume that the ideal decision of group is the mean of all individual experts’ decision. Thus the ideal decision matrix denoted by R* can be determined by using the following formula as:

231 232

233

p

where ˛ ˜ ∗ij = 1/p



p dk k=1 ij

1/p

˛ ˜k k=1 ij

and using Eq. (7), we obtain a∗ij = 1 −

p k=1

(1 − akij )

1/p

, b∗ij = 1 −

p k=1

(1 − bkij )

1/p

, cij∗ =



p ck k=1 ij

1/p

, dij∗ =

(i ∈ M, j ∈ N).

235

Then, according to Eq. (14) the consensus index between the expert ek and the ideal expert e* for all alternatives with respect to the attribute xj is defined as:

236

(16)

234

237 238 239

240

   ≥ e ≥ m ([A ] k ) ∩ ([A ]e∗ )   i xj i xj 1   , k ∈ P, j ∈ N GCk∗ (xj ) =  ek ≥ m e∗ ≥  i=1 ([Ai ]xj ) ∪ ([Ai ]xj ) 

where m is the total number of alternatives. Furthermore, the weighted sum of the consensus indices between the experts ek and e* for all alternatives and all attributes can be defined as below GCk∗ =

n 

GCk∗ (xj )wj =

m

j=1 241

242

j=1

Thus, the overall consensus index is defined as:

GC =

p 

k GCk∗ =

k=1 243 244 245

246

247 248

   ≥ e ≥ ([Ai ]xkj ) ∩ ([Ai ]e∗ xj )  , k∈P wj   ek ≥ e∗ ≥  ([A ] ) ∪ ([A ] )   i xj i xj i=1

m n 1 

p n  

k wj GCk∗ (xj ) =

k=1 j=1

   ≥ e ≥ ([Ai ]xkj ) ∩ ([Ai ]e∗ xj )   k wj   ek ≥ e∗ ≥  ([A ] ) ∪ ([A ] )   i i x i=1 xj j

p n m 1 

m k=1 j=1

(17)

(18)

It is clear that the opinion of individual expert ek is more consistent with the opinion of the ideal expert e*, the more important the opinion of the expert ek in the MAGDM process. This is to say, the bigger the consensus index between the expert ek and e* is, the more the weight of the expert ek should be signed. Consequently, the weight of the kth (k ∈ P) expert can be obtained using the following formula: k =

GCk∗

p

GCk∗ k=1

n

=

j=1

GCk∗ (xj )wj

p n k=1

j=1

GCk∗ (xj )wj

(19)

On the other hand, there are some real-life situations that the information about the weights of the experts is not completely unknown but partially known. For these cases, let  be the set of information known of weights, consequently, we have to choose the experts’ weight Please cite this article in press as: X. Zhang, Z. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.08.073

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252

253

254 255 256 257 258 259 260 261 262 263 264

vector  to maximize overall consensus between all the experts and the ideal expert for all the alternatives with all the attributes from the perspective of the ranking of decision information. To this end, we construct a non-linear programming model to determine the experts’ weights as below:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

 

    ek ≥ e∗ ≥  ([Ai ]xj ) ∪ ([Ai ]xj ) 

1     max GC() = k wj  m

p

s.t.  ∈ ,

p

n

m

k=1

j=1

i=1

 k=1 k

= 1, k ≥0,

≥

e

([Ai ]xkj ) ∩ ([Ai ]e∗ xj )

268

269

270

271 272 273

274

In the following, we introduce a novel decision method combining the fuzzy TOPSIS and the MCGP for solving the MAGDM problems with IVIFNs in which the weights of the attributes are completely known but the weights of experts are partially known or completely unknown. The developed method involves the following steps: Step 1. For a MAGDM problem with IVIFNs, we construct the decision matrix Rk = (˛ ˜ kij ) , where all the arguments ˛ ˜ kij = m×n

([akij , bkij ], [cijk , dijk ])(i ∈ M, j ∈ N, k ∈ P) are IVIFNs, given by the expert ek ∈ E, for the alternative Ai ∈ A with respect to the attribute xj ∈ X. ∗

T

Step 2. Determine the optimal weight vector  = (∗1 , ∗2 , . . ., ∗p ) of experts. If the information about the experts’ weights is completely unknown, then we can obtain the expert’s weights by using Eq. (19); if the information about the expert’s weights is partly known, then we employ Model (M-4) to obtain the experts’ weights. Step 3. Utilize the IVIFWA operator to aggregate the decision information. ∗ T In the MAGDM process, after obtaining the optimal weight vector  = (∗1 , ∗2 , . . ., ∗p ) of experts, we usually need to aggregate all individual decision matrices Rk (k ∈ P) into a collective decision R by using the following formula:

where ˛ ˜ ij =

p

 ˛ ˜k k=1 k ij

and by Eq. (7), aij = 1 −

p 

 (1 − akij ) k ,

A+ = A− =

!

277

278 279

280 281

282 283

 (1 − bkij ) k , cij

=

 p k  cijk

k=1 (IVIF-PIS) A+

k=1

and dij =

 p k  dijk

(i ∈ M, j ∈ N).

k=1

" #          j ∈ N = x1 , [a+ , b+ ], [c+ , d+ ] , x2 , [a+ , b+ ], [c+ , d+ ] , . . ., xn , [a+n , b+n ], [cn+ , dn+ ]

(21)

" #          j ∈ N = x1 , [a− , b− ], [c− , d− ] , x2 , [a− , b− ], [c− , d− ] , . . ., xn , [a−n , b−n ], [cn− , dn− ]

(22)

xj , max˛ ˜ ij i

!

xj , min˛ ˜ ij i

1

1

1

1

1

1

1

2

1

2

2

2

2

2

2

2

Step 5. Calculate the separation measures Di+ and Di− of each alternative Ai (i ∈ M) from the IVIF-PIS A+ and the IVIF-NIS A− , respectively. Using Eq. (10), the separation measures Di+ and Di− of each alternative Ai (i ∈ M) from the IVIF-PIS A+ and the IVIF-NIS A− , respectively, are obtained by using the following formula: Di+ =

n 

d(˛ ˜ ij , ˛ ˜+ )wj = j

Di−

=

n 

wj

j=1

n 

d(˛ ˜ ij , ˛ ˜− )wj j

j=1 276

bij = 1 −

p 

and the interval-valued intuitionistic fuzzy NIS (IVIF-NIS) A− . Step 4. Identify the interval-valued intuitionistic fuzzy PIS + − The IVIF-PIS A and the IVIF-NIS A can be defined, respectively, as follows:

j=1

275

(M-4)

4. An approach to MAGDM with interval-valued intuitionistic fuzzy information

k=1

267

≥

k∈P

265

266

7

=

n 

wj

  1 4

  1 4

2

2

2

2

2

2

2

2

(aij − a+ ) + (bij − b+ ) + (cij − cj+ ) + (dij − dj+ ) j j

(aij − a− ) + (bij − b− ) + (cij − cj− ) + (dij − dj− ) j j

 ,

i∈M

(23)

,

i∈M

(24)



j=1

Step 6. Calculate the relative closeness index CIi of the alternative Ai (i ∈ M) to the IVIF-PIS A+ by using the following formula: CIi =

Di−

Di+

+ Di−

(25)

It is easily seen that 0 ≤ CIi ≤ 1(i ∈ M). Step 7. Rank the alternatives and select the best alternatives (and determine their corresponding optimum quantities if needed). (1) For some MAGDM problems, the manager just needs to select the best alternative(s), then we rank the alternatives Ai (i ∈ M) according to the relative closeness index CIi (i ∈ M) and select the most desirable one(s). In general, the bigger the relative closeness index CIi , the better the alternative Ai (i ∈ M). The alternative Ai is closer to the IVIF-PIS A+ and farther from the IVIF-NIS A− as the CIi approaches 1. If CIi = 0, it means that the alternative Ai is farthest from the IVIF-PIS A+ and is Please cite this article in press as: X. Zhang, Z. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.08.073

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Fig. 1. Schematic diagram of the proposed model for MAGDM with IVIFNs.

284 285

closest to the IVIF-NIS A− . Conversely, if CIi = 1, it means that the alternative Ai is closest to the IVIF-PIS A+ and is farthest from the IVIF-NIS A− . Thus, the alternative with the maximal relative closeness index is the best alternative, namely,

 286

287 288 289 290

291 292 293 294

∗

A =

$ Ai :

% i : CIi = max CIl

(26)

1≤l≤m

(2) For some other MAGDM problems, the manager can not only need to select the best alternatives but also need to determine the corresponding optimum quantities for the optimal alternatives under some tangible constraints, then according to the relative closeness indices CIi (i ∈ M) to the IVIF-PIS A+ , we use Model (M-3) to construct the corresponding MCGP model for finding the best alternatives and determining their optimum quantities.

In this study, for the sake of simplicity and without the loss of generality, we take the supplier selection problems for example. To find the optimal suppliers and their optimum order quantities, similar to Ref. [40], the relative closeness indices CIi of the suppliers Ai (i ∈ M) are considered as the supplier weights (or priority values) in an objective function to allocate order quantities among suppliers such that the total value of procurement (TVP) is maximized. Please cite this article in press as: X. Zhang, Z. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.08.073

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Table 1 Interval-valued intuitionistic fuzzy decision matrix R1 .

A1 A2 A3 A4 A5

x1

x2

x3

x4

([0.5,0.6],[0.2,0.3]) ([0.3,0.5],[0.4,0.5]) ([0.6,0.7],[0.2,0.3]) ([0.5,0.7],[0.1,0.2]) ([0.1,0.4],[0.3,0.5])

([0.3,0.4],[0.4,0.6]) ([0.1,0.3],[0.2,0.4]) ([0.3,0.4],[0.4,0.5]) ([0.2,0.4],[0.5,0.6]) ([0.7,0.8],[0.1,0.2])

([0.4,0.5],[0.3,0.5]) ([0.7,0.8],[0.1,0.2]) ([0.5,0.8],[0.1,0.2]) ([0.4,0.6],[0.2,0.3]) ([0.5,0.6],[0.2,0.3])

([0.3,0.5],[0.4,0.5]) ([0.1,0.2],[0.7,0.8]) ([0.1,0.2],[0.5,0.8]) ([0.2,0.3],[0.4,0.6]) ([0.2,0.3],[0.5,0.6])

x1

x2

x3

x4

([0.4,0.5],[0.2,0.4]) ([0.3,0.4],[0.4,0.6]) ([0.6,0.7],[0.1,0.2]) ([0.5,0.6],[0.1,0.3]) ([0.1,0.3],[0.3,0.5])

([0.3,0.5],[0.4,0.5]) ([0.1,0.3],[0.3,0.7]) ([0.3,0.4],[0.4,0.5]) ([0.2,0.3],[0.6,0.7]) ([0.6,0.8],[0.1,0.2])

([0.4,0.6],[0.3,0.4]) ([0.6,0.8],[0.1,0.2]) ([0.7,0.8],[0.1,0.2]) ([0.4,0.6],[0.3,0.4]) ([0.5,0.6],[0.2,0.4])

([0.3,0.4],[0.4,0.6]) ([0.1,0.2],[0.6,0.8]) ([0.1,0.2],[0.7,0.8]) ([0.3,0.4],[0.4,0.6]) ([0.2,0.4],[0.5,0.6])

Table 2 Interval-valued intuitionistic fuzzy decision matrix R2 .

A1 A2 A3 A4 A5

295

The corresponding Model (M-5) can be shown as: Mind1+ + d1− + d2+ + d2− + d3+ + d3− + e1+ + e1− + e2+ + e2−

⎧ m  ⎪ ⎪ CIi xi − d1+ + d1− = y1 (goal constraint, the more the better) ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ y1 − e1+ + e1− = g1,max ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ g ≤ y1 ≤ g1,max ⎪ ⎪ 1,min ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ Pi xi − d2+ + d2− = y2 (budgeting constraint, the less the better) ⎪ ⎪ ⎪ ⎪ ⎨ i=1

296

s.t.

(M-5)

y2 − e2+ + e2− = g2,min

⎪ ⎪ ⎪ g2,min ≤ y2 ≤ g2,max ⎪ ⎪ ⎪ ⎪ ⎪ m ⎪  ⎪ ⎪ ⎪ xi − d3+ + d3− = DC(demand constraint) ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ 0 ≤ xi ≤ SCCi 0, i ∈ M(suppliers capacity constraints) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d1+ , d1− , d2+ , d2− , d3+ , d3− , e1+ , e1− , e2+ , e2− ≥0 ⎪ ⎪ ⎩ x∈F

299

where CIi is the relative closeness index of the ith supplier, and Pi is the sale price of the ith supplier. Then DC represents the total purchase from xi , and SCCi is the capacity of the ith supplier. Other variables are defined as in Model (M-3). The schematic diagram of the proposed method for solving the MAGDM problems with IVIFNs is provided in Fig. 1.

300

5. Illustrative example

297 298

302

In this section, we consider an illustrative example modified from Ref. [40] to demonstrate the implementation process of the proposed method. The comparison analysis of computational results is also conducted to show its superiority.

303

5.1. Description of the problem and the analysis process

301

304 305 306 307 308 309 310 311

312

The company Formosa Watch Co., Ltd. (FWCL) is a large, well-known manufacturer that sells watches in its own chain stores in Asia. To develop new products, its board of directors wishes to select material suppliers to purchase key components in order to achieve the competitive advantage in the market. A decision committee including four experts E = {e1 , e2 , e3 , e4 } has been formed to select a supplier from five qualified suppliers A = {A1 , A2 , A3 , A4 , A5 } according to the following four attributes: (1) x1 is the quality of product; (2) x2 is the relationship closeness; (3) x3 is the delivery capabilities; (4) x4 is the experience time. The weights of the experts and the attributes in practical MAGDM process are usually determined through a direct assignment by the managers (decision makers) or by the AHP approach [41]. Here we assume that the weight vector of the attributes is completely known in advance as w = (w1 , w2 , w3 , w4 )T = (0.3, 0.2, 0.3, 0.2)T . And the information about the experts’ weights is partially known as:  = {2 ≥1.51 , 0.01 ≤ 1 − 3 ≤ 0.1, 0.1 ≤ 3 ≤ 0.3, 4 − 2 ≥1 − 3 ,

4 k=1

k = 1, k ≥0,

k = 1, 2, 3, 4}

Please cite this article in press as: X. Zhang, Z. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.08.073

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10 Table 3 Interval-valued intuitionistic fuzzy decision matrix R3 .

A1 A2 A3 A4 A5

x1

x2

x3

x4

([0.4,0.7],[0.1,0.2]) ([0.3,0.5],[0.3,0.4]) ([0.6,0.7],[0.1,0.2]) ([0.5,0.6],[0.1,0.3]) ([0.3,0.5],[0.4,0.5])

([0.4,0.5],[0.2,0.4]) ([0.2,0.4],[0.4,0.5]) ([0.4,0.5],[0.3,0.4]) ([0.1,0.2],[0.7,0.8]) ([0.6,0.7],[0.2,0.3])

([0.2,0.4],[0.3,0.4]) ([0.6,0.8],[0.1,0.2]) ([0.5,0.7],[0.1,0.3]) ([0.5,0.7],[0.2,0.3]) ([0.6,0.8],[0.1,0.2])

([0.3,0.4],[0.2,0.3]) ([0.1,0.2],[0.6,0.8]) ([0.1,0.3],[0.5,0.7]) ([0.2,0.3],[0.5,0.7]) ([0.1,0.2],[0.6,0.8])

x1

x2

x3

x4

([0.6,0.7],[0.2,0.3]) ([0.3,0.4],[0.3,0.4]) ([0.7,0.8],[0.1,0.2]) ([0.5,0.6],[0.1,0.3]) ([0.1,0.2],[0.5,0.7])

([0.4,0.5],[0.4,0.5]) ([0.1,0.2],[0.2,0.3]) ([0.3,0.4],[0.5,0.6]) ([0.2,0.3],[0.4,0.6]) ([0.6,0.7],[0.1,0.2])

([0.4,0.5],[0.3,0.4]) ([0.6,0.7],[0.1,0.3]) ([0.5,0.8],[0.1,0.2]) ([0.4,0.5],[0.2,0.3]) ([0.5,0.6],[0.3,0.4])

([0.3,0.4],[0.4,0.5]) ([0.1,0.3],[0.6,0.7]) ([0.1,0.2],[0.5,0.8]) ([0.2,0.3],[0.4,0.5]) ([0.3,0.4],[0.5,0.6])

Table 4 Interval-valued intuitionistic fuzzy decision matrix R4 .

A1 A2 A3 A4 A5

Table 5 Interval-valued intuitionistic fuzzy ideal decision matrix R*.

A1 A2 A3 A4 A5

A1 A2 A3 A4 A5

313 314 315 316 317 318 319 320 321 322

323

324 325

330 331

332

x3

x4

([0.3553,0.5051],[0.3000,0.4229]) ([0.6278,0.7787],[0.1000,0.2213]) ([0.5599,0.7787],[0.1000,0.2213]) ([0.4267,0.6064],[0.2213,0.3224]) ([0.4209,0.6636],[0.1861,0.3130])

([0.3000,0.4267],[0.3364,0.4949]) ([0.1000,0.2263],[0.6236,0.7737]) ([0.1000,0.2263],[0.5439,0.7737]) ([0.2263,0.3265],[0.4229,0.5958]) ([0.2031,0.3299],[0.5233,0.6447])

5×4

([A1 ]xe11 )≥ = {A1 , A2 , A5 }, ([A2 ]ex11 )≥ = {A2 , A5 }, ([A3 ]ex11 )≥ = {A1 , A2 , A3 , A5 }, ([A4 ]ex11 )≥ = {A1 , A2 , A3 , A4 , A5 }, ([A5 ]ex11 )≥ = {A5 } and the set of the dominance classes A/(Rxe11 )≥ under the expert e1 for the alternative set A with respect to the attribute x1 can be obtained using Eq. (13) as: e

≥

= {([A1 ]ex11 )≥ , ([A2 ]ex11 )≥ , ([A3 ]ex11 )≥ , ([A4 ]ex11 )≥ , ([A5 ]ex11 )≥ } = {{A1 , A2 , A5 }, {A2 , A5 }, {A1 , A2 , A3 , A5 }, {A1 , A2 , A3 , A4 , A5 }, {A5 }}

Similar logic is used to determine the set of the dominance class A/(Rxe11 )≥ under the ideal expert e* for the alternative set A with respect to the attribute x1 as: A

329

([0.3519,0.4767],[0.3364,0.4949]) ([0.1261,0.3036],[0.2632,0.4527]) ([0.3265,0.4267],[0.3936,0.4949]) ([0.1761,0.2800],[0.5384,0.6701]) ([0.6278,0.7551],[0.1189,0.2213])

(k = 1, 2, 3, 4)): In the following, we employ the proposed method to solve the above selection problem. By Steps 1 and 2, we utilize Model (M-4) to determine the optimal weights of the experts. Before that, we first calculate the ideal decision matrix R* using Eq. (15). The results are listed in Table 5. According to Definitions 4–6, the ranking of the assessment values of the alternative set A with respect to the attribute x1 provided by the expert e1 can be easily derived from Table 1 as below: Using Eq. (12), the dominance classes of the alternative Ai (i = 1, 2, 3, 4, 5) with respect to the attribute x1 under the expert e1 can be calculated as:

(Rx11 ) 327 328

x2

([0.4820,0.6337],[0.1682,0.2913]) ([0.3000,0.4523],[0.3464,0.4681]) ([0.6278,0.7289],[0.1189,0.2213]) ([0.5000,0.6278],[0.1000,0.2711]) ([0.1548,0.3598],[0.3663,0.5439])

The assessments of five possible alternatives Ai (i = 1, 2, 3, 4, 5) with respect to the attributes xj (j = 1, 2, 3, 4) provided by the experts ek (k = 1, 2, 3, 4), are expressed by the IVIFNs ˛ ˜ kij , listed in Tables 1–4 (i.e., the interval-valued intuitionistic fuzzy decision matrices Rk = (˛ ˜ kij )

A 326

x1

(Rxe∗1 )≥

≥

≥

≥

≥

≥

e∗ e∗ e∗ e∗ = {([A1 ]e∗ x1 ) , ([A2 ]x1 ) , ([A3 ]x1 ) , ([A4 ]x1 ) , ([A5 ]x1 ) } = {{A1 , A2 , A5 }, {A2 , A5 }, {A1 , A2 , A3 , A4 , A5 }, {A1 , A2 , A4 , A5 }, {A5 }}

Using Eq. (16), we can obtain the consensus index GC1∗ (x1 ) between the expert e1 and the ideal expert e* for the alternatives with respect to the attribute x1 as:

  ≥  ≥ m ([A ]e1 ) ∩ ([A ]e∗ )     i i x x 1 1 1 1 3 2 4 4 1  = GC1∗ (x1 ) = = 0.92. + + + +  5 5 3 2 5 5 1 e1 ≥ e∗ ≥  i=1 ([Ai ]x1 ) ∪ ([Ai ]x1 ) 

Please cite this article in press as: X. Zhang, Z. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.08.073

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Table 6 GC between the individual expert’s opinions and the ideal expert’s opinion. GCk∗

Attributes

GC1∗ (·) GC2∗ (·) GC3∗ (·) GC4∗ (·)

x1

x2

x3

x4

0.92 1 0.9 0.9

0.783 0.583 1 0.87

1 0.61 0.87 0.77

1 0.76 0.77 0.7

Table 7 Interval-valued intuitionistic fuzzy collective decision matrix R.

A1 A2 A3 A4 A5

A1 A2 A3 A4 A5

333 334 335 336

337 338

339

340

341

342 343 344

349

([0.3519,0.4814],[0.3506,0.4970]) ([0.1199,0.2915],[0.2577,0.4515]) ([0.3202,0.4204],[0.4058,0.5071]) ([0.1819,0.2744],[0.5253,0.6637]) ([0.6224,0.7551],[0.1141,0.2160])

x3

x4

([0.3663,0.5159],[0.3000,0.4183]) ([0.6224,0.7732],[0.1000,0.2268]) ([0.5710,0.7840],[0.1000,0.2160]) ([0.4204,0.5942],[0.2259,0.3270]) ([0.4409,0.6494],[0.1988,0.3310])

([0.3000,0.4215],[0.3506,0.5062]) ([0.1000,0.2324],[0.6188,0.7676]) ([0.1000,0.2200],[0.5531,0.7800]) ([0.2314,0.3316],[0.4173,0.5839]) ([0.2151,0.3464],[0.5176,0.6337])

Then, using model (M-4), we establish the following objective programming model: max GC() = 0.91691 + 0.73992 + 0.8753 + 0.7984 ⎧ ⎪ 2 ≥1.51 ⎪ ⎪ ⎪ ⎨ 0.01 ≤ 1 − 3 ≤ 0.1 0.1 ≤ 3 ≤ 0.3 s.t. 4 − 2 ≥1 − 3 ⎪ ⎪ ⎪ ⎪ ⎩ 1 + 2 + 3 + 4 = 1 k ≥0, k = 1, 2, 3, 4 By solving the above model, we can obtain the optimal weight vector of the experts as below: ∗

 = (∗1 , ∗2 , ∗3 , ∗4 ) = (0.2, 0.3, 0.19, 0.31)T T

By Step 3, we employ Eq. (20) to aggregate all individual decision matrices Rk (k = 1, 2, 3, 4) into a collective decision matrix R. The results are shown in Table 7: By Step 4, we utilize Eqs. (21) and (22) to determine the IVIFPIS A+ and the IVIFNIS A− , which are obtained as follows:

A− =

348

([0.4898,0.6296],[0.1753,0.3208]) ([0.3000,0.4412],[0.3464,0.4724]) ([0.6341,0.7354],[0.1149,0.2169]) ([0.5000,0.6244],[0.1000,0.2766]) ([0.1420,0.3364],[0.3712,0.5550])

GC1∗ = 0.9169, GC2∗ = 0.7399, GC3∗ = 0.875, GC4∗ = 0.798

345

347

x2

Analogously, we can obtain the following calculation results shown in Table 6. Afterwards, we proceed to calculate the weighted sum of consensus indices between the individual expert’s opinion and the ideal expert’s opinion for all attributes from the perspective of the ranking of decision data. Using Eq. (17), the weighted sum of the consensus indices between the expert ek (k = 1, 2, 3, 4) and the ideal expert e* for all attributes can be calculated, respectively, as follows:

A+ =

346

x1

 

 

 

 

 

([0.5000, 0.6244], [0.1000, 0.2766]) , ([0.6224, 0.7551], [0.1141, 0.2160]) ,

([0.1420, 0.3364], [0.3712, 0.5550]) , ([0.1819, 0.2744], [0.5253, 0.6637]) ,

 

T

 

T

([0.6224, 0.7732], [0.1000, 0.2268]) , ([0.3000, 0.4215], [0.3506, 0.5062])

([0.3663, 0.5159], [0.3000, 0.4183]) , ([0.1000, 0.2324], [0.6188, 0.7676])

; .

By Step 5, we utilize Eqs. (23) and (24) to calculate the separation measures Di+ and Di− of each alternative Ai (i = 1, 2, 3, 4, 5) from the IVIF-PIS A+ and the IVIF-NIS A− , respectively. The results obtained are shown as follows: D1+ = 0.1347, D2+ = 0.1825, D3+ = 0.1412, D4+ = 0.1515, D5+ = 0.1535, D1− = 0.1647, D2− = 0.1345, D3− = 0.2155, D4− = 0.1448, D5− = 0.1429.

350 351 352

353

By Step 6, we employ Eq. (25) to compute the relative closeness indices of the alternative Ai (i = 1, 2, 3, 4, 5) with respect to the IVIF-PIS A+ as follows: CI1 = 0.5497,CI2 = 0.4243,CI3 = 0.6041,CI4 = 0.4887,CI5 = 0.4821 By Step 7, we finally rank the alternatives Ai (i = 1, 2, 3, 4, 5). As mentioned previously, there may exist two cases: Please cite this article in press as: X. Zhang, Z. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.08.073

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356 357 358 359 360 361

362 363 364

365 366 367 368

369 370 371

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Case 1. The manager of the company just needs to find the best supplier(s). Obviously, we just need to rank CIi (i = 1, 2, 3, 4, 5) in descending order: CI3  CI1  CI4  CI5  CI2 and obtain the maximum value CI3 . Therefore, the best supplier A3 is chosen. Case 2. The manager of the company cannot only need to find the best suppliers but also to determine their corresponding optimum quantities. According to the sales record in the last 5 years and the sales forecast by FWCL, the CEO and top managers of FWCL have established three goals as: (1) The TVP of at least 1500 units from procurement, and the more the better. (2) The total cost of procurement of less than 83,200 thousand dollars, and the less the better. (3) For seeking differentiation strategy (i.e., quality leadership), maintain the current procurement level of less than 7000 units. The coefficients of variables in model are given by FWCL’s database calculated from the last 5 years record. The unit material cost for the suppliers Ai (i = 1, 2, 3, 4, 5) are $12, $14, $15, $10 and $12.5, respectively, and the capacities of the five candidate suppliers Ai (i = 1, 2, 3, 4, 5) are 2700, 3000, 2600, 3100, and 280,000 units, respectively. The functions and parameters related to the FWCL’s supplier selection problem are listed below: f1 (x) = 0.5497x1 + 0.4243x2 + 0.6041x3 + 0.4887x4 + 0.4821x5 ≥ 1500 and ≤ 3500 (TVP goal, the more the better) f2 (x) = 12x1 + 14x2 + 15x3 + 10x4 + 12.5x5 ≥ 66000 and ≤ 83200 (cost goal, the less the better) f3 (x) = x1 + x2 + x3 + x4 + x5 ≤ 7000 (procurement level goal) Consequently, according to the closeness indices obtained from Step 6, we utilize Model (M-5) to construct the corresponding MCGP model as follows: Min z = d1+ + d1− + d2+ + d2− + d3+ + d3− + e1+ + e1− + e2+ + e2−

⎧ 0.5497x1 + 0.4243x2 + 0.6041x3 + 0.4887x4 + 0.4821x5 − d1+ + d1− = y1 ⎪ ⎪ ⎪ y1 − e1+ + e1− = 3500 ⎪ ⎪ ⎪ ⎪ 1500 ≤ y1 ≤ 3500 ⎪ ⎪ + − ⎪ ⎪ ⎨ 12x1 ++ 14x−2 + 15x3 + 10x4 + 12.5x5 − d2 + d2 = y2

374

s.t.

y2 − e2 + e2 = 66000

66, 000 ≤ y2 ≤ 83, 200 ⎪ ⎪ ⎪ x + x2 + x3 + x4 + x5 − d3+ + d3− ≤ 7000 ⎪ ⎪ ⎪ x1 ≤ 2700, ⎪ x2 ≤ 3000, x3 ≤ 2600, x4 ≤ 3100, x5 ≤ 2800 1 ⎪ ⎪ ⎪ xi ≥0, i = 1, 2, . . ., 5 ⎪ ⎩ + − + − + − + − + − d1 , d1 , d2 , d2 , d3 , d3 , e1 , e1 , e2 , e2 ≥0

375 376 377

378

379

380 381 382 383 384

385

where all the decision variables and the corresponding coefficients are defined as in Model (M-5). The above model can be easily solved using the LINGO 11.0 software. Its optimal solutions, namely, the optimum quantities of the optimal suppliers are calculated as below: ¯ A4 (x4 = 3100), A5 (x5 = 0) and TVP = 3103.871. A1 (x1 = 2700), A2 (x2 = 0), A3 (x3 = 173), 5.2. Comparison analysis of the obtained results In Ref. [11], Yue proposed a method based on distance measures for determining the experts’ weights in interval-valued intuitionistic fuzzy MAGDM under the assumption that the expert’s weights are completely unknown. To compare Yue’s approach [11] with the proposed method, here we utilize it to solve the aforementioned selection problem. Firstly, we calculate the similarity measure between each individual decision ek (k = 1, 2, 3, 4) and the ideal decision e* of the group by using the following formula [11]:

m n

∗

s(ek , e ) =

m n i=1

386

387

389 390 391

j=1

d(˛ ˜ kij , ˛ ˜ ∗ij ) +

d(˛ ˜ kij , ˛ ˜ ∗c ) ij

m n i=1

j=1

d(˛ ˜ kij , ˛ ˜ ∗c ) ij

(27)

where ˛ ˜ ∗c = ([cij∗ , dij∗ ], [a∗ij , b∗ij ]).and then determine the experts’ weights by using the following formula [11]: ij k =

s(ek , e∗ )

p

k=1 388

j=1

i=1

(28)

s(ek , e∗ )

The optimal weights of the experts are obtained as follows: ∗

 = (∗1 , ∗2 , ∗3 , ∗4 ) = (0.2592, 0.2544, 0.2380, 0.2484)T . T

After obtaining the weight vector of experts, we can aggregate all individual decision matrices into the collective decision matrix by using Eq. (20). Furthermore, we can obtain the overall evaluations of the alternatives by aggregating all elements in each line of a collective Please cite this article in press as: X. Zhang, Z. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.08.073

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13

Table 8 Overall evaluations scores and ranking of alternatives by Yue’s method. Alternatives

Overall evaluation

Scores

Ranking

A1 A2 A3 A4 A5

([0.4221,0.5687],[0.2312,0.3639]) ([0.3637,0.5306],[0.2541,0.4111]) ([0.4743,0.6347],[0.1949,0.3341]) ([0.3716,0.5140],[0.2370,0.3999]) ([0.3683,0.5597],[0.2561,0.3982])

0.3057 0.2291 0.5800 0.2487 0.2737

2 5 1 4 3

Table 9 The separation measures, closeness and ranking of alternatives by our method. Alternatives

Si+

Si−

CIi

Ranking

A1 A2 A3 A4 A5

0.1538 0.2031 0.1152 0.1801 0.1766

0.2034 0.2665 0.2897 0.2404 0.2464

0.5694 0.5676 0.7155 0.5717 0.5825

4 5 1 3 2

Fig. 2. The pictorial representation of the rankings of the alternatives obtained by two distinct methods.

392 393 394 395 396

397

398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417

decision by using Eq. (20). According to Definitions 4–6, we can calculate the scores of all alternatives and meanwhile obtain the ranking of the alternatives on the basis of their scores. The overall evaluations, scores and the ranking of alternatives are listed in Table 8. Therefore, the final ranking of the alternatives should be A3  A1  A5  A4  A2 , and the best alternative is the alternative A3 . To make this comparison fair, here we also assume that the information about experts’ weights in the above example is completely unknown and then utilize our approach to solve this problem. Using Eq. (19), it can easily derive the optimal weight vector: ∗

 = (∗1 , ∗2 , ∗3 , ∗4 ) = (0.275, 0.222, 0.263, 0.24)T . T

which is different from that obtained by Yue [11]. The main reason is that the former considers the problem from the perspective of the magnitude of decision information, while the latter considers it from the perspective of the ranking of decision information. Moreover, we can calculate the separation measures and the relative closeness index by using Eqs. (23)–(25), respectively, and further obtain the ranking of alternatives according to the obtained relative closeness indices. The separation measures, the relative closeness index and the ranking of alternatives are organized in Table 9: To provide a better view of the comparison results, we put the results of the rankings of the alternatives obtained by the proposed method and Yue’s method, respectively, into Fig. 2. From Fig. 2, we clearly know that the ranking of the alternatives obtained by Yue’s method is different from that obtained by the proposed method. Although the main reason is that the techniques used to obtain the weights of experts in Ref. [11] and in the proposed approach are different, the different ranking approach used in Ref. [11] and in our approach may be another reason. The former just employs the IVIFWA operator to aggregate decision information and to rank the alternatives according to the score function and the accuracy function, while the latter uses the fuzzy TOPSIS method to rank the alternatives. Additionally, for some MAGDM problems where the manager needs to obtain simultaneously the optimal alternatives and their optimum quantities, the proposed method integrated with the MCGP can better assign the optimum quantities based on multiple aspiration levels provided by experts (i.e., various constraints as shown in the above example); while Yue’s method [11] fails to identify the optimum quantities of the optimal alternatives. Therefore, it is not hard to see that the proposed method, compared with Yue’s method [11], has at least three desirable advantages: (1) it is capable of considering the decision problems from the ranking and the magnitude of decision data two aspects and thus utilizes the decision data more sufficiently; (2) it can not only reduce the influence of unjust arguments on the decision results, but also avoid losing or distorting the original decision information in the process of aggregation; (3) it allows experts to set multiple aspiration levels for the MAGDM problems in order to further identify the optimum quantities of the optimal alternatives. Please cite this article in press as: X. Zhang, Z. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.08.073

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On the other hand, it is worthwhile to note that Park et al. [12] also extended the TOPSIS method to solve the MAGDM problem in an interval-valued intuitionistic fuzzy environment, for convenience, we call it Park’s method. Compared to Park’s method [12], our approach defines the consensus index from the perspective of the ranking of decision information and constructs an optimal model based on maximizing consensus to derive the expert’s weights; while Park’s method [12] just assumes that all the experts have equal importance in the process of MAGDM, which may reduce the reasonability of the final result, and meanwhile Park’s method fails to consider the decision information from the perspective of the ranking, which plays an important role in determining the expert’s weights and will have an important effect on the final decision making results. Moreover, for some MAGDM problems where the manager needs to obtain simultaneously the optimal alternatives and their optimum quantities, Park’s method [12] also fails to identify the optimum quantities of the optimal alternatives. Consequently, it is easy to see that the developed method, compared with Park’s method [12], has the following three desirable advantages: (1) it can determine objectively the weights of the experts, which avoids the subjective randomness of selecting the weights; (2) it takes full advantage of the decision information from the perspectives of both the ranking and the magnitude; (3) it can further identify the optimum quantities of the optimal alternatives.

430

6. Conclusions

418 419 420 421 422 423 424 425 426 427 428

431 432 433 434 435 436 437 438 439 440 441 442 443 444

In the MAGDM process, the experts may have vague knowledge about the degree of an alternative satisfying the attribute and cannot provide their assessments with exact numerical values. It is more suitable to express their assessments by means of IVIFNs instead of numerical values. In this article, we have developed a novel approach, an integrated idea of maximizing consensus and fuzzy TOPSIS approach, for solving the MAGDM problems in which the attribute values take the form of IVIFNs and the weights of attributes are completely known but the weights of the experts are partially known or completely unknown. There are three key issues being addressed in the proposed approach. The first one is to define the consensus index from the perspective of the ranking of decision information and to construct an optimal model based on the maximizing consensus in order to derive the experts’ weights. The second one is to extend the TOPSIS method to identify the optimal alternatives from the perspective of magnitude of decision information. The third one is to integrate the MCGP to determine the optimum quantities of the optimal alternatives. Compared with Yue’s method [11] and Park’s method [12], the prominent characteristic of the developed approach is that it cannot only determine objectively the experts’ weights and reduce the influence of unjust arguments on the final decision results, but also take full advantage of the decision information from the perspectives of both the ranking and the magnitude, and at the same time can also allow experts to set multiple aspiration levels for MAGDM problems in order to determine the optimum quantities. In the future, it is worth continuing this research in several directions:

449

• The developed approach should be extended to other MAGDM problems where the decision information takes the form of linguistic information [42,43], hesitant fuzzy information [44], etc. • A decision support system based on the proposed approach, similar to Refs. [45,46], should be developed to solve the practical problems. • Some other methods, such as social network analysis method [47], should be integrated into the proposed method to deal with more complex decision problem.

450

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Please cite this article in press as: X. Zhang, Z. Xu, Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.08.073