Neural Comput & Applic DOI 10.1007/s00521-017-3254-7
ORIGINAL ARTICLE
Group decision making with interval fuzzy preference relations based on DEA and stochastic simulation Jinpei Liu1 • Qin Xu2 • Huayou Chen3 • Ligang Zhou3,4 • Jiaming Zhu3 Zhifu Tao3
•
Received: 27 December 2016 / Accepted: 15 October 2017 Ó The Natural Computing Applications Forum 2017
Abstract This paper proposes an integrated approach to group decision making with interval fuzzy preference relations using data envelopment analysis (DEA) and stochastic simulation. A novel output-oriented CCR DEA model is proposed to obtain the priority vector for the consistency fuzzy preference relation, in which each of the alternatives is viewed as a decision-making unit. Meanwhile, we design a consistency adjustment algorithm for the inconsistent fuzzy preference relation. Furthermore, we build an optimization model to get the weights of each fuzzy preference relation based on maximizing group consensus. Then, an input-oriented DEA model is introduced to obtain the final priority vector of the alternatives. Finally, a stochastic group & Jinpei Liu
[email protected] Qin Xu
[email protected] Huayou Chen
[email protected] Ligang Zhou
[email protected] Jiaming Zhu
[email protected] Zhifu Tao
[email protected] 1
School of Business, Anhui University, Hefei 230601, Anhui, China
2
MOE Key Laboratory of Intelligence Computing and Signal Processing, Anhui University, Hefei 230601, Anhui, China
3
School of Mathematical Sciences, Anhui University, Hefei 230601, Anhui, China
4
Institute of Manufacturing Development, Nanjing University of Information Science and Technology, Nanjing 210044, China
preference analysis method is developed by analyzing the judgments space, which is carried out by Monte Carlo simulation. A numerical example demonstrates that the proposed method is effective. Keywords Group decision making (GDM) Interval fuzzy preference relation Data envelopment analysis (DEA) Stochastic simulation
1 Introduction Group decision-making (GDM) problems can be defined as decision situations where a group of decision makers or experts try to achieve a common solution to a problem consisting of two or more possible alternatives [9]. In the group decision-making process, each decision maker provides his/her preference information which includes multiplicative preference relation [27], fuzzy preference relation [8], interval multiplicative preference relation [19, 21], interval fuzzy preference relation [2], linguistic preference relation [5], intuitionistic fuzzy preference relation [10], hesitant fuzzy preference relation [31], and interval-valued hesitant fuzzy preference relation [13]. The fuzzy preference relations were widely used to express decision-makers’ preferences on decision alternatives. How to derive priority vector from the fuzzy preference relation has become a hot issue of study. Since then, many prioritization methods with fuzzy preference relation have been developed, such as the eigenvector method [17], the goal programming methods [6], the least deviation method (LDM) [26], the Chi square method (CSM) [16], and fuzzy linear programming method (FLPM) [29]. Wu [18] proposed an integrated method of multi-attribute decision making (MADM) based on data envelopment analysis (DEA).
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Neural Comput & Applic
In group decision-making process, because of the decision-makers’ time pressure and vague knowledge, the decision makers sometimes cannot estimate their preference information over alternatives with exact numerical values, and use interval fuzzy preference relations. To solve this problem, Chen and Zhou [4] developed a group decisionmaking method with interval fuzzy preference relations based on consistency induced generalized continuous ordered weighted averaging (C-IGCOWA) operator. Xu and Cai [25] developed a method based on the uncertain power average operators for group decision making with interval fuzzy preference relations. Gong et al. [7] constructed two multi-objective optimization models in consensus interval preference decision making. These methods all translate the interval preference information into a single representative value or obtain the priority vector based on some programming models. However, each of these methods may lose information which given by decision makers. Recently, Zhu and Xu [30] developed a stochastic preference analysis (SPA) method for single interval preference relation, which can avoid information loss. The SPA method brings in several outcomes including the expected priority vector and confidence degree to aid the decision makers to make better decisions. In fact, an interesting and important issue to be solved is how to develop a stochastic group preference analysis (SGPA) method for group decision making with interval fuzzy preference relations. Up to now, there have been no investigations about this issue. In this paper, we propose a new approach to group decision making with interval fuzzy preference relations using data envelopment analysis (DEA) and stochastic simulation, which can avoid information loss during decision making. A new output-oriented CCR DEA model is proposed to get the priority vector for the consistency fuzzy preference relation, in which each of the alternatives is viewed as a decision-making unit (DMU). Meanwhile, a consistency adjustment algorithm is designed for the inconsistent fuzzy preference relations. Then, we build an optimization model to yield the weights of each fuzzy preference relation based on maximizing group consensus. Furthermore, an input-oriented DEA model is developed to obtain the final priority vector of the alternatives, and a stochastic group preference analysis (SGPA) method is developed by analyzing the judgments space, which is carried out by Monte Carlo simulation. Finally, a numerical example is shown to verify the proposed method. The rest of this paper is organized as follows. Section 2 introduces some basic concepts, such as fuzzy preference relation, interval fuzzy preference relation, and additive consistency. In Sect. 3, a CCR DEA model for the priority vector from fuzzy preference relation is proposed. A consistency adjustment algorithm of fuzzy preference relation
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is developed in Sect. 4. In Sect. 5, an input-oriented DEA model is introduced to obtain the final priority vector of the alternatives, and a stochastic group preference analysis (SGPA) method is developed by analyzing the judgments space, which is carried out by Monte Carlo simulation. A numerical example is illustrated to show the applications of our group decision-making method in Sect. 6. In the last section, we draw our conclusions.
2 Preliminary In this section, we briefly review some fundamental concepts including fuzzy preference relation, additive consistent fuzzy preference relation [15], and interval fuzzy preference relation [24]. Group decision-making problems are characterized by the participation of two or more experts in decision making, in which the set of alternatives to the problem is presented. Unless otherwise mentioned, it is assumed that X ¼ fx1 ; x2 ; . . .; xn g is the set of alternatives, and D ¼ fd1 ; d2 ; . . .; dm g is the set of decision makers or experts. In the decision-making process, each expert provides his/her opinions over alternatives in X ¼ fx1 ; x2 ; . . .; xn g by means of preference relation. Let w ¼ ðw1 ; w2 ; . . .; wn ÞT be the priority vector, where wi reflects the importance degree of the alternative xi . All the wi (i ¼ 1; 2; . . .; n) are no less than zeros and sum to one, i.e., n X wi ¼ 1: wi 0; i ¼ 1; 2; . . .; n; i¼1
Definition 1 [12]. A fuzzy preference relation R on a set of alternatives X is represented by a fuzzy set on the product set X X, which is characterized by the membership function: lR : X X ! ½0; 1: When the number of alternatives is finite, the preference relation R is represented by an n n matrix R ¼ ðrij Þnn , where rij ¼ lR ðxi ; xj Þ, i; j ¼ 1; 2; . . .; n. rij denotes the preference degree of the alternative xi over the alternative xj with rij þ rji ¼ 1; rii ¼ 0:5; 0 rij 1 for all i; j ¼ 1; 2; . . .; n: Definition 2 [15]. A fuzzy preference relation R ¼ ðrij Þnn is additive consistent if rij ¼ rik rjk þ 0:5;
for all i; j; k ¼ 1; 2; . . .; n;
ð1Þ
and such a fuzzy preference relation is given by [11, 22] rij ¼ 0:5ðwi wj þ 1Þ;
for all i; j ¼ 1; 2; . . .; n:
ð2Þ
Neural Comput & Applic
In 2004, Xu [24] developed the interval fuzzy preference relation, which can be defined as follows: Definition 3 [24, 28]. An interval fuzzy preference relation R~ is defined as R~ ¼ ð~ rij Þnn , which satisfies that r~ij ¼ ½rijL ; rijU ; 0 rijL rijU 1; rijL þ rjiU ¼ rijU þ rjiL ¼ 1; for all i; j ¼ 1; 2; . . .; n; where r~ij indicates the interval-valued preference degree of the alternative xi over the alternative xj , rijL and rijU are the lower and upper limits of r~ij , respectively.
3.1 Data envelopment analysis Data envelopment analysis (DEA) is a nonparametric programming technique, which has been successfully employed for assessing the relative performance of a set of decision-making units (DMUs) [3]. Suppose that there are n DMUs, denoted as DMUi (i ¼ 1; 2; . . .; n), to be evaluated. Each DMU has s different inputs xil and m different outputs yik (xil 0, yik 0, l ¼ 1; 2; . . .; s, k ¼ 1; 2; . . .; m). Let Xi ¼ ðxi1 ; xi2 ; . . .; xis ÞT be the input vector and Yi ¼ ðyi1 ; yi2 ; . . .; yim ÞT be the output vector of DMUi . Then, the following output-oriented CCR model can be used to evaluate the efficiency of DMUp (p ¼ 1; 2; . . .; n):
ð3Þ
Denoting l ¼ ðl1 ; l2 ; . . .; ln ÞT , Xi ¼ ðxi1 ; xi2 ; . . .; xis ÞT , 0 1 x11 x12 . . . x1s B x21 x22 . . . x2s C C and Yi ¼ ðyi1 ; yi2 ; . . .; yim ÞT , X ¼ B @... A ... xn1 xn2 . . . xns 0 1 y11 y12 . . . y1m B y21 y22 . . . y2m C C; then the model (3) can be Y¼B @... A ... yn1 yn1 . . . ynm rewritten as max bp 8 T > < Y l bp Y p s:t: X T l Xp > : l 0; bp free
ciency score of DMUp , and DMUp is not efficient if bp [ 1. Meanwhile, the input-oriented DEA model to evaluate alternative DMUp (p ¼ 1; 2; . . .; n) is as follows: min hp 8 T > < Y l Yp s:t: X T l hp Xp > : l 0; hp free
3 Priority vector based on DEA
max bp 8 Pn > < Pi¼1 li yik bp ypk ; k ¼ 1; 2; . . .; m n s:t: i¼1 li xil xpl ; l ¼ 1; 2; . . .; s > : bp free; li 0; i ¼ 1; 2; . . .; n
proportion for which DMUi is used to construct the composite unit, and an efficiency score can be generated for the DMUp by maximizing outputs with limited inputs. According to model (4), the composite unit consumes at most the same inputs as DMUp and produces at least bp times outputs of DMUp with bp 1. The inverse of optimal . objective function value of model (4), 1 bp , is the effi-
ð4Þ
In model (4), an imaginary composite unit is constructed, which outperforms each DMUi , li represents the
ð5Þ
In model (5), an imaginary composite unit is also constructed that outperforms each DMU and an efficiency score can be generated for the DMUp by minimizing inputs with limited outputs. The composite unit produces at least the same outputs as DMUp and invests at most hp times inputs of DMUp with hp 1. The optimal objective function value, hp , is the efficiency score of DMUp in model (5), and DMUp is not efficient if hp \1. 3.2 Deriving priority weight vector using DEA In this section, we propose a CCR DEA model to derive the priority weight vector from the fuzzy preference relation. For a fuzzy preference relation matrix R ¼ ðrij Þnn on the alternative set X ¼ fx1 ; x2 ; . . .; xn g, each alternative may be viewed as a decision-making unit (DMU), each column of R ¼ ðrij Þnn can be viewed as an output. A dummy input that takes the value of 0.5 for all the alternatives is employed. The relationships between inputs, outputs, and the fuzzy preference matrix R ¼ ðrij Þnn are shown in Table 1. We build the following output-oriented CCR model to evaluate the efficiency score of alternative xp (p ¼ 1; 2; . . .; n). max bp 8 Pn li rik bp rpk ; k ¼ 1; 2; . . .; n > < i¼1 Pn s:t: 0:5 i¼1 li 0:5 > : bp free; li 0; i ¼ 1; 2; . . .; n
ð6Þ
Theorem 1 For an additive consistent fuzzy preference matrix R ¼ ðrij Þnn , the optimal solution of model (6) is bp (p ¼ 1; 2; . . .; n) and r : f1; 2; . . .; ng ! f1; 2; . . .; ng is a permutation, such that brð1Þ brð2Þ brðnÞ . Then,
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Neural Comput & Applic Table 1 Inputs and outputs of DEA about R ¼ ðrij Þnn Output 1
Output 2
Output n
Dummy input
DMU1
r11
r12
r1n
0.5
DMU2
r21
r22
r2n
0.5
DMUn
rn1
rn2
rnn
0.5
n n X X 1 wp w1 þ 1 n þ 1 nw1 ¼ : ¼ b w w1 þ 1 wn w1 þ 1 p¼1 p p¼1 n
From Eqs. (9) and (10), we have n nþ1 1X 1 max fbp g w1 ¼ : 1 p n n n p¼1 bp
ð10Þ
ð11Þ
According to Eqs. (8), (9), we also have the priority vector satisfies that wrð1Þ wrð2Þ wrðnÞ , and it can be derived by the following formulas: Pn 1 1 (1) wrð1Þ ¼ nþ1 p¼1 b . n n brð1Þ p
(2)
wrðpÞ ¼ wrð1Þ þ
brð1Þ brðpÞ brðpÞ ,
for all p ¼ 2; . . .; n.
Proof If R ¼ ðrij Þnn is an additive consistent fuzzy preference relation, from Eq. (2), the model (6) can be rewritten as max bp 8 Pn < Pi¼1 li ðwi wk þ 1Þ bp ðwp wk þ 1Þ; k ¼ 1; 2; . . .; n; n l 1; s:t: : i¼1 i bp free; li 0; i ¼ 1; 2; . . .; n:
max fbp g bp
wp ¼ w1 þ
p¼1;2;;n
bp
;
for all p ¼ 2; . . .; n
ð12Þ
According to the Eqs. (11) and (12), the expressions of Theorem 1 are established. In fact, the Theorem 1 provides a new method for deriving the priority weight vector using the proposed DEA model for the additive fuzzy preference relation. Example 1 For an additive consistent fuzzy preference matrix: 0 1 0:5 0:525 0:55 0:525 B 0:475 0:5 0:525 0:5 C C: R¼B @ 0:45 0:475 0:5 0:475 A 0:475 0:5 0:525 0:5
ð7Þ Without loss of generality, it is assumed that w1 w2 wn . Then, we have wi wk þ 1 0 and wp wk þ 1 0. Thus, the objective function bp is maxiPn mal if and only if the constraint i¼1 li 1 reduces to Pn equality, i.e., i¼1 li ¼ 1. Since 0 w1 wk þ 1 w2 wk þ 1 wn wk þ 1 for any k 2 f1; 2; . . .; ng, from the first constraint of model (7), it is obvious that the optimal solution is l1 ¼ l2 ¼ ¼ ln1 ¼ 0 and ln ¼ 1. Then, the first constraint k þ1 of model (7) becomes wwnp w wk þ1 bp ; k ¼ 1; 2; . . .; n. Thus, the optimal objective function value of model (7) follows that wn w1 þ 1 wn w2 þ 1 1 bp ¼ min ; ; . . .; wp w1 þ 1 wp w2 þ 1 wp wn þ 1 wn w1 þ 1 : ¼ wp w1 þ 1 ð8Þ Since w1 w2 wn , we have b1 b2 bn . Therefore, max fbp g ¼ b1 ¼ wn w1 þ 1:
1pn
ð9Þ w w þ1
The efficiency score of alternative xp is b1 ¼ wpn w11 þ1. p Then, the sum of efficiency scores of all alternatives is
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According to model (5), we can evaluate the efficiency score of alternative x2 from the following linear programming. max bp
8 0:5l1 þ 0:475l2 þ 0:45l3 þ 0:475l4 0:475bp > > > > > 0:525l1 þ 0:5l2 þ 0:475l3 þ 0:5l4 0:5bp > > > < 0:55l1 þ 0:525l2 þ 0:5l3 þ 0:525l4 0:525bp s:t: > 0:525l1 þ 0:5l2 þ 0:475l3 þ 0:5l4 0:5bp > > Pn > > > > i¼1 li 1 > : bp free; l1 ; l2 ; l3 ; l4 0
The optimal solution of above linear programming is l1 ¼ 1,b2 ¼ 1:0476, l2 ¼ l3 ¼ l4 ¼ 0. The efficiency score of alternative x2 is 1 b2 ¼ 0:9545. Similar, alternatives x1 , x2 , and x3 can be evaluated, and b1 ¼ 1, b3 ¼ 1:1, b4 ¼ 1:0476. So, we have b1 \b2 ¼ b4 \b3 , rð1Þ ¼ 3; rð2Þ ¼ 2; rð3Þ ¼ 4 and rð4Þ ¼ 1. According to Theorem 1, we have w3 w2 w4 w1 , and the priority vector of alternative set X ¼ fx1 ; x2 ; x3 ; x4 g is that T w ¼ ð0:30; 0:25; 0:20; 0:25Þ .
Neural Comput & Applic
4 The consistency adjustment of fuzzy preference relation In fact, Theorem 1 cannot work for inconsistent fuzzy preference relation. So it is necessary for us to design a consistency adjustment algorithm. According to the Definition 2, the additive consistency index of additive fuzzy preference can be given as follows: Definition 4 Assume that R ¼ ðrij Þnn is a fuzzy preference relation given by an expert, we define the additive consistency index of R as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u n X n n X X u 2 1 2 CIðRÞ ¼ t ðrik rjk rij þ 0:5Þ ; nðn 1Þ i¼1 j¼iþ1 n k¼1 ð13Þ where 0 CIðRÞ 1. The smaller the value of CIðRÞ, the stronger the degree of additive consistency of the fuzzy preference relation R ¼ ðrij Þnn is. It is obvious that CIðRÞ ¼ 0 if and only if R is an additive consistent fuzzy preference relation. Definition 5 Let R ¼ ðrij Þnn be a fuzzy preference relation given by an expert. Given a threshold value CI, if the additive consistency index of R satisfies CIðRÞ CI:
Then, the fuzzy preference relation R is of acceptable consistency. The value of CI can be determined according to the decision-makers’ preferences and the practical situations [1, 20]. In this paper, it is supposed that CI¼0:1. For a given fuzzy preference relation R, the following theorem provides a method to obtain a corresponding additive consistent fuzzy preference relation. Theorem 2 For a fuzzy preference relation R ¼ ðrij Þnn , let R ¼ ð rij Þnn , where Q Q 1=n n 1=n 0:5 nl¼1 rjl l¼1 ðril Þ þ 0:5; rij ¼ Pn Q n ð14Þ 1=n h¼1 l¼1 ðrhl Þ for all i; j ¼ 1; 2; . . .; n:
Then, R is an additive consistent fuzzy preference relation.
Proof
Since 0 rij 1, for all i; j ¼ 1; 2; . . .; n, we have
Q Qn 1=n n 1=n Qn 1=n 0:5 ð r Þ il l¼1 l¼1 rjl 0:5 l¼1 rjl 0:5 Pn Q Pn Q n n 1=n 1=n h¼1 l¼1 ðrhl Þ h¼1 l¼1 ðrhl Þ Q 0:5 n ðril Þ1=n þ 0:5 Pn Ql¼1 þ 0:5: n 1=n h¼1 l¼1 ðrhl Þ
Thus, we can get Q Q 1=n n 1=n 0:5 nl¼1 rjl l¼1 ðril Þ 0 rij ¼ þ 0:5 1 Pn Q n 1=n h¼1 l¼1 ðrhl Þ for all i; j ¼ 1; 2; . . .; n: Furthermore, for any i; j; k ¼ 1; 2; . . .; n, it follows that 1 Q ðril Þ1=n nl¼1 ðrkl Þ1=n þ 0:5A Pn Qn 1=n h¼1 l¼1 ðrhl Þ 0 Qn 1=n Qn 1 0:5 l¼1 ðrkl Þ1=n l¼1 rjl @ þ 0:5A þ 0:5 P n Qn 1=n h¼1 l¼1 ðrhl Þ Q Q Q 1=n Qn n 1=n 0:5 nl¼1 ðrkl Þ1=n nl¼1 rjl þ l¼1 ðrkl Þ1=n l¼1 ðril Þ ¼ P n Qn 1=n h¼1 l¼1 ðrhl Þ 0
rik rjk þ 0:5 ¼ @
0:5
Q n
l¼1
þ 0:5 Q Q 1=n n 1=n 0:5 nl¼1 rjl l¼1 ðril Þ þ 0:5 ¼ rij : ¼ Pn Qn 1=n h¼1 l¼1 ðrhl Þ
From the Definition 2, R is an additive consistent fuzzy preference relation. If the given fuzzy preference relation does not satisfy the acceptable consistency, from the Theorem 2, a consistency adjusting algorithm is proposed as follows: Algorithm 1 Input The original individual fuzzy preference relation R ¼ ðrij Þnn , the controlling parameter h 2 ð0; 1Þ, and the threshold CI.Output The adjusted fuzzy ðCÞ
preference relation RðCÞ ¼ ðrij Þnn and the consistency index CIðRðCÞ Þ. ð0Þ
Step 1 Let Rð0Þ ¼ ðrij Þnn ¼ ðrij Þnn , and c ¼ 0. Step 2 Calculate the consistency index CIðRðcÞ Þ, based on Eq. (13). Step 3 If CIðRðcÞ Þ CI, then go to step 6; otherwise, go to the next step. ðcÞ Step 4 Let RðcÞ ¼ ð rij Þnn , where
Qn ðcÞ 1=n Qn ðcÞ 1=n 0:5 l¼1 rjl l¼1 ril ðcÞ þ 0:5: rij ¼ Pn Qn ðcÞ 1=n r h¼1 l¼1 hl
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Neural Comput & Applic
Step 5 Apply the following strategy to construct a new ðcþ1Þ
fuzzy preference relation Rðcþ1Þ ¼ ðrij ðcþ1Þ
rij
ðcÞ
Þnn .
ðcÞ
¼ hrij þ ð1 hÞ rij ;
ð15Þ
where h 2 ð0; 1Þ.Let c ¼ c þ 1, and return to Step 2. Step 6 End.
Thus, it is obvious that CIðRðcÞ Þ¼hc CIðRð0Þ Þ. Moreover, from h 2 ð0; 1Þ, we can obtain that limc!1 CIðRðcÞ Þ ¼ 0. According to Theorem 3, it is obvious that any fuzzy preference relation with unacceptable consistency can be transformed into an acceptable consistent fuzzy preference relation.
From Algorithm 1, we have the following theorem. Theorem 3 Algorithm 1 is convergent. That is, CIðRðcþ1Þ Þ CIðRðcÞ Þ and lim CIðRðcÞ Þ ¼ 0. c!1
Proof ðcþ1Þ
rij
From Step 5 of Algorithm 1 and Eq. (15), we have ðcÞ
ðcÞ
ðcÞ
¼ hrij þ ð1 hÞ rij ¼ hrij þ ð1 hÞ 0 Q 1=n Q 1=n 1 ðcÞ ðcÞ n n 0:5 r r l¼1 l¼1 il jl B C B þ 0:5C @ A 1=n Pn Q n ðcÞ h¼1 l¼1 rhl
From Eq. (13), we obtain CIðRðcþ1Þ Þ ¼ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u n X n n X u 2 1X ðcþ1Þ ðcþ1Þ ðcþ1Þ 2 t ðr rjk rij þ 0:5Þ : nðn 1Þ i¼1 j¼iþ1 n k¼1 ik
ð16Þ where ðcþ1Þ
rik
ðcþ1Þ
ðcþ1Þ
ðcÞ
rjk rij þ 0:5 ¼ hrik þ ð1 hÞ
1 Qn ðcÞ 1=n Qn ðcÞ 1=n 0:5 r r l¼1 l¼1 il kl C B ðcÞ B þ 0:5C A hrjk @ Pn Qn ðcÞ 1=n r h¼1 l¼1 hl 1 0 Q 1=n Q 1=n ðcÞ ðcÞ n 0:5 nl¼1 rkl l¼1 rjl C B ð1 hÞB þ 0:5C A @ Pn Qn ðcÞ 1=n h¼1 l¼1 rhl 1 0 Q 1=n Q 1=n ðcÞ ðcÞ n 0:5 nl¼1 rjl l¼1 ril C B ðcÞ hrij ð1 hÞB þ 0:5C A @ Pn Qn ðcÞ 1=n h¼1 l¼1 rhl ðcÞ ðcÞ ðcÞ þ 0:5 ¼ h rik rjk rij þ 0:5 0
Therefore, CIðRðcþ1Þ Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u n X n n X X u 2 1 ðcþ1Þ ðcþ1Þ ðcþ1Þ 2 ¼t ðr rjk rij þ 0:5Þ nðn 1Þ i¼1 j¼iþ1 n k¼1 ik vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u n X n n X X u 2 1 ðcÞ ðcÞ ðcÞ 2 ðr rjk rij þ 0:5Þ ¼ ht nðn 1Þ i¼1 j¼iþ1 n k¼1 ik ¼ h CIðRðcÞ Þ:
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5 The group decision-making method Let D ¼ fd1 ; d2 ; . . .; dm g be the set of decision makers or experts, and R~ðkÞ ¼ ð~ rij;ðkÞ Þnn be the interval fuzzy preference relation provided by expert dk (k ¼ 1; 2; . . .; m) on the alternative set X ¼ fx1 ; x2 ; . . .; xn g. For each of the matrices of the interval fuzzy preference relation rij;ðkÞ Þnn , discrete values for the judgments are R~ðkÞ ¼ ð~ randomly generated from the uniform distribution of the provided interval values. Assume that rij;ðkÞ is randomly L U generated from the uniform distribution of ½rij;ðkÞ ; rij;ðkÞ , (i j; i ¼ 1; 2; . . .; n; j ¼ 1; 2; . . .; n; k ¼ 1; 2; . . .; m), and rji;ðkÞ ¼ 1 rij;ðkÞ . Then, we obtain the fuzzy preference relation RðkÞ ¼ rij;ðkÞ nn , which is generated from R~ðkÞ ¼ ð~ rij;ðkÞ Þnn provided by dk (k ¼ 1; 2; . . .; m). Suppose that k ¼ ðk1 ; k2 ; . . .; km ÞT is the weighting vector, in which kk can reflect the importance degree of RðkÞ ¼ rij;ðkÞ nn . Moreover, the weighting vector satisfies kk 0 and Pm k¼1 kk ¼ 1. 5.1 Determining weights of RðkÞ based on group consensus In this subsection, we develop an optimal model to calculate the weights of RðkÞ ( k ¼ 1; 2; . . .; m). Definition 6 Let RðkÞ ¼ rij;ðkÞ nn (k ¼ 1; 2; . . .; m) be fuzzy preference relations. If m X kk rij;ðkÞ ; for all i; j ¼ 1; 2; . . .; n ð17Þ rijc ¼ k¼1
then Rc ¼ rijc is called the synthetic preference relann tion of RðkÞ ¼ rij;ðkÞ nn , and kk is the weights of RðkÞ , P which satisfies kk 0 and m k¼1 kk ¼ 1. The corresponding additive consistent fuzzy preference RðkÞ ¼ ð rij;ðkÞ Þnn of RðkÞ ¼ rij;ðkÞ nn can be obtained by Eq. (14), where Q 1=n Qn 1=n n 0:5 l¼1 rjl;ðkÞ l¼1 ril;ðkÞ rij;ðkÞ ¼ 1=n Pn Q n ð18Þ h¼1 l¼1 rhl;ðkÞ þ 0:5; for all i; j ¼ 1; 2; . . .; n:
Neural Comput & Applic
If the weighting vector of RðkÞ ¼ ð rij;ðkÞ Þnn is T k ¼ ðk1 ; k2 ; . . .; km Þ , from Definition 6, we can get the , where synthetic preference relation Rc ¼ rijc nn
rijc ¼
m X
kk rij;ðkÞ ; for all i; j ¼ 1; 2; . . .; n:
ð19Þ
k¼1
Theorem 4 The synthetic preference relation Rc ¼ is an additive consistent fuzzy preference relation. rijc nn
Proof According to Theorem 2, RðkÞ ¼ ð rij;ðkÞ Þnn is additive consistent, so for all i; j; s ¼ 1; 2; . . .; n and k ¼ 1; 2; . . .; m, we have 0 rij;ðkÞ 1, rij;ðkÞ þ rji;ðkÞ ¼ 1, P and ris;ðkÞ rjs;ðkÞ þ 0:5 ¼ rij;ðkÞ . Since rijc ¼ m k¼1 kk rij;ðkÞ , Pm kk 0 and k¼1 kk ¼ 1, it is obvious that 0 rijc 1 and rijc
þ
rjic
¼
m X
kk rij;ðkÞ þ rji;ðkÞ ¼
m X
k¼1
kk ¼ 1:
k¼1
Moreover, we can get m m X X kk ris;ðkÞ kk rjs;ðkÞ þ 0:5 risc rjsc þ 0:5 ¼ k¼1
¼
m X
k¼1
kk ris;ðkÞ rjs;ðkÞ þ 0:5
k¼1
¼
m X
kk rij;ðkÞ ¼ rijc :
k¼1
Thus, R ¼ rijc is an additive consistent fuzzy nn preference relation. The definition of compatibility degree between two fuzzy preference relations was given by Xu [23] as follows: Definition 7 Assume that A ¼ aij nn and B ¼ bij nn are two fuzzy preference relations. The compatibility degree between A and B is defined as follows: n X n 1X CðA; BÞ ¼ 2 jaij bij j: n i¼1 j¼1 c
Inspired by Definition 7, a new definition of compatibility degree is defined as follows: Definition 8 Let A ¼ aij nn and B ¼ bij nn be two fuzzy preference relations, then n X n 1X ðaij bij Þ2 ð20Þ CðA; BÞ ¼ 2 n i¼1 j¼1 is called the compatibility degree between A and B.
As we can see, the compatibility degree of fuzzy preference relations A and B can reflect the total difference between A and B. According to Definition 8, the compatibility degree between Rc and Rc can be written as follows: !2 n X n m m X X X 1 CðRc ; Rc Þ ¼ 2 kk rij;ðkÞ kk rij;ðkÞ n i¼1 j¼1 k¼1 k¼1 !2 n X n m X 1X ¼ 2 kk rij;ðkÞ rij;ðkÞ n i¼1 j¼1 k¼1 ¼
m X m X
n X n 1X rij;ðk1 Þ rij;ðk1 Þ 2 n i¼1 j¼1 k1 ¼1 k2 ¼1 ! rij;ðk2 Þ rij;ðk2 Þ :
k k1 k k2
ð21Þ The compatibility degree between Rc and Rc , which reflects group consensus, can measure the agreement between the individual fuzzy preference relation and the synthetic preference relation. The smaller the value of CðRc ; Rc Þ, the better the group consensus is. P P Let E ¼ ðek1 k2 Þmm , where ek1 k2 ¼ n12 ni¼1 nj¼1 rij;ðk1 Þ rij;ðk1 Þ Þ rij;ðk2 Þ rij;ðk2 Þ , i; j ¼ 1; 2; . . .; n. Then, E is called the compatibility information matrix. In particular, if 2 P P k1 ¼ k2 ¼ k, then ekk ¼ n12 ni¼1 nj¼1 rij;ðkÞ rij;ðkÞ . Therefore, we can obtain the following optimal model to calculate the weights vector of RðkÞ (k ¼ 1; 2; ; m) based on group consensus: min CðRc ; Rc Þ ¼ kT Ek T Q k¼1 s:t: k0
ð22Þ
where Q ¼ ð1; 1; . . .; 1ÞT and k ¼ ðk1 ; k2 ; . . .; km ÞT . By solving the model (22), we can obtain the optimal P solution kk satisfying kk 0 and m k¼1 kk ¼ 1, which can reflect the importance degree of RðkÞ ( k ¼ 1; 2; . . .; m). 5.2 The final priority vector derived by DEA In group decision making, although we can get the priority vector wðkÞ ¼ ðw1k ; w2k ; . . .; wnk ÞT from fuzzy preference relation RðkÞ ¼ rij;ðkÞ nn using Theorem 1, and calculate the weights of RðkÞ based on model (22) (k ¼ 1; 2; . . .; m), we need to develop a method to derive the final priority vector. In this subsection, a dual DEA model is developed to derive the final priority weight vector from kk and the priority vectors wðkÞ ¼ ðw1k ; w2k ; ; wnk ÞT (k ¼ 1; 2; . . .; m). The dual model of the input-oriented DEA is as follows:
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Neural Comput & Applic Table 2 Inputs, outputs, and the priority vectors
Output 1 (d1 )
Output 2 (d2 )
Output m (dm )
Dummy input
DMU1
w11
w12
w1m
1=n
DMU2
w21
w22
w2m
1=n
DMUn
wn1
wn2
wnm
1=n
max Z ¼ vT Yp 8 T > < u Xp ¼ 1 s:t: uT X T vT Y T 0 ; > : u 0; v 0
ð23Þ
where u ¼ ðu1 ; u2 ; . . .; us ÞT , v ¼ ðv1 ; v2 ; . . .; vm ÞT . For alternative set X ¼ fx1 ; x2 ; . . .; xn g, each alternative may be viewed as a decision-making unit, each priority vector wðkÞ ¼ ðw1k ; w2k ; . . .; wnk ÞT can be viewed as an output. The dummy inputs take the value of 1=n for all the alternatives. The relationships among inputs, outputs, and the priority vectors wðkÞ ¼ ðw1k ; w2k ; . . .; wnk ÞT are shown in Table 2. From Table 2 and dual model (23), we build the following DEA model to evaluate the efficiency score of the alternative xp ðp ¼ 1; 2; . . .; nÞ. max Zp ¼ m1 wp1 þ m2 wp2 þ þ mm wpm 8 1 > > > u1 n ¼ 1 > > > > < v1 w11 þ v2 w12 þ vm w1m 1 s: t: v1 w21 þ v2 w22 þ vm w2m 1 > > > > > > > : v1 wn1 þ v2 wn2 þ vm wnm 1 u1 ; v1 ; v2 ; . . .; vm 0
ð24Þ
Because each output has different importance degree with the weighting vector k ¼ ðk1 ; k2 ; ; km ÞT . Let vi ¼ v1 kk1i (i ¼ 1; 2; . . .; m), then model (24) can be rewritten as follows: m1 k1 wp1 þ k2 wp2 þ þ km wpm max Zp ¼ k1 8 1 > > u1 ¼ 1 > > > n > > m1 > > > ðk1 w11 þ k2 w12 þ þ km w1m Þ 1 > > > < k1 m1 s: t: ðk1 w21 þ k2 w22 þ þ km w2m Þ 1 > k > 1 > > > > > m > > 1 ðk1 wn1 þ k2 wn2 þ þ km wnm Þ 1 > > k > > : 1 u1 ; v1 ; v2 ; . . .; vm 0
max
i¼1;2;;n
m1 ðk1 wi1 þ k2 wi2 þ þ km wim Þ ¼ 1: k1
Then, the optimal objective function value of model (25) P ki is Zp ¼ m1 m i¼1 k1 wpi . Thus, the final weight of each alternative can be calculated by Z wi ¼ Pn i
p¼1
Zp
; i ¼ 1; 2; . . .; n:
ð26Þ
5.3 The stochastic group decision-making method Since fuzzy preference relation is a special case of interval fuzzy preference relation, we propose a stochastic group preference analysis (SGPA) method by analyzing the group preference relations space. Let n be a set of random variables, where n ¼ fnij;ðkÞ : i; j ¼ 1; 2; . . .; n; i\j; k ¼ 1; 2; . . .; mg, nij;ðkÞ satisfies L U uniform distribution in r~ij;ðkÞ ¼ ½rij;ðkÞ ; rij;ðkÞ . R~ ¼
r~ij;ðkÞ j i; j ¼ 1; 2; . . .; n; k ¼ 1; 2; . . .; m is the interval preference relation, and f ðnij;ðkÞ Þ is the probability density function of nij;ðkÞ , where nij;ðkÞ is the judgments of xi over xj provided by expert dk (k ¼ 1; 2; . . .; m). For n, it is assumed that wðnÞ is the corresponding final priority vector. Inspired by Zhu and Xu [30], we provide a ranking function, which is defined as follows: n X ðnÞ ðnÞ Ranki ðwðnÞ Þ ¼ 1 þ dðwk [ wi Þ ¼ r; ð27Þ k¼1
where the rank of xi is r, dðtrueÞ ¼ 1, and dðfalseÞ ¼ 0. Moreover, the Eq. (27) defines a group preference relations space Rri ðnÞ.
ðnÞ ~ Rri ðnÞ ¼ fn 2 Rjrank i ðw Þ ¼ 1 þ
n X
ðnÞ
ðnÞ
dðwk [ wi Þ ¼ rg:
k¼1
ð25Þ
ð28Þ
If we evaluate different alternatives based on model (25), the DEA models have the same optimal solution m1 , which satisfies
According to the final priority weights vector wðnÞ and the probability density function f ðnÞ, we can obtain the expected priority vector of the alternative set X ¼ fx1 ; x2 ; . . .; xn g:
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Neural Comput & Applic
ðnÞ
Eðw Þ ¼
Z
ðnÞ
f ðnÞw dn:
ð29Þ
R~
From EðwðnÞ Þ, we further get the expected rank of xi ði ¼ 1; 2; ; nÞ, which can be computed as follows: n X dðEðwðnÞ Þk [ EðwðnÞ Þi Þ; ð30Þ Rankei ¼ 1 þ k¼1
where EðwðnÞ Þk is the kth component of EðwðnÞ Þ. Moreover, a probability measure is defined to describe the confidence degree of the expected rank, which can reflect the probability that the group receives the expected rank. Z e pi ¼ f ðnÞdn: ð31Þ n: ranki ðwðnÞ Þ¼Eðri Þ
In conclusion, to get the expected priority vector and the associated confidence degree in group decision making with interval fuzzy preference relations, the following steps are involved: Step 1 The expert group fd1 ; d2 ; . . .; dm g give their rij;ðkÞ Þnn interval fuzzy preference relations R~ðkÞ ¼ ð~ (i; j ¼ 1; 2; . . .; n; k ¼ 1; 2; . . .; m) on the alternative set X ¼ fx1 ; x2 ; . . .; xn g, respectively. Step 2 Obtain the fuzzy preference relation matrices RðkÞ ¼ rij;ðkÞ nn , (k ¼ 1; 2; . . .; m), which generated rij;ðkÞ Þ , where rij;ðkÞ is randomly generfrom R~ðkÞ ¼ ð~ nn
ated from the uniform distribution of
L U ½rij;ðkÞ ; rij;ðkÞ ,
n P o 2k \d , where e is simulation error, d j0 ¼ min j e12 1 k k¼j 2 is the expected accuracy. Then, the required number of iterations is N ¼ 2j0 1 . Especially, when e ¼ 0:01, d ¼ 0:072, then j0 ¼ 11 and N ¼ 210 ¼ 1024: The process is shown in Fig. 1.
6 Numerical example In this section, the proposed group decision-making method is used to evaluate four candidates fx1 ; x2 ; x3 ; x4 g for a chair professor position. The numerical example is implemented in Matlab. Assume that there are four interval fuzzy preference relations R~ð1Þ , R~ð2Þ , R~ð3Þ , and R~ð4Þ given by the experts d1 , d2 , d3 , and d4 , respectively, which are shown as follows: 0 1 ½0:5; 0:5 ½0:6; 0:7 ½0:6; 0:7 ½0:3; 0:4 B ½0:3; 0:4 ½0:5; 0:5 ½0:5; 0:6 ½0:2; 0:3 C C R~ð1Þ ¼ B @ ½0:3; 0:4 ½0:4; 0:5 ½0:5; 0:5 ½0:5; 0:6 A ½0:6; 0:7 ½0:7; 0:8 ½0:4; 0:5 ½0:5; 0:5 0
R~ð2Þ
0
R~ð3Þ
(i j;
i ¼ 1; 2; . . .; n; j ¼ 1; 2; . . .; n; k ¼ 1; 2; . . .; m), and rji;ðkÞ ¼ 1 rij;ðkÞ . Step 3 Transform RðkÞ ¼ rij;ðkÞ nn (k ¼ 1; 2; . . .; m) into acceptable consistent fuzzy preference relations ðCÞ ðCÞ (k ¼ 1; 2; . . .; m) according to AlgoRðkÞ ¼ rij;ðkÞ nn
rithm 1, and calculate k ¼ ðk1 ; k2 ; . . .; km ÞT , which is the weight vector of fRð1Þ ; Rð2Þ ; . . .; RðmÞ g according to model (22). Step 4 Compute the priority vector wðkÞ ¼ ðCÞ ðCÞ ðw1k ; w2k ; . . .; wnk ÞT of RðkÞ ¼ rij;ðkÞ according to nn
Theorem 1 (k ¼ 1; 2; . . .; m). Step 5 Obtain the final priority vector w ¼ ðw1 ; w2 ; . . .; wn ÞT according to model (25) and Eq. (26). Step 6 Repeat the Step 2-Step 5 N times, and execute stochastic simulation to get the outcomes of SGPA, i.e., EðwðnÞ Þ, rankei and pei . In order to determine the proper number of iterations in stochastic simulation process, Pula [14] introduced a general method based on Kolmogorov’s inequality. Let
½0:5; B ½0:4; ¼B @ ½0:3; ½0:6; ½0:5; B ½0:3; ¼B @ ½0:1; ½0:5; 0
R~ð4Þ
½0:5; B ½0:6; ¼B @ ½0:2; ½0:3;
0:5 0:5 0:4 0:7
½0:5; ½0:5; ½0:3; ½0:6;
0:6 0:5 0:3 0:8
½0:6; ½0:7; ½0:5; ½0:4;
0:7 0:7 0:5 0:5
½0:3; ½0:2; ½0:5; ½0:5;
1 0:4 0:4 C C 0:6 A 0:5
0:5 0:5 0:3 0:7
½0:5; ½0:5; ½0:3; ½0:6;
0:7 0:5 0:5 0:8
½0:7; ½0:5; ½0:5; ½0:4;
0:9 0:7 0:5 0:5
½0:3; ½0:2; ½0:5; ½0:5;
1 0:5 0:4 C C 0:6 A 0:5
0:5 0:7 0:3 0:4
½0:6; ½0:5; ½0:3; ½0:7;
0:7 0:5 0:4 0:8
½0:7; ½0:6; ½0:5; ½0:2;
0:8 0:7 0:5 0:3
½0:3; ½0:2; ½0:7; ½0:5;
1 0:4 0:3 C C 0:8 A 0:5
Let the threshold value CI ¼ 0:1, and execute stochastic group preference analysis according to Step 1–Step 6. According to Monte Carlo simulation, the obtained EðwðnÞ Þ are shown in Table 3 with different number of iterations. From Table 3, when the number of iterations is equal or greater than 160, EðwðnÞ Þ is robust. The outcomes of SGPA are shown in Table 4 with the number of iterations being 1024. Moreover, Fig. 2 shows the confidence degrees of the four candidates for each rank by a three-dimensional column chart. From the expected priority vector EðwðnÞ Þ with the number of iterations being 500, the rank of four candidates is x4 x1 x2 x3 . As we can see, x4 is the best candidate. Meanwhile, if the expected rank has high confidence degree, the expected rank should be reliable. Particularly, x4 is the best alternative with 0.930 confidence degree.
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Neural Comput & Applic
Obtain the priority vector based on Theorem 1
Consistency adjustment according to Algorithm 1
R(1)
R(1) = ( rij ,(1) )
R(2)
R(2) = ( rij ,(2) )
R( m )
R( m ) = ( rij ,( m ) )
n× n
n×n
n× n
(Γ) Γ) R(1) = ( rij(,(1) )
n× n
(Γ) ) R(2) = ( rij(,Γ(2) )
(Γ) (m)
R
= (r
n×n
)
(Γ) ij , ( m ) n×n
Obtain the weight vector λ = (λ1 , λ2 , , λm )T of preference relationships R( k ) , k = 1, 2, , m according to model (22)
w(1) = ( w11 ,
, wn1 )T
w(2) = ( w12 ,
, wn 2 )T
w( m ) = ( w1m ,
, wnm )
T
Obtain the expected (ξ ) priority vector E ( w ) and the associated e confidence degree pi according to stochastic simulation
Obtain the final weights w = ( w1 , w2 , , wn )T according to Model (25) and Eq. (26)
Fig. 1 Group decision-making method based on DEA and stochastic simulation
Table 3 Expected priority vector EðwðnÞ Þ with different number of iterations Number of iterations
EðwðnÞ Þ EðwðnÞ Þ1
40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 640 680 720 760 800 840 880 920 960 1000
Table 4 Outcomes of SGPA with the number of iterations being 1024
0.2649 0.2664 0.2658 0.2657 0.2659 0.2661 0.2663 0.2666 0.2666 0.2664 0.2665 0.2667 0.2668 0.2669 0.2669 0.2669 0.2668 0.2667 0.2666 0.2665 0.2666 0.2666 0.2667 0.2666 0.2666
EðwðnÞ Þ2 0.2251 0.2251 0.2261 0.2263 0.2268 0.2270 0.2258 0.2253 0.2252 0.2256 0.2252 0.2251 0.2251 0.2251 0.2251 0.2246 0.2250 0.2248 0.2247 0.2253 0.2253 0.2252 0.2254 0.2255 0.2255
EðwðnÞ Þ3 0.2219 0.2193 0.2193 0.2192 0.2187 0.2185 0.2190 0.2195 0.2195 0.2194 0.2194 0.2193 0.2194 0.2194 0.2194 0.2197 0.2194 0.2197 0.2198 0.2195 0.2195 0.2196 0.2195 0.2195 0.2195
EðwðnÞ Þ4 0.2879 0.2891 0.2887 0.2886 0.2885 0.2882 0.2887 0.2885 0.2886 0.2886 0.2887 0.2888 0.2886 0.2884 0.2884 0.2885 0.2885 0.2886 0.2887 0.2885 0.2884 0.2884 0.2883 0.2884 0.2884
EðwðnÞ Þ
Rankei
pei
x1
0.2666
2
0.908
x2
0.2255
3
0.550
x3
0.2195
4
0.572
x4
0.2884
1
0.930
QðyÞ ¼ y2 , the attitudinal character of Q is k ¼ 13. Then, the expected value fuzzy preference relations FQ ðR~ðkÞ Þ of R~ðkÞ (k ¼ 1; 2; 3; 4) are derived based on OWA operator as follows: 0 1 0:5 0:6333 0:6333 0:3333 B 0:3667 0:5 0:5333 0:2333 C B C FQ ðR~ð1Þ Þ ¼ B C; @ 0:3667 0:4667 0:5 0:5333 A 0:6667 0:7667 0:4667 0:5 1 0:5 0:6 0:6333 0:3333 B 0:4 0:5 0:7 0:2667 C B C FQ ðR~ð2Þ Þ ¼ B C; @ 0:3667 0:3 0:5 0:5333 A 0
0:6667 0:7333 0:4667 0:5 1 0:5 0:5667 0:7667 0:3667 B 0:4333 0:5 0:5667 0:2667 C B C FQ ðR~ð3Þ Þ ¼ B C; @ 0:2333 0:4333 0:5 0:5333 A 0
0:6333 0:7333 0:4667 0:5 1 0:5 0:6333 0:7333 0:3333 B 0:3667 0:5 0:6333 0:2333 C B C FQ ðR~ð4Þ Þ ¼ B C: @ 0:2667 0:3667 0:5 0:7333 A 0
0:6667 0:7667 0:2667 Chen and Zhou [4] proposed an approach to group decision making based on IGCOWA operator. If we apply Chen and Zhou’s method to this example, when the BUM function
123
0:5
From Eqs. (31) and (32) in Chen and Zhou [4], it is assumed that c ¼ 0:5, we get the collective fuzzy preference relations matrix A ¼ ðaij Þ44 , i.e.,
Neural Comput & Applic Fig. 2 Confidence degrees of the four candidates
0
0:5 0:5996 B 0:4004 0:5 A¼B @ 0:3488 0:4115 0:6735 0:7469
0:6512 0:5885 0:5 0:4481
1 0:3265 0:2531 C C: 0:5519 A 0:5
According to A, we obtain the priority vector w ¼ ð0:2564; 0:2284; 0:2343; 2806ÞT . Thus, x4 x1 x2 x3 . The obtained result is in accordance with those given by the proposed method in this paper. From the above examples, one can see that our proposed method and Chen and Zhou’s method can derive the same ranking result. Chen and Zhou’s method translates four interval fuzzy preference relations into a collective fuzzy preference relations matrix, and this procedure may lead to decision information loss. On the other hand, sometimes the collective fuzzy preference relations matrix is not consistent. The group decision-making method based on compatibility given in Zhou et al. [28] also has these weaknesses. Meanwhile, our proposed method has several desirable properties. First, it can avoid information loss by stochastic simulation. Second, the weights of preference relation generated from each interval fuzzy preference relations are derived by group consensus. Third, our method includes an effective consistency adjustment algorithm, it can address
inconsistent cases. Finally, the confidence degrees of ranking results are given.
7 Conclusion We have proposed a new approach to group decision making with interval fuzzy preference relations using data envelopment analysis (DEA) and stochastic simulation. The advantage of the developed approach is that it can avoid information loss. A new output-oriented CCR DEA model is proposed to obtain the priority vector for the consistency fuzzy preference relation, where each of the alternatives is viewed as a decision-making unit (DMU). Meanwhile, a consistency adjustment algorithm is designed for the inconsistent fuzzy preference relations. Then, we build an optimization model to yield the weights of each fuzzy preference relation based on maximizing group consensus. Moreover, an input-oriented DEA model is introduced to obtain the final priority vector of the alternatives. We further develop a stochastic group preference analysis (SGPA) method by analyzing the judgments space, which is carried out by Monte Carlo simulation. The proposed approach is verified by a numerical example.
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Neural Comput & Applic
Future research should, in the authors’ opinion, focus on further extensions of the SGPA such as the use of interval linguistic group decision making. Acknowledgements The work was supported by National Natural Science Foundation of China (Nos. 71501002, 61502003, 71371011, 71771001, 71701001) and Anhui Provincial Natural Science Foundation (Nos. 1508085QG149, 1608085QF133). Compliance with ethical standards Conflict of interest The authors declare that they have no conflict of interest.
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