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ORIGINAL ARTICLE
Group decision making with interval fuzzy preference relations
based on DEA and stochastic simulation
Jinpei Liu
1
•Qin Xu
2
•Huayou Chen
3
•Ligang Zhou
3,4
•Jiaming Zhu
3
•
Zhifu Tao
3
Received: 27 December 2016 / Accepted: 15 October 2017
ÓThe Natural Computing Applications Forum 2017
Abstract This paper proposes an integrated approach to
group decision making with interval fuzzy preference rela-
tions 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-ori-
ented DEA model is introduced to obtain the final priority
vector of the alternatives. Finally, a stochastic group
preference analysis method is developed by analyzing the
judgments space, which is carried out by Monte Carlo sim-
ulation. A numerical example demonstrates that the pro-
posed 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 pro-
vides his/her preference information which includes mul-
tiplicative 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 decisionalternatives. 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).
&Jinpei Liu
liujinpei2012@163.com
Qin Xu
xuqin2013@aliyun.com
Huayou Chen
huayouc@126.com
Ligang Zhou
shuiqiaozlg@126.com
Jiaming Zhu
502470059@qq.com
Zhifu Tao
zhifut_0514@163.com
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
123
Neural Comput & Applic
DOI 10.1007/s00521-017-3254-7
In group decision-making process, because of the deci-
sion-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 decision-
making 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 program-
ming 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 deci-
sion 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 con-
sistency adjustment algorithm of fuzzy preference relation
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 con-
cepts including fuzzy preference relation, additive consis-
tent 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 mak-
ing, in which the set of alternatives to the problem is
presented. Unless otherwise mentioned, it is assumed that
X¼fx1;x2;...;xngis the set of alternatives, and D¼
fd1;d2;...;dmgis the set of decision makers or experts. In
the decision-making process, each expert provides his/her
opinions over alternatives in X¼fx1;x2;...;xngby means
of preference relation. Let w¼ðw1;w2;...;wnÞTbe the
priority vector, where wireflects the importance degree of
the alternative xi. All the wi(i¼1;2;...;n) are no less
than zeros and sum to one, i.e.,
wi0;i¼1;2;...;n;X
n
i¼1
wi¼1:
Definition 1 [12]. A fuzzy preference relation Ron a set
of alternatives Xis represented by a fuzzy set on the pro-
duct set XX, which is characterized by the membership
function:
lR:XX!½0;1:
When the number of alternatives is finite, the preference
relation Ris represented by an nnmatrix R¼ðrij Þnn,
where rij ¼lRðxi;xjÞ,i;j¼1;2;...;n.rij denotes the
preference degree of the alternative xiover the alternative
xjwith
rij þrji ¼1;rii ¼0:5;0rij 1 for all i;j¼1;2;...;n:
Definition 2 [15]. A fuzzy preference relation R¼
ðrijÞnnis additive consistent if
rij ¼rik rjk þ0:5;for all i;j;k¼1;2;...;n;ð1Þ
and such a fuzzy preference relation is given by [11,22]
rij ¼0:5ðwiwjþ1Þ;for all i;j¼1;2;...;n:ð2Þ
Neural Comput & Applic
123
In 2004, Xu [24] developed the interval fuzzy prefer-
ence relation, which can be defined as follows:
Definition 3 [24,28]. An interval fuzzy preference rela-
tion ~
Ris defined as ~
R¼ð
~
rijÞnn, which satisfies that
~
rij ¼½rL
ij;rU
ij ;0rL
ij rU
ij 1;rL
ij þrU
ji ¼rU
ij þrL
ji ¼1;
for all i;j¼1;2;...;n;
where ~
rij indicates the interval-valued preference degree of
the alternative xiover the alternative xj,rL
ij and rU
ij are the
lower and upper limits of ~
rij, respectively.
3 Priority vector based on DEA
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 nDMUs, denoted as DMUi
(i¼1;2;...;n), to be evaluated. Each DMU has sdifferent
inputs xil and mdifferent outputs yik (xil 0, yik 0,
l¼1;2;...;s,k¼1;2;...;m). Let Xi¼ðxi1;xi2;...;xisÞT
be the input vector and Yi¼ðyi1;yi2;...;yimÞTbe the out-
put vector of DMUi. Then, the following output-oriented
CCR model can be used to evaluate the efficiency of
DMUp(p¼1;2;...;n):
max bp
s:t:Pn
i¼1liyik bpypk;k¼1;2;...;m
Pn
i¼1lixil xpl;l¼1;2;...;s
bpfree;li0;i¼1;2;...;n
8
>
<
>
:
ð3Þ
Denoting l¼ðl1;l2;...;lnÞT,Xi¼ðxi1;xi2;...;xisÞT,
Yi¼ðyi1;yi2;...;yimÞT,X¼
x11 x12 ... x1s
x21 x22 ... x2s
... ...
xn1xn2... xns
0
B
B
@1
C
C
Aand
Y¼
y11 y12 ... y1m
y21 y22 ... y2m
... ...
yn1yn1... ynm
0
B
B
@1
C
C
A;then the model (3) can be
rewritten as
max bp
s:t:
YTlbpYp
XTlXp
l0;bpfree
8
>
<
>
:
ð4Þ
In model (4), an imaginary composite unit is con-
structed, which outperforms each DMUi,lirepresents the
proportion for which DMUiis used to construct the com-
posite unit, and an efficiency score can be generated for the
DMUpby maximizing outputs with limited inputs.
According to model (4), the composite unit consumes at
most the same inputs as DMUpand produces at least bp
times outputs of DMUpwith bp1. The inverse of optimal
objective function value of model (4), 1.b
p, is the effi-
ciency score of DMUp, and DMUpis not efficient if
bp[1.
Meanwhile, the input-oriented DEA model to evaluate
alternative DMUp(p¼1;2;...;n) is as follows:
min hp
s:t:
YTlYp
XTlhpXp
l0;hpfree
8
>
<
>
:
ð5Þ
In model (5), an imaginary composite unit is also con-
structed that outperforms each DMU and an efficiency
score can be generated for the DMUpby minimizing inputs
with limited outputs. The composite unit produces at least
the same outputs as DMUpand invests at most hptimes
inputs of DMUpwith hp1. The optimal objective func-
tion value, h
p, is the efficiency score of DMUpin model
(5), and DMUpis 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 R¼ðrij Þnnon the
alternative set X¼fx1;x2;...;xng, each alternative may be
viewed as a decision-making unit (DMU), each column of
R¼ðrijÞnncan 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 R¼ðrij Þnnare shown in
Table 1.
We build the following output-oriented CCR model to
evaluate the efficiency score of alternative xp
(p¼1;2;...;n).
max bp
s:t:Pn
i¼1lirik bprpk;k¼1;2;...;n
0:5Pn
i¼1li0:5
bpfree;li0;i¼1;2;...;n
8
>
<
>
:
ð6Þ
Theorem 1 For an additive consistent fuzzy preference
matrix R ¼ðrijÞnn,the optimal solution of model (6)is b
p
(p¼1;2;...;n)and r:f1;2;...;ng!f1;2;...;ngis a
permutation, such that b
rð1Þb
rð2Þ b
rðnÞ.Then,
Neural Comput & Applic
123
the priority vector satisfies that wrð1Þwrð2Þ wrðnÞ,
and it can be derived by the following formulas:
(1) wrð1Þ¼nþ1
n1
nb
rð1ÞPn
p¼1
1
b
p.
(2) wrðpÞ¼wrð1Þþb
rð1Þb
rðpÞ
b
rðpÞ, for all p¼2;...;n.
Proof If R¼ðrijÞnnis an additive consistent fuzzy
preference relation, from Eq. (2), the model (6) can be
rewritten as
max bp
s:t:Pn
i¼1liðwiwkþ1Þbpðwpwkþ1Þ;k¼1;2;...;n;
Pn
i¼1li1;
bpfree;li0;i¼1;2;...;n:
8
<
:ð7Þ
Without loss of generality, it is assumed that
w1w2 wn. Then, we have wiwkþ10 and
wpwkþ10. Thus, the objective function bpis maxi-
mal if and only if the constraint Pn
i¼1li1 reduces to
equality, i.e., Pn
i¼1li¼1.
Since 0 w1wkþ1w2wkþ1 wn
wkþ1 for any k2f1;2;...;ng, from the first constraint of
model (7), it is obvious that the optimal solution is l
1¼
l
2¼¼l
n1¼0 and l
n¼1. Then, the first constraint
of model (7) becomes wnwkþ1
wpwkþ1bp;k¼1;2;...;n. Thus,
the optimal objective function value of model (7) follows
that
b
p¼min wnw1þ1
wpw1þ1;wnw2þ1
wpw2þ1;...;1
wpwnþ1
¼wnw1þ1
wpw1þ1:
ð8Þ
Since w1w2 wn, we have b
1b
2 b
n.
Therefore,
max
1pnfb
pg¼b
1¼wnw1þ1:ð9Þ
The efficiency score of alternative xpis 1
b
p¼wpw1þ1
wnw1þ1.
Then, the sum of efficiency scores of all alternatives is
X
n
p¼1
1
b
p
¼X
n
p¼1
wpw1þ1
wnw1þ1¼nþ1nw1
wnw1þ1:ð10Þ
From Eqs. (9) and (10), we have
w1¼nþ1
nmax
1pnfb
pg1
nX
n
p¼1
1
b
p
:ð11Þ
According to Eqs. (8), (9), we also have
wp¼w1þ
max
p¼1;2;;nfb
pgb
p
b
p
;for all p¼2;...;nð12Þ
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:
R¼
0:50:525 0:55 0:525
0:475 0:50:525 0:5
0:45 0:475 0:50:475
0:475 0:50:525 0:5
0
B
B
@1
C
C
A:
According to model (5), we can evaluate the efficiency
score of alternative x2from the following linear
programming.
max bp
s:t:
0:5l1þ0:475l2þ0:45l3þ0:475l40:475bp
0:525l1þ0:5l2þ0:475l3þ0:5l40:5bp
0:55l1þ0:525l2þ0:5l3þ0:525l40:525bp
0:525l1þ0:5l2þ0:475l3þ0:5l40:5bp
Pn
i¼1li1
bpfree;l1;l2;l3;l40
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
The optimal solution of above linear programming is
l
1¼1,b
2¼1:0476, l
2¼l
3¼l
4¼0. The efficiency
score of alternative x2is 1b
2¼0:9545. Similar, alterna-
tives x1,x2, and x3can be evaluated, and b
1¼1, b
3¼1:1,
b
4¼1:0476. So, we have b
1\b
2¼b
4\b
3,rð1Þ¼
3;rð2Þ¼2;rð3Þ¼4 and rð4Þ¼1. According to Theo-
rem 1, we have w3w2w4w1, and the priority vector
of alternative set X¼fx1;x2;x3;x4gis that
w¼ð0:30;0:25;0:20;0:25ÞT.
Table 1 Inputs and outputs of DEA about R¼ðrijÞnn
Output 1 Output 2 Output nDummy input
DMU1r11 r12 r1n0.5
DMU2r21 r22 r2n0.5
DMUnrn1rn2 rnn 0.5
Neural Comput & Applic
123
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 R¼ðrijÞnnis a fuzzy prefer-
ence relation given by an expert, we define the additive
consistency index of Ras follows:
CIðRÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
nðn1ÞX
n
i¼1X
n
j¼iþ1
1
nX
n
k¼1
ðrik rjk rij þ0:5Þ2
!
v
u
u
t;
ð13Þ
where 0 CIðRÞ1.
The smaller the value of CIðRÞ, the stronger the degree
of additive consistency of the fuzzy preference relation
R¼ðrijÞnnis. It is obvious that CIðRÞ¼0 if and only if R
is an additive consistent fuzzy preference relation.
Definition 5 Let R¼ðrijÞnnbe a fuzzy preference
relation given by an expert. Given a threshold value CI, if
the additive consistency index of Rsatisfies
CIðRÞCI:
Then, the fuzzy preference relation Ris 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 ¼ðrijÞnn,
let
R¼ð
rijÞnn,where
rij ¼
0:5Qn
l¼1ril
ðÞ
1=nQn
l¼1rjl
1=n
Pn
h¼1Qn
l¼1rhl
ðÞ
1=nþ0:5;
for all i;j¼1;2;...;n:
ð14Þ
Then,
Ris an additive consistent fuzzy preference
relation.
Proof Since 0 rij 1, for all i;j¼1;2;...;n, we have
0:50:5Qn
l¼1rjl
1=n
Pn
h¼1Qn
l¼1rhl
ðÞ
1=n
0:5Qn
l¼1ril
ðÞ
1=nQn
l¼1rjl
1=n
Pn
h¼1Qn
l¼1rhl
ðÞ
1=n
þ0:50:5Qn
l¼1ril
ðÞ
1=n
Pn
h¼1Qn
l¼1rhl
ðÞ
1=nþ0:5:
Thus, we can get
0
rij ¼
0:5Qn
l¼1ril
ðÞ
1=nQn
l¼1rjl
1=n
Pn
h¼1Qn
l¼1rhl
ðÞ
1=nþ0:51
for all i;j¼1;2;...;n:
Furthermore, for any i;j;k¼1;2;...;n, it follows that
rik
rjk þ0:5¼
0:5Qn
l¼1ril
ðÞ
1=nQn
l¼1rkl
ðÞ
1=n
Pn
h¼1Qn
l¼1rhl
ðÞ
1=nþ0:5
0
@1
A
0:5Qn
l¼1rjl
1=nQn
l¼1rkl
ðÞ
1=n
Pn
h¼1Qn
l¼1rhl
ðÞ
1=nþ0:5
0
@1
Aþ0:5
¼
0:5Qn
l¼1ril
ðÞ
1=nQn
l¼1rkl
ðÞ
1=nQn
l¼1rjl
1=nþQn
l¼1rkl
ðÞ
1=n
Pn
h¼1Qn
l¼1rhl
ðÞ
1=n
þ0:5
¼
0:5Qn
l¼1ril
ðÞ
1=nQn
l¼1rjl
1=n
Pn
h¼1Qn
l¼1rhl
ðÞ
1=nþ0:5¼
rij:
From the Definition 2,
Ris an additive consistent fuzzy
preference relation.
If the given fuzzy preference relation does not satisfy
the acceptable consistency, from the Theorem 2, a con-
sistency adjusting algorithm is proposed as follows:
Algorithm 1 Input The original individual fuzzy prefer-
ence relation R¼ðrijÞnn, the controlling parameter
h2ð0;1Þ, and the threshold CI.Output The adjusted fuzzy
preference relation RðCÞ¼ðrðCÞ
ij Þnnand the consistency
index CIðRðCÞÞ.
Step 1 Let Rð0Þ¼ðrð0Þ
ij Þnn¼ðrijÞnn, and c¼0.
Step 2 Calculate the consistency index CIðRðcÞÞ, based
on Eq. (13).
Step 3 If CIðRðcÞÞCI, then go to step 6; otherwise, go
to the next step.
Step 4 Let
RðcÞ¼ð
rðcÞ
ij Þnn, where
rðcÞ
ij ¼
0:5Qn
l¼1rðcÞ
il
1=nQn
l¼1rðcÞ
jl
1=n
Pn
h¼1Qn
l¼1rðcÞ
hl
1=nþ0:5:
Neural Comput & Applic
123
Step 5 Apply the following strategy to construct a new
fuzzy preference relation Rðcþ1Þ¼ðrðcþ1Þ
ij Þnn.
rðcþ1Þ
ij ¼hrðcÞ
ij þð1hÞ
rðcÞ
ij ;ð15Þ
where h2ð0;1Þ.Let c¼cþ1, and return to Step 2.
Step 6 End.
From Algorithm 1, we have the following theorem.
Theorem 3 Algorithm 1 is convergent. That is,
CIðRðcþ1ÞÞCIðRðcÞÞand lim
c!1 CIðRðcÞÞ¼0.
Proof From Step 5 of Algorithm 1 and Eq. (15), we have
rðcþ1Þ
ij ¼hrðcÞ
ij þð1hÞ
rðcÞ
ij ¼hrðcÞ
ij þð1hÞ
0:5Qn
l¼1rðcÞ
il
1=nQn
l¼1rðcÞ
jl
1=n
Pn
h¼1Qn
l¼1rðcÞ
hl
1=nþ0:5
0
B
B
@1
C
C
A
From Eq. (13), we obtain
CIðRðcþ1ÞÞ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
nðn1ÞX
n
i¼1X
n
j¼iþ1
1
nX
n
k¼1
ðrðcþ1Þ
ik rðcþ1Þ
jk rðcþ1Þ
ij þ0:5Þ2
!
v
u
u
t:
ð16Þ
where
rðcþ1Þ
ik rðcþ1Þ
jk rðcþ1Þ
ij þ0:5¼hrðcÞ
ik þð1hÞ
0:5Qn
l¼1rðcÞ
il
1=nQn
l¼1rðcÞ
kl
1=n
Pn
h¼1Qn
l¼1rðcÞ
hl
1=nþ0:5
0
B
B
@1
C
C
AhrðcÞ
jk
ð1hÞ
0:5Qn
l¼1rðcÞ
jl
1=nQn
l¼1rðcÞ
kl
1=n
Pn
h¼1Qn
l¼1rðcÞ
hl
1=nþ0:5
0
B
B
@1
C
C
A
hrðcÞ
ij ð1hÞ
0:5Qn
l¼1rðcÞ
il
1=nQn
l¼1rðcÞ
jl
1=n
Pn
h¼1Qn
l¼1rðcÞ
hl
1=nþ0:5
0
B
B
@1
C
C
A
þ0:5¼hrðcÞ
ik rðcÞ
jk rðcÞ
ij þ0:5
Therefore,
CIðRðcþ1ÞÞ
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
nðn1ÞX
n
i¼1X
n
j¼iþ1
1
nX
n
k¼1
ðrðcþ1Þ
ik rðcþ1Þ
jk rðcþ1Þ
ij þ0:5Þ2
!
v
u
u
t
¼hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
nðn1ÞX
n
i¼1X
n
j¼iþ1
1
nX
n
k¼1
ðrðcÞ
ik rðcÞ
jk rðcÞ
ij þ0:5Þ2
!
v
u
u
t
¼hCIðRðcÞÞ:
Thus, it is obvious that CIðRðcÞÞ¼hcCIðRð0ÞÞ. Moreover,
from h2ð0;1Þ, we can obtain that limc!1 CIðRðcÞÞ¼0.
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.
5 The group decision-making method
Let D¼fd1;d2;...;dmgbe the set of decision makers or
experts, and ~
RðkÞ¼ð
~
rij;ðkÞÞnnbe the interval fuzzy pref-
erence relation provided by expert dk(k¼1;2;...;m)on
the alternative set X¼fx1;x2;...;xng. For each of the
matrices of the interval fuzzy preference relation
~
RðkÞ¼ð
~
rij;ðkÞÞnn, discrete values for the judgments are
randomly generated from the uniform distribution of the
provided interval values. Assume that rij;ðkÞis randomly
generated from the uniform distribution of ½rL
ij;ðkÞ;rU
ij;ðkÞ,
(ij;i¼1;2;...;n;j¼1;2;...;n;k¼1;2;...;m), and
rji;ðkÞ¼1rij;ðkÞ. Then, we obtain the fuzzy preference
relation RðkÞ¼rij;ðkÞ
nn, which is generated from ~
RðkÞ¼
ð~
rij;ðkÞÞnnprovided by dk(k¼1;2;...;m). Suppose that
k¼ðk1;k2;...;kmÞTis the weighting vector, in which kk
can reflect the importance degree of RðkÞ¼rij;ðkÞ
nn.
Moreover, the weighting vector satisfies kk0 and
Pm
k¼1kk¼1.
5.1 Determining weights of RðkÞbased on group
consensus
In this subsection, we develop an optimal model to cal-
culate the weights of RðkÞ(k¼1;2;...;m).
Definition 6 Let RðkÞ¼rij;ðkÞ
nn(k¼1;2;...;m)be
fuzzy preference relations. If
rc
ij ¼X
m
k¼1
kkrij;ðkÞ;for all i;j¼1;2;...;nð17Þ
then Rc¼rc
ij
nnis called the synthetic preference rela-
tion of RðkÞ¼rij;ðkÞ
nn, and kkis the weights of RðkÞ,
which satisfies kk0 and Pm
k¼1kk¼1.
The corresponding additive consistent fuzzy preference
RðkÞ¼ð
rij;ðkÞÞnnof RðkÞ¼rij;ðkÞ
nncan be obtained by
Eq. (14), where
rij;ðkÞ¼
0:5Qn
l¼1ril;ðkÞ
1=nQn
l¼1rjl;ðkÞ
1=n
Pn
h¼1Qn
l¼1rhl;ðkÞ
1=n
þ0:5;for all i;j¼1;2;...;n:
ð18Þ
Neural Comput & Applic
123
If the weighting vector of
RðkÞ¼ð
rij;ðkÞÞnnis
k¼ðk1;k2;...;kmÞT, from Definition 6, we can get the
synthetic preference relation
Rc¼
rc
ij
nn, where
rc
ij ¼X
m
k¼1
kk
rij;ðkÞ;for all i;j¼1;2;...;n:ð19Þ
Theorem 4 The synthetic preference relation
Rc¼
rc
ij
nnis an additive consistent fuzzy preference relation.
Proof According to Theorem 2,
RðkÞ¼ð
rij;ðkÞÞnnis
additive consistent, so for all i;j;s¼1;2;...;nand
k¼1;2;...;m, we have 0
rij;ðkÞ1,
rij;ðkÞþ
rji;ðkÞ¼1,
and
ris;ðkÞ
rjs;ðkÞþ0:5¼
rij;ðkÞ. Since
rc
ij ¼Pm
k¼1kk
rij;ðkÞ,
kk0 and Pm
k¼1kk¼1, it is obvious that 0
rc
ij 1 and
rc
ij þ
rc
ji ¼X
m
k¼1
kk
rij;ðkÞþ
rji;ðkÞ
¼X
m
k¼1
kk¼1:
Moreover, we can get
rc
is
rc
js þ0:5¼X
m
k¼1
kk
ris;ðkÞX
m
k¼1
kk
rjs;ðkÞþ0:5
¼X
m
k¼1
kk
ris;ðkÞ
rjs;ðkÞþ0:5
¼X
m
k¼1
kk
rij;ðkÞ¼
rc
ij:
Thus,
Rc¼
rc
ij
nnis an additive consistent fuzzy
preference relation.
The definition of compatibility degree between two
fuzzy preference relations was given by Xu [23] as follows:
Definition 7 Assume that A¼aij
nnand B¼bij
nn
are two fuzzy preference relations. The compatibility
degree between Aand Bis defined as follows:
CðA;BÞ¼ 1
n2X
n
i¼1X
n
j¼1
jaij bijj:
Inspired by Definition 7, a new definition of compati-
bility degree is defined as follows:
Definition 8 Let A¼aij
nnand B¼bij
nnbe two
fuzzy preference relations, then
CðA;BÞ¼ 1
n2X
n
i¼1X
n
j¼1
ðaij bijÞ2ð20Þ
is called the compatibility degree between Aand B.
As we can see, the compatibility degree of fuzzy pref-
erence relations Aand Bcan reflect the total difference
between Aand B. According to Definition 8, the compat-
ibility degree between
Rcand Rccan be written as follows:
CðRc;
RcÞ¼ 1
n2X
n
i¼1X
n
j¼1X
m
k¼1
kkrij;ðkÞX
m
k¼1
kk
rij;ðkÞ
!
2
¼1
n2X
n
i¼1X
n
j¼1X
m
k¼1
kkrij;ðkÞ
rij;ðkÞ
!
2
¼X
m
k1¼1X
m
k2¼1
kk1kk2
1
n2X
n
i¼1X
n
j¼1
rij;ðk1Þ
rij;ðk1Þ
rij;ðk2Þ
rij;ðk2Þ
!:
ð21Þ
The compatibility degree between
Rcand 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;
RcÞ, the better the group consensus is.
Let E¼ek1k2
ðÞ
mm, where ek1k2¼1
n2Pn
i¼1Pn
j¼1rij;ðk1Þ
rij;ðk1ÞÞrij;ðk2Þ
rij;ðk2Þ
,i;j¼1;2;...;n. Then, Eis
called the compatibility information matrix. In particular, if
k1¼k2¼k, then ekk ¼1
n2Pn
i¼1Pn
j¼1rij;ðkÞ
rij;ðkÞ
2.
Therefore, we can obtain the following optimal model to
calculate the weights vector of RðkÞ(k¼1;2;;m) based
on group consensus:
min CðRc;
RcÞ¼kTEk
s:t:QTk¼1
k0
ð22Þ
where Q¼ð1;1;...;1ÞTand k¼ðk1;k2;...;kmÞT.
By solving the model (22), we can obtain the optimal
solution k
ksatisfying k
k0 and Pm
k¼1k
k¼1, which can
reflect the importance degree of RðkÞ(k¼1;2;...;m).
5.2 The final priority vector derived by DEA
In group decision making, although we can get the priority
vector wðkÞ¼ðw1k;w2k;...;wnk ÞTfrom fuzzy preference
relation RðkÞ¼rij;ðkÞ
nnusing Theorem 1, and calculate
the weights of RðkÞbased on model (22)(k¼1;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 kkand the pri-
ority vectors wðkÞ¼ðw1k;w2k;;wnkÞT(k¼1;2;...;m).
The dual model of the input-oriented DEA is as follows:
Neural Comput & Applic
123
max Z¼vTYp
s:t:
uTXp¼1
uTXTvTYT0
u0;v0
8
>
<
>
:;ð23Þ
where u¼ðu1;u2;...;usÞT,v¼ðv1;v2;...;vmÞT. For
alternative set X¼fx1;x2;...;xng, each alternative may be
viewed as a decision-making unit, each priority vector
wðkÞ¼ðw1k;w2k;...;wnkÞTcan be viewed as an output. The
dummy inputs take the value of 1=nfor all the alternatives.
The relationships among inputs, outputs, and the priority
vectors wðkÞ¼ðw1k;w2k;...;wnkÞTare shown in Table 2.
From Table 2and dual model (23), we build the fol-
lowing DEA model to evaluate the efficiency score of the
alternative xpðp¼1;2;...;nÞ.
max Zp¼m1wp1þm2wp2þþmmwpm
s:t:
u11
n¼1
v1w11 þv2w12 þvmw1m1
v1w21 þv2w22 þvmw2m1
v1wn1þv2wn2þvmwnm 1
u1;v1;v2;...;vm0
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
ð24Þ
Because each output has different importance degree with
the weighting vector k¼ðk1;k2;;kmÞT.Letvi¼v1ki
k1
(i¼1;2;...;m), then model (24) can be rewritten as follows:
max Zp¼m1
k1
k1wp1þk2wp2þþkmwpm
s:t:
u11
n¼1
m1
k1
k1w11 þk2w12 þþkmw1m
ðÞ1
m1
k1
k1w21 þk2w22 þþkmw2m
ðÞ1
m1
k1
k1wn1þk2wn2þþkmwnm
ðÞ1
u1;v1;v2;...;vm0
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:ð25Þ
If we evaluate different alternatives based on model
(25), the DEA models have the same optimal solution m
1,
which satisfies
max
i¼1;2;;n
m
1
k1
k1wi1þk2wi2þþkmwim
ðÞ
¼1:
Then, the optimal objective function value of model (25)
is Z
p¼m
1Pm
i¼1
ki
k1wpi. Thus, the final weight of each
alternative can be calculated by
wi¼Z
i
Pn
p¼1Z
p
;i¼1;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 nbe a set of random variables, where n¼fnij;ðkÞ:
i;j¼1;2;...;n;i\j;k¼1;2;...;mg,nij;ðkÞsatisfies
uniform distribution in ~
rij;ðkÞ¼½rL
ij;ðkÞ;rU
ij;ðkÞ.~
R¼
~
rij;ðkÞji;j¼1;2;...;n;k¼1;2;...;m
is the interval
preference relation, and fðnij;ðkÞÞis the probability density
function of nij;ðkÞ, where nij;ðkÞis the judgments of xiover xj
provided by expert dk(k¼1;2;...;m).
For n, it is assumed that wðnÞis the corresponding final
priority vector. Inspired by Zhu and Xu [30], we provide a
ranking function, which is defined as follows:
RankiðwðnÞÞ¼1þX
n
k¼1
dðwðnÞ
k[wðnÞ
iÞ¼r;ð27Þ
where the rank of xiis r,dðtrueÞ¼1, and dðfalseÞ¼0.
Moreover, the Eq. (27) defines a group preference relations
space Rr
iðnÞ.
Rr
iðnÞ¼fn2~
RjrankiðwðnÞÞ¼1þX
n
k¼1
dðwðnÞ
k[wðnÞ
iÞ¼rg:
ð28Þ
According to the final priority weights vector wðnÞand
the probability density function fðnÞ, we can obtain the
expected priority vector of the alternative set
X¼fx1;x2;...;xng:
Table 2 Inputs, outputs, and
the priority vectors Output 1 (d1) Output 2 (d2) Output m(dm) Dummy input
DMU1w11 w12 w1m1=n
DMU2w21 w22 w2m1=n
DMUnwn1wn2 wnm 1=n
Neural Comput & Applic
123
EðwðnÞÞ¼Z~
R
fðnÞwðnÞdn:ð29Þ
From EðwðnÞÞ, we further get the expected rank of
xiði¼1;2;;nÞ, which can be computed as follows:
Ranke
i¼1þX
n
k¼1
dðEðwðnÞÞk[EðwðnÞÞiÞ;ð30Þ
where EðwðnÞÞkis the kth component of EðwðnÞÞ.
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.
pe
i¼Zn:rankiðwðnÞÞ¼EðriÞ
fðnÞdn:ð31Þ
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;...;dmggive their
interval fuzzy preference relations ~
RðkÞ¼ð
~
rij;ðkÞÞnn
(i;j¼1;2;...;n;k¼1;2;...;m) on the alternative set
X¼fx1;x2;...;xng, respectively.
Step 2 Obtain the fuzzy preference relation matrices
RðkÞ¼rij;ðkÞ
nn,(k¼1;2;...;m), which generated
from ~
RðkÞ¼ð
~
rij;ðkÞÞnn, where rij;ðkÞis randomly gener-
ated from the uniform distribution of ½rL
ij;ðkÞ;rU
ij;ðkÞ,(ij;
i¼1;2;...;n;j¼1;2;...;n;k¼1;2;...;m), and
rji;ðkÞ¼1rij;ðkÞ.
Step 3 Transform RðkÞ¼rij;ðkÞ
nn(k¼1;2;...;m) into
acceptable consistent fuzzy preference relations
RðCÞ
ðkÞ¼rðCÞ
ij;ðkÞ
nn(k¼1;2;...;m) according to Algo-
rithm 1, and calculate k¼ðk1;k2;...;kmÞT, which is the
weight vector of fRð1Þ;Rð2Þ;...;RðmÞgaccording to
model (22).
Step 4 Compute the priority vector wðkÞ¼
ðw1k;w2k;...;wnkÞTof RðCÞ
ðkÞ¼rðCÞ
ij;ðkÞ
nnaccording to
Theorem 1 (k¼1;2;...;m).
Step 5 Obtain the final priority vector w¼
ðw1;w2;...;wnÞTaccording to model (25) and Eq. (26).
Step 6 Repeat the Step 2-Step 5 Ntimes, and execute
stochastic simulation to get the outcomes of SGPA, i.e.,
EðwðnÞÞ, ranke
iand pe
i.
In order to determine the proper number of iterations in
stochastic simulation process, Pula [14] introduced a gen-
eral method based on Kolmogorov’s inequality. Let
j0¼min j1
e2P1
k¼j
2k
2k\d
no
, where eis simulation error, d
is the expected accuracy. Then, the required number of
iterations is N¼2j01. Especially, when e¼0:01,
d¼0:072, then j0¼11 and N¼210 ¼1024:
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;x4g
for a chair professor position. The numerical example is
implemented in Matlab.
Assume that there are four interval fuzzy preference
relations ~
Rð1Þ,~
Rð2Þ,~
Rð3Þ, and ~
Rð4Þgiven by the experts d1,
d2,d3, and d4, respectively, which are shown as follows:
~
Rð1Þ¼
½0:5;0:5½0:6;0:7½0:6;0:7½0:3;0:4
½0:3;0:4½0:5;0:5½0:5;0:6½0:2;0:3
½0:3;0:4½0:4;0:5½0:5;0:5½0:5;0:6
½0:6;0:7½0:7;0:8½0:4;0:5½0:5;0:5
0
B
B
@1
C
C
A
~
Rð2Þ¼
½0:5;0:5½0:5;0:6½0:6;0:7½0:3;0:4
½0:4;0:5½0:5;0:5½0:7;0:7½0:2;0:4
½0:3;0:4½0:3;0:3½0:5;0:5½0:5;0:6
½0:6;0:7½0:6;0:8½0:4;0:5½0:5;0:5
0
B
B
@1
C
C
A
~
Rð3Þ¼
½0:5;0:5½0:5;0:7½0:7;0:9½0:3;0:5
½0:3;0:5½0:5;0:5½0:5;0:7½0:2;0:4
½0:1;0:3½0:3;0:5½0:5;0:5½0:5;0:6
½0:5;0:7½0:6;0:8½0:4;0:5½0:5;0:5
0
B
B
@1
C
C
A
~
Rð4Þ¼
½0:5;0:5½0:6;0:7½0:7;0:8½0:3;0:4
½0:6;0:7½0:5;0:5½0:6;0:7½0:2;0:3
½0:2;0:3½0:3;0:4½0:5;0:5½0:7;0:8
½0:3;0:4½0:7;0:8½0:2;0:3½0:5;0:5
0
B
B
@1
C
C
A
Let the threshold value CI ¼0:1, and execute stochastic
group preference analysis according to Step 1–Step 6.
According to Monte Carlo simulation, the obtained EðwðnÞÞ
are shown in Table 3with different number of iterations.
From Table 3, when the number of iterations is equal or
greater than 160,EðwðnÞÞis robust. The outcomes of SGPA are
shown in Table 4with the number of iterations being 1024.
Moreover, Fig. 2shows the confidence degrees of the
four candidates for each rank by a three-dimensional col-
umn chart.
From the expected priority vector EðwðnÞÞwith the
number of iterations being 500, the rank of four candidates
is x4x1x2x3. As we can see, x4is the best candi-
date. Meanwhile, if the expected rank has high confidence
degree, the expected rank should be reliable. Particularly,
x4is the best alternative with 0.930 confidence degree.
Neural Comput & Applic
123
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
QðyÞ¼y2, the attitudinal character of Q is k¼1
3. Then, the
expected value fuzzy preference relations FQð~
RðkÞÞof ~
RðkÞ
(k¼1;2;3;4) are derived based on OWA operator as follows:
FQð~
Rð1ÞÞ¼
0:50:6333 0:6333 0:3333
0:3667 0:50:5333 0:2333
0:3667 0:4667 0:50:5333
0:6667 0:7667 0:4667 0:5
0
B
B
B
@1
C
C
C
A;
FQð~
Rð2ÞÞ¼
0:50:60:6333 0:3333
0:40:50:70:2667
0:3667 0:30:50:5333
0:6667 0:7333 0:4667 0:5
0
B
B
B
@1
C
C
C
A;
FQð~
Rð3ÞÞ¼
0:50:5667 0:7667 0:3667
0:4333 0:50:5667 0:2667
0:2333 0:4333 0:50:5333
0:6333 0:7333 0:4667 0:5
0
B
B
B
@1
C
C
C
A;
FQð~
Rð4ÞÞ¼
0:50:6333 0:7333 0:3333
0:3667 0:50:6333 0:2333
0:2667 0:3667 0:50:7333
0:6667 0:7667 0:2667 0:5
0
B
B
B
@1
C
C
C
A:
From Eqs. (31) and (32) in Chen and Zhou [4], it is
assumed that c¼0:5, we get the collective fuzzy prefer-
ence relations matrix A¼ðaijÞ44, i.e.,
(1)
R
(2)
R
()m
R
()
(1) ,(1)ij nn
Rr
×
=
()
(2) ,(2)ij nn
Rr
×
=
()
() ,()mijm
nn
Rr
×
=
Obtain the weight vector
of preference relationships
according to model (22)
12
(, , , )
T
m
λλλ λ
=
()
,1,2,,
k
Rk m=
Consistency adjustment
according to Algorithm 1
()
() ()
(1) ,(1)ij nn
Rr
ΓΓ
×
=
()
() ()
(2) ,(2)ij nn
Rr
ΓΓ
×
=
()
() ()
() ,()mijm
nn
Rr
ΓΓ
×
=
Obtain the priority vector
based on Theorem 1
(1) 11 1
(,, )
T
n
ww w=
(2) 12 2
(,, )
T
n
www=
() 1
(,, )
T
mmnm
www=
Obtain the final weights
according to Model (25)
and Eq. (26)
12
(, ,, )
T
n
www w=
Obtain the expected
priority vector
and the associated
confidence degree
according to stochastic
simulation
()
()Ew
ξ
e
i
p
Fig. 1 Group decision-making method based on DEA and stochastic simulation
Table 3 Expected priority vector EðwðnÞÞwith different number of
iterations
Number of iterations EðwðnÞÞ
EðwðnÞÞ1EðwðnÞÞ2EðwðnÞÞ3EðwðnÞÞ4
40 0.2649 0.2251 0.2219 0.2879
80 0.2664 0.2251 0.2193 0.2891
120 0.2658 0.2261 0.2193 0.2887
160 0.2657 0.2263 0.2192 0.2886
200 0.2659 0.2268 0.2187 0.2885
240 0.2661 0.2270 0.2185 0.2882
280 0.2663 0.2258 0.2190 0.2887
320 0.2666 0.2253 0.2195 0.2885
360 0.2666 0.2252 0.2195 0.2886
400 0.2664 0.2256 0.2194 0.2886
440 0.2665 0.2252 0.2194 0.2887
480 0.2667 0.2251 0.2193 0.2888
520 0.2668 0.2251 0.2194 0.2886
560 0.2669 0.2251 0.2194 0.2884
600 0.2669 0.2251 0.2194 0.2884
640 0.2669 0.2246 0.2197 0.2885
680 0.2668 0.2250 0.2194 0.2885
720 0.2667 0.2248 0.2197 0.2886
760 0.2666 0.2247 0.2198 0.2887
800 0.2665 0.2253 0.2195 0.2885
840 0.2666 0.2253 0.2195 0.2884
880 0.2666 0.2252 0.2196 0.2884
920 0.2667 0.2254 0.2195 0.2883
960 0.2666 0.2255 0.2195 0.2884
1000 0.2666 0.2255 0.2195 0.2884
Table 4 Outcomes of SGPA
with the number of iterations
being 1024
EðwðnÞÞRanke
ipe
i
x10.2666 2 0.908
x20.2255 3 0.550
x30.2195 4 0.572
x40.2884 1 0.930
Neural Comput & Applic
123
A¼
0:50:5996 0:6512 0:3265
0:4004 0:50:5885 0:2531
0:3488 0:4115 0:50:5519
0:6735 0:7469 0:4481 0:5
0
B
B
@1
C
C
A:
According to A, we obtain the priority vector
w¼ð0:2564;0:2284;0:2343;2806ÞT. Thus, x4x1
x2x3. 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 pref-
erence 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-ori-
ented 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.
Fig. 2 Confidence degrees of
the four candidates
Neural Comput & Applic
123
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 Foun-
dation (Nos. 1508085QG149, 1608085QF133).
Compliance with ethical standards
Conflict of interest The authors declare that they have no conflict of
interest.
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