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Fuzzy Optimization and Decision Making
https://doi.org/10.1007/s10700-018-9297-0
Group decision making based on DEA cross-efficiency
with intuitionistic fuzzy preference relations
Jinpei Liu1,2 ·Jingmiao Song1·Qin Xu3·Zhifu Tao4·Huayou Chen5
© Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract
The aim of this paper is to investigate a novel approach to group decision making
based on DEA cross-efficiency with intuitionistic fuzzy preference relations, which
can avoid information distortion and obtain more credible decision making results.
An interval transform function is defined, which can transform an intuitionistic fuzzy
preference relation into an interval multiplicative preference relation. Then, an inter-
val transform function based data envelopment analysis model is developed to obtain
the ranking vector of consistent intuitionistic fuzzy preference relation, in which each
of the alternatives is viewed as a decision making unit. Moreover, for any intuition-
istic fuzzy preference relations, we propose two DEA cross-efficiency models to get
the cross-efficiency values of all alternatives, and we can calculate the normalized
intuitionistic fuzzy priority weight vector of the intuitionistic fuzzy preference rela-
tion based on the cross-efficiency values. A goal programming model is investigated
to derive the weight vector of decision makers. A step-by-step procedure for group
decision making approach based on DEA cross-efficiency with intuitionistic fuzzy
preference relations is presented. Finally, numerical examples are given to illustrate
the validity and applicability of the proposed method. This is the first attempt of
employing the DEA cross-efficiency to the group decision making with intuitionistic
fuzzy preference relations.
Keywords Group decision making ·Intuitionistic fuzzy preference relation ·DEA
cross-efficiency ·Interval transform function
1 Introduction
Group decision making can be simplified as a group of decision makers to make a
choice from a set of alternatives for action in accordance with the opinions provided
BJinpei Liu
liujinpei2012@163.com
Extended author information available on the last page of the article
123
J. Liu et al.
by the members. In group decision making process, the decision makers often need to
provide their preferences over a set of alternatives or criteria by pairwise comparison.
The analytic hierarchy process (AHP) furnishes a convenient framework to derive
multiplicative preference relations (MPRs), which uses a 1/9-9 ratio scale to measure
the intensity of the pairwise comparison (Saaty 1980). Besides, the 0–1 scale is adopted
by the fuzzy preference relation (Herrera-Viedma et al. 2004;Xu2007a) to express
the decision maker’s preference information.
However, in many real decision making environment, it is very difficult for the
decision makers to express their preferences using a single crisp value due to that (1)
the decision makers are lack of knowledge and information associated with the problem
(Xu and Cai 2010) and (2) the decision makers are not confident about their judgment,
i.e., there exists hesitation regarding their preferences (Wu et al. 2018). To circumvent
this issue, Atanassov’s intuitionistic fuzzy set (IFS) was introduced in Atanassov
(1986), which is an extension of Zadeh’s fuzzy set and uses membership degree,
non-membership degree and hesitation degree to express decision maker’s subjective
preference. This IFS provides a powerful framework, which can not only characterize
vagueness, uncertainty, but also represent inevitably imprecise or not totally reliable
judgments well. From then on, the IFSs have been proven to be very meaningful and
practical, and attached great importance by many scholars (Xu and Liao 2015).
Based on IFS, the intuitionistic fuzzy preference relation (IFPR) was defined in Xu
(2007b), which is more powerful and useful than multiplicative preference relation
and the fuzzy preference relation. Xu (2007b) firstly defined the concepts of con-
sistent IFPR, incomplete IFPR and acceptable IFPR. How to derive priority vector
from the fuzzy preference relation has become a hot issue of study. Since then, many
prioritization methods for IFPRs have been developed. Wang (2013) proposed a lin-
ear goal programming model for deriving intuitionistic fuzzy weights from IFPR by
minimizing its deviation from the converted additive consistent IFPR. Besides, a con-
vergent iterative algorithm to improve the consistency of IFPR was developed by Xu
and Xia (2014). Liao et al. (2015) and Meng et al. (2017) introduced two consistency
adjustment methods to yield the acceptable consistent IFPRs respectively. However,
these methods of deriving consistent IFPRs have changed the original evaluation infor-
mation given by the decision makers, which may distort the given decision making
information and make the decision result less reliable.
In addition, Ramanathan (2006) introduced the data envelopment analytic hierarchy
process (DEAHP) model for weight derivation from multiplicative preference relation,
but the DEAHP model is not applicable for some inconsistent MPRs. Then, Wang
et al. (2008) proposed the DEA/AR model, which can overcome the shortcoming of
the DEAHP model. Furthermore, Liu et al. (2017) developed a novel output-oriented
data envelopment analysis (DEA) model to obtain the priority vector for the consistent
fuzzy preference relation. The existing research shows that the DEA cross-efficiency
can overcome the problem of extreme weights and incomplete ranking in traditional
DEA by mutual evaluation of decision making units (DMUs) (Liang et al. 2008). In
fact, an interesting and important issue to be solved is how to develop a DEA cross-
efficiency model to derive the priority weight vector from IFPR, which does not require
repairing the IFPR. Up to now, there has been no investigation about this issue.
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Group decision making based on DEA cross-efficiency with…
Recently, Wu et al. (2018) developed the definition of the multiplicative transitivity
of IFPR, and discovered the substantial relationship between interval-valued fuzzy
preference relation and IFPR. Another interesting issue is to discover the substantial
relationship between IFPR and interval multiplicative preference relations (IMPRs).
In this paper, we will propose a new approach to group decision making with IFPRs
based on DEA cross-efficiency, which can avoid information distortion and obtain
more credible decision making result. An interval transform function is defined to
establish the substantial relationship between IFPR and IMPR. Based on the interval
transform function, we develop two DEA cross-efficiency models to get the peer-
evaluation cross-efficiency values of all alternatives, in which each of the alternatives
is viewed as a decision making unit. Then, the normalized intuitionistic fuzzy weight
vector for the IFPR is derived based on the cross-efficiency values. Furthermore, a
goal programming model is investigated to yield the weights of decision makers.
Finally, a novel approach to group decision making based on DEA cross-efficiency
with IFPRs is proposed, which does not need consistency adjustments and can yield a
normalized intuitionistic fuzzy weight vector. In particular, when the IFPRs given by
the experts have a poor consistency, our method has better applicability and can derive
more reasonable ranking result than some known methods. This is the first attempt of
employing the theory of DEA cross-efficiency for the ranking of preference relations.
Hopefully, the proposed work is not only a generalization of existing theory but also
an initial step for the development of group decision making based on DEA cross-
efficiency.
The paper is organized as follows. Some basic concepts are introduced in Sect. 2,
such as the multiplicative preference relation, interval multiplicative preference rela-
tion, the intuitionistic fuzzy preference relation and the consistency of MPR, IMPR
and IFPR. In Sect. 3, we define the interval transform function, develop two DEA
cross-efficiency models to get the peer-evaluation cross-efficiency values of all alter-
natives and propose the deriving method of the normalized intuitionistic fuzzy weight
vector based on cross-efficiency values. In Sect. 4, a goal programming model is inves-
tigated to yield the weights of decision makers and a step-by-step procedure for group
decision making approach is presented. There are numerical examples and compar-
ison analysis in Sect. 5to illustrate the advantages, validity and applicability of the
proposed methods. Finally, Sect. 6concludes the paper, and discusses some future
research directions.
2 Preliminaries
In this section, we provide a brief review on some basic concepts, including multiplica-
tive preference relation (MPR), interval multiplicative preference relation (IMPR), the
intuitionistic fuzzy preference relation (IFPR) and the consistency of MPR, IMPR and
IFPR.
For a decision making problem, it is assumed that X{x1,x2,...,xn}is a finite set
of alternatives. In the process of decision making, the decision maker usually provides
pairwise judgments on any two alternatives over the set X.
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J. Liu et al.
Definition 1 (Saaty 1980) A reciprocal multiplicative preference relation Ron Xis
characterized by a matrix R(rij)n×n⊂X×Xwith
rij ·rji 1,rij ≥0,rii 1,i,j1,2,...,n,
where rij is interpreted as the ratio of the preference intensity of alternative xito that
of xj.
The reciprocal multiplicative preference relation is also called the multiplicative
preference relation. And the consistency of MPR has been proposed by Saaty (1977)
as follows:
Definition 2 (Saaty 1977)AMPR R(rij)n×nis consistent if
rik ·rkj rij,i,j,k1,2,...,n.(1)
For a MPR R(rij)n×n, and w(w1,w
2,···,w
n)Tis a normalized priority
weight vector of R, then Ris consistent if and only if
rij wiwj,i,j1,2,...,n,(2)
where wi>0, i1,2,···,n, and n
i1wi1.
In order to express the decision makers’ uncertain preferences, Saaty and Vargas
(1987) introduced the definition of interval multiplicative preference relation.
Definition 3 (Saaty and Vargas 1987; Xia and Chen 2015)AnIMPR ˜
Ris defined as
follows:
˜
R(˜rij)n×n⎛
⎜
⎜
⎝
[1,1] [r−
12,r+
12]··· [r−
1n,r+
1n]
[r−
21,r+
21][1,1] ··· [r−
2n,r+
2n]
··· ··· ··· ···
[r−
n1,r+
n1][r−
n2,r+
n2]··· [1,1]
⎞
⎟
⎟
⎠
,
where r−
ij,r+
ij >0, such that r−
ij ≤r+
ij,r−
ijr+
ji 1 and r+
ijr−
ji 1, i,j1,2,...,n.
When r−
ij r+
ij for all i,j1,2,...,n,˜
Rdegenerates to a MPR.
The consistency of IMPR was defined as follows.
Definition 4 (Wang 2015)AnIMPR ˜
R(˜rij)n×nwith ˜rij [r−
ij,r+
ij](i,j
1,2,...,n) is consistent if it satisfies the following transitivity condition:
r−
ikr+
ikr−
kjr+
kj r−
ijr+
ij,i,j,k1,2,...,n.(3)
According to Definition 4, it is evident that IMPR ˜
R(˜rij)n×nwith ˜rij [r−
ij,r+
ij]
(i,j1,2,...,n) is consistent if and only if
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Group decision making based on DEA cross-efficiency with…
ρ(˜rik)ρ(˜rkj)ρ(˜rij),i,j,k1,2,...,n,(4)
where ρ(˜rij)r−
ij ·r+
ij.
In 1986, intuitionistic fuzzy set was introduced by Atanassov (1986), which was
defined as follows.
Definition 5 (Atanassov 1986)Let X{x1,x2,...,xn}be an ordinary finite non-
empty set. An IFS ˜
Aover Xis represented as ˜
A{<x,μ˜
A(x),ν˜
A(x)>|x∈X},
where μ˜
A:X→[0,1], ν˜
A:X→[0,1] and μ˜
A(x)+ν˜
A(x)≤1, for all x∈X.
The numbers μ˜
A(x) and ν˜
A(x) represent the membership degree and non-membership
degree of the element x∈Xto ˜
A, respectively. π˜
A(x)1−μ˜
A(x)−ν˜
A(x) is called
the hesitation degree of xto ˜
A. Obviously, 0 ≤π˜
A(x)≤1 for all x∈X.Ifπ˜
A(x)0,
we have μ˜
A(x)+ν˜
A(x)1. Then, ˜
Ais degraded to an ordinary fuzzy set.
For convenience, αμα,ν
αis called an intuitionistic fuzzy value (IFV) (Saaty
1977), where μα,ν
α∈[0,1] and μα+να≤1.
Definition 6 (Meng et al. 2017)Letαμα,ν
αbe an IFV, then the score function
and accuracy function of αare defined by S(α)uα−vαand H(α)uα+vα,
respectively. Suppose α1and α2are two IFVs, then
(1) If S(α1)>S(α2), then α1is larger than α2, denoted by α1>α
2;
(2) If S(α1)S(α2), then
(a) If H(α1)>H(α2), then α1>α
2;
(b) If H(α1)H(α2), then α1α2, which means that α1is equal to α2.
BasedonIFV,Xu(2007b) gave a standard definition of intuitionistic fuzzy prefer-
ence relation, which is shown as follows:
Definition 7 (Xu 2007b) An IFPR over a finite set of alternatives X
{x1,x2,...,xn}is represented by a matrix ˜
P(˜pij)n×n, where ˜pij <
(xi,xj),μ(xi,xj),ν(xi,xj),π(xi,xj)>for all i,j1,2,...,n. For convenience,
let ˜pij μij,ν
ijbe an IFV, where μij defines the certainty degree that the
alternative xiis preferred to xj,νij indicates the certainty degree that the alter-
native xiis non-preferred to xjand 0 ≤μij,ν
ij ≤1, 0 ≤μij +νij ≤1,
μij νji,μji νij,μii νii 0.5 for all i,j1,2,···,n.
Definition 8 (Wu et al. 2018) An IFPR ˜
P(˜pij)n×nwith ˜pij μij,ν
ij
(i,j1,2,...,n) is multiplicative consistent (multiplicative transitive) if and only
if ∀i,j,k1,2,...,n:
μijμjkμki μikμkjμji ,
(1 −νij)(1 −νjk)(1 −νki )(1 −νik)(1 −νkj)(1 −νji).
Definition 9 (Wang 2013) An intuitionistic fuzzy weight vector ˜w
(˜w1,˜w2,···,˜wn)Twith ˜wiwμ
i,w
ν
iT,0 ≤wμ
i,w
v
i≤1 and wμ
i+wν
i≤1
for i1,2,...,nis said to be normalized if it satisfies the following conditions:
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J. Liu et al.
n
j1,ji
wμ
j≤wν
i,w
μ
i+n−2≥
n
j1,ji
wν
j,i1,2,...,n.
3 Deriving priority vector of IFPR based on DEA cross-efficiency
In this section, we investigate a new approach of deriving the normalized intuition-
istic fuzzy priority weight vector from IFPR based on DEA cross-efficiency. Unless
otherwise mentioned, it is assumed that Ωis the set of interval values and Ψis the set
of intuitionistic fuzzy values (IFVs).
3.1 Interval transform function of IFPR
In order to extract effective information of the intuitionistic fuzzy value sufficiently, we
propose the definition of interval transform function, which can be defined as follows.
Definition 10 Let αμα,ν
αbe an IFV with μα,ν
α>0. Then, an interval trans-
form function of IFV is a mapping τ:Ψ+→Ω+, such that
τ(α)τ(μα,ν
α)μα
1−μα
,1−να
να,(5)
where Ψ+{
μα,ν
α|μα>0,ν
α>0,μ
α+να≤1}is the set of IFVs, in which
both the membership degree and the non-membership degree of IFVs are all positive.
The interval transform function defined by Definition 10 can transform an IFV into
an interval value. Then, we show a characterized theorem for the interval transform
function.
Theorem 1 Let ˜
P(˜pij)n×nμij,ν
ijn×nbe an IFPR and μij,ν
ij >0.
(1) ˜
H[h−
ij,h+
ij]n×nτ(˜pij)n×nis an IMPR.
(2) If ˜
P is multiplicative consistent, then ˜
H[h−
ij,h+
ij]n×nτ(˜pij)n×nis a
consistent IMPR.
Proof Let ˜
P(˜pij)n×nμij,ν
ijn×nbe an IFPR, μij,ν
ij >0 and ˜
H
τ(˜pij)n×n. From Definition 10,wehave[h−
ij,h+
ij]τ(˜pij)[μij
1−μij ,1−νij
νij ]. Since
μij,ν
ij >0, μij +νij ≤1, μij νji,μji νij and μii νii 0.5 for all
i,j1,2,...,n,wehaveh−
ij μij
1−μij ,h+
ij 1−νij
νij >0, 1 −μij ≥νij and
1−νij ≥μij. Thus, h−
ij μij
1−μij ≤1−νij
νij h+
ij,
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Group decision making based on DEA cross-efficiency with…
h−
ijh+
ji μij
1−μij
·1−νji
νji
μij
1−μij
·1−μij
μij
1,
h−
jih+
ij μji
1−μji
·1−νij
νij
μji
1−μji
·1−μji
μji
1,(6)
and
h−
ii ,h+
iiτ(˜pii )μii
1−μii
,1−νii
νii 0.5
1−0.5,1−0.5
0.5[1,1].(7)
According to Eqs. (6), (7) and Definition 3, it is obvious that ˜
H[h−
ij,h+
ij]n×n
is an IMPR.
If ˜
Pis multiplicative consistent, from Definition 8,wehave
μijμjkμki μikμkj μji
(1 −νij)(1 −νjk)(1 −νki )(1 −νik)(1 −νkj )(1 −νji)for all i,k,j1,2,...,n.
(8)
Thus, one can obtain
μijμjkμki (1 −νij)(1 −νjk)(1 −νki)μikμkjμji(1 −νik)(1 −νkj )(1 −νji).
(9)
Equation (9) can be rewritten as
μik
1−νki
·1−νik
μki
·μkj
1−νjk
·1−νkj
μjk
μij
1−νji
·1−νij
μji
.(10)
By Eq. (10), it is obvious that
h−
ik ·h+
ik ·h−
kj ·h+
kj h−
ijh+
ij,i,k,j1,2,...,n.(11)
According to Definition 4,˜
Hτ(˜pij)n×nis a consistent IMPR.
Theorem 1shows that the interval transform function can not only transform the
IFPR into IMPR, but also ensure the consistency of preference relation.
3.2 DEA model of IFPR
DEA is a powerful tool to evaluate the relative efficiency of a set of homogenous deci-
sion making units (DMUs) with multiple inputs and multiple outputs by establishing
a nonparametric linear programming model. In this section, we propose a DEA model
to evaluate the relative efficiency of each alternative from IFPR.
In decision making process, decision maker offers an IFPR ˜
P(˜pij)n×n
μij,ν
ijn×nover the alternative set X{x1,x2,...,xn}, where μij,ν
ij >0.
Each alternative xican be viewed as an independent DMU, i1,2,...,n. When the
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J. Liu et al.
Table 1 Inputs and outputs of DEA based on ˜
P(˜pij)n×nμij ,ν
ijn×n
Output 1 Output 2 ··· Output nDummy
input
x1DMU1τ(˜p11 )τ(˜p12 ) ··· τ(˜p1n)τ(˜p0)
x2DMU2τ(˜p21 )τ(˜p22 ) ··· τ(˜p2n)τ(˜p0)
··· ··· ··· ··· ··· ··· ···
xnDMUnτ(˜pn1)τ(˜pn2) ··· τ(˜pnn )τ(˜p0)
decision maker believes that alternative xiis better than xj, for any k∈{1,2,...,n},
the certainty degree that the alternative xiis preferred to xkis greater than the certainty
degree that the alternative xjis preferred to xk, i.e., μik ≥μjk, and certainty degree
that the alternative xiis non-preferred to xkis smaller than the certainty degree that the
alternative xjis non-preferred to xk, i.e., νik ≤νjk.Thus,wehave ˜pik μik,ν
ik≥
˜pjk μjk,νjk. Furthermore, since μik
1−μik ≥μjk
1−μjk and 1−νik
νik ≥1−νjk
νjk , one can
obtain τ(˜pik)≥τ(˜pjk). Hence, each row of the IFPR corresponds to a DMU, and
each column of the IFPR can be viewed as an output. Since DEA calculations cannot be
made without inputs, to ensure the fairness of decision making, each decision making
unit is given the same intuitionistic fuzzy input variable ˜p00.5,0.5. Then, the
relationships between inputs, outputs and IFPR ˜
P(˜pij)n×nμij,ν
ijn×nare
shown in Table 1.
Based on above analysis and the interval transform function τgiven by Defini-
tion 10, we can develop an output-oriented CCR DEA model to evaluate the relative
efficiency of alternative xk(k1,2,···,n) as follows:
Max
μ,φ φk
s.t.⎧
⎨
⎩n
i1μiτ(˜pij)≥φkτ(˜pkj),j1,2,...,n,
n
i1μiτ(˜p0)≤τ(˜p0),
μi≥0,i1,2,...,n.
(12)
For an IFPR ˜
P(˜pij)n×nμij,ν
ijn×nwith μij,ν
ij >0, if μij+νij1, then
μij
1−μij 1−νij
νij . It follows that τ(˜pij)[h−
ij,h+
ij][μij
1−μij ,1−νij
νij ] degenerates to a real
number. Denoting hij h−
ij h+
ij μij
1−μij 1−νij
νij ,˜
Hτ(˜pij)n×ndegenerates to
aMPRH(hij)n×n. Based on Model (12), we have the following Theorem.
Theorem 2 Let ˜
P(˜pij)n×nμij,ν
ijn×nbe a multiplicative consistent IFPR
with μij,ν
ij >0, μij+νij1, and φ∗
kbe the optimal objective function value of
Model (12). Then, the normalized priority vector of ˜
Hτ(˜pij)n×ncan be derived
by
wk(φ∗
k)−1n
l1
(φ∗
l)−1,k1,2,...,n.(13)
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Group decision making based on DEA cross-efficiency with…
Proof Let ˜
P(˜pij)n×nμij,ν
ijn×nbe a multiplicative consistent IFPR
with μij,ν
ij >0, μij+νij1. According to Theorem 1,˜
H[h−
ij,h+
ij]n×n
τ(˜pij)n×nis a consistent IMPR. Then, we have
h−
ijh+
ji 1,h−
ikh+
ikh−
kjh+
kj h−
ijh+
ij,i,j,k1,2,...,n.(14)
Denoting hij h−
ij h+
ij μij
1−μij 1−νij
νij ,fromEq.(14), one can obtain
hijhji 1,hikhkj hij,i,j,k1,2,...,n.
Consequently, ˜
Hτ(˜pij)n×ndegenerates to consistent MPR H(hij)n×n,
where hij τ(˜pij). According to Eq. (2), the priority weight vector of ˜
H
τ(˜pij)n×nsatisfies
hij τ(˜pij)wiwj,i,j1,2,...,n.(15)
From Eq. (15), Model (12) can be rewritten as
Max
μ,φ φk
s.t.⎧
⎨
⎩n
i1μiwi/wk≥φk,j1,2,...,n,
n
i1μi≤1,
μi≥0,i1,2,...,n.
(16)
Without loss of generality, it is assumed that w1≤w2≤ ··· ≤ wn. The objective
function φkis maximal if and only if the constraint n
i1μi≤1 reduces to equality,
i.e., n
i1μi1.
For any k∈{1,2,···,n},byμk1−μ1−··· −μk−1−μk+1 − ···− μn,the
first constraint of Model (16) becomes
1+
n
ik,i1
μiwiwk−1≥φk.(17)
Because w1wk−1≤w2wk−1≤ ··· ≤ wnwk−1, it follows that φkis
maximal if and only if μ1μ2 ··· μn−10, μn1. Thus, the optimal
objective function value of Model (12)isφ∗
kwnwk.
Furthermore, since wi>0, i1,2,...,n, and n
i1wi1, we have wk
(φ∗
k)−1n
l1(φ∗
l)−1,k1,2,...,n.
In fact, Theorem 1provides a method of deriving the priority weight vector of
consistent IFPR using the proposed DEA model. However, when μij+νij 1, this
method is not applicable. In order to solving this problem, based on interval aggregation
function ρ, we propose the following CCR DEA model:
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J. Liu et al.
Max
μ,φ φk
s.t.⎧
⎨
⎩n
i1μiρτ(˜pij)≥φkρτ(˜pkj),j1,2,...,n,
n
i1μiρ(τ(˜p0))≤ρ(τ(˜p0)),
μi≥0,i1,2,...,n.
(18)
where τ(˜pij)[h−
ij,h+
ij][μij
1−μij ,1−νij
νij ] and ρτ(˜pij)h−
ij ·h+
ijμij(1−νij)
νij(1−μij).
In case of ˜
P(˜pij)n×nμij,ν
ijn×nbeing a multiplicative consistent IFPR,
according to Theorem 1, it is evident that ˜
H[h−
ij,h+
ij]n×nτ(˜pij)n×nis a
consistent IMPR. Moreover, by Eq. (4), we have ρ(τ(˜pik)) ·ρ(τ(˜pkj)) ρ(τ(˜pij)),
i,j,k1,2,...,n, which implies that ˆ
Hρ(τ(˜pij))n×nis a consistent MPR.
According to the proof of Theorem 2, we have the following proposition.
Proposition 1 Let ˜
P(˜pij)n×nμij,ν
ijn×nbe a multiplicative consistent IFPR
with μij,ν
ij >0and φ∗
kbe the optimal objective function value of Model (18). Then,
the priority weight vector of ˆ
Hρ(τ(˜pij))n×ncan be derived by
wk(φ∗
k)−1n
l1
(φ∗
l)−1,k1,2,...,n.(19)
For any multiplicative consistent IFPR ˜
P(˜pij)n×nμij,ν
ijn×nover the
alternative set X{x1,x2,...,xn}, the inverse of optimal objective function value of
model (18), (φ∗
k)−1, is the efficiency score of alternative xk,k1,2,...,n. Moreover,
the corresponding priority weight vector of ˜
Pcan be derived by Proposition 1.
3.3 Deriving the intuitionistic fuzzy weights based on DEA cross-efficiency
It is worth noting that deriving priority weight vector based Model (18) is not applicable
for inconsistent IFPR. To resolve this problem, we will develop a new method of
deriving the normalized intuitionistic fuzzy priority weight vector from an IFPR based
on DEA cross-efficiency. The cross-efficiency DEA uses the DMUs to evaluate each
other, and the final efficiency evaluation value of the DMU is obtained from the self-
evaluation and the peer-evaluation (Liang et al. 2008).
Based on Table 1and interval aggregation function ρ, the input-oriented CCR DEA
model is formulated as the following linear programming:
Max αkk
n
r1
ωrkρ(τ(˜pkr ))
s.t.⎧
⎨
⎩n
r1ωrkρτ(˜pjr)−ηkρ(τ(˜p0))≤0,j1,2,...,n,
ηkρ(τ(˜p0))1,
ωrk,η
k≥0,k1,2,...,n.
(20)
123
Group decision making based on DEA cross-efficiency with…
For each alternative xk(k1,2,...,n) under evaluation, we can obtain
the optimal objective value (efficiency value) α∗
kk and a set of optimal weights
η∗
k,ω
∗
1k,ω
∗
2k,...,ω
∗
nk associated with inputs and outputs.
Note that the optimal weights obtained from Model (20) may not be unique. Inspired
by the benevolent DEA cross-efficiency model and aggressive DEA cross-efficiency
model (Doyle and Green 1994), we propose two new DEA cross-efficiency models
for IFPR. Since τ(˜pij)[h−
ij,h+
ij][μij
1−μij ,1−νij
νij ] and τ(˜p0)1, we construct
the following two mathematical programming model for each alternative xk(k
1,2,···,n).
Max
n
j1
n
r1
urkh−
jr
s.t.⎧
⎪
⎪
⎨
⎪
⎪
⎩
vk·n1,
n
r1urkh+
kr −α∗
kkvk0,
n
r1urkh+
kr −vk≤0,j1,2,...,n;j k,
vk,urk ≥0,k1,2,...,n.
(21)
Min
n
j1
n
r1
urkh−
jr
s.t.⎧
⎪
⎪
⎨
⎪
⎪
⎩
vk·n1,
n
r1urkh+
kr −α∗
kkvk0,
n
r1urkh+
kr −vk≤0,j1,2,...,n;j k,
vk,urk ≥0,k1,2,...,n.
(22)
In Models (21) and (22), α∗
kk is the target efficiency of alternative xk, which is
obtained from Model (20). As can be seen, Model (21) aims to maximize the lower
bound of overall cross-efficiency values of all alternatives and Model (22)aimsto
minimize the lower bound of overall cross-efficiency values of all alternatives. The
second equality constraint of Models (21) and (22), n
r1urkh+
kr −α∗
kkvk0, is used
to ensure that the upper bound efficiency value of alternative xkis equal to its optimal
efficiency α∗
kk, i.e., n
r1urkh+
kr vkα∗
kk.
By solving Models (21) and (22), we can obtain the optimal weights u∗
(u∗
1k,u∗
2k,...,u∗
nk)Tand v∗
k, which are the associated output weights and input weight.
Thus, the cross-efficiency of alternative xj(j1,2,...,n) using the optimal weights
of alternative xk(k1,2,...,n) can be calculated as in Eq. (23),
θ−
jk
n
r1
u∗
rkh−
jrv∗
kand θ+
jk
n
r1
u∗
rkh+
jrv∗
k,j1,2,...,n,(23)
where θ−
jk and θ+
jk are the lower bound and upper bound of the cross-efficiency,
respectively. Therefore, we get an interval cross-efficiency value of alternative xj,
i.e., ˜
θjk [θ−
jk,θ+
jk], which represents the peer-evaluation of alternative xkto alter-
native xj.
123
J. Liu et al.
In Models (21) and (22), while alternative xk(k1,2,...,n) is changed, one can
get the interval cross-efficiency matrix, namely,
Θ(˜
θjk)n×n⎛
⎜
⎜
⎝
[θ−
11,θ+
11][θ−
12,θ+
12]··· [θ−
1n,θ+
1n]
[θ−
21,θ+
21][θ−
22,θ+
22]··· [θ−
2n,θ+
2n]
··· ··· ··· ···
[θ−
n1,θ+
n1][θ−
n2,θ+
n2]··· [θ−
nn,θ+
nn]
⎞
⎟
⎟
⎠
.
Furthermore, we aggregate the cross-efficiency values of alternative xj(j
1,2,...,n) based on Eq. (24), and obtain the average cross-efficiency value ˜
θj.
˜
θjθ−
j,θ+
j1
n
n
k1
θ−
jk,1
n
n
k1
θ+
jk ,j1,2,...,n.(24)
Based on the average cross-efficiency values of all alternatives, we can calculate
the intuitionistic fuzzy priority weight vector ˜w(˜w1,˜w2,..., ˜wn)Twith ˜wj
!wμ
j,w
ν
j"(j1,2,...,n), based on the following formulas:
wμ
jθ−
j
ψ,w
ν
j1
ψ
n
r1,r j
θ+
r,j1,2,...,n,(25)
where ψn
r1θ+
r+1
n−2max
k#θ+
k−θ−
k$.
It is worth noting that the intuitionistic fuzzy priority weight vector obtained from
Eq. (25) satisfies the following theorem.
Theorem 3 For any IFPR ˜
P(˜pij)n×n, its intuitionistic fuzzy priority weight vector
˜w(˜w1,˜w2,..., ˜wn)T,which is obtained from Eq. (25), is normalized.
Proof According to Formula (25), since 0 ≤θ−
j≤θ+
j≤1, it follows that wμ
j,
wν
i∈[0,1] and
wμ
j+wν
jθ−
j+n
r1,r jθ+
r
n
r1θ+
r+1
n−2max
k#θ+
k−θ−
k$≤θ−
j+n
r1,r jθ+
r
n
r1θ+
r
≤1.
Note that n
i1,i jθ−
i
n
r1θ+
r+1
n−2max
k#θ+
k−θ−
k$≤n
r1,r jθ+
r
n
r1θ+
r+1
n−2max
k#θ+
k−θ−
k$,wehave
n
i1,i jwμ
i≤wν
j.
For any j1,2,...,n,wehave
wμ
j+n−2
θ−
j+(n−2) n
r1θ+
r+max
k#θ+
k−θ−
k$
n
r1θ+
r+1
n−2max
k#θ+
k−θ−
k$,
and
123
Group decision making based on DEA cross-efficiency with…
n
i1,i j
wν
in
i1,i jn
r1,riθ+
r
n
r1θ+
r+1
n−2max
k#θ+
k−θ−
k$(n−1) n
r1θ+
r−n
i jθ+
i
n
r1θ+
r+1
n−2max
k#θ+
k−θ−
k$.
Thus, on can get
wμ
j+n−2−
n
i1,i j
wν
i
max
k#θ+
k−θ−
k$−(θ+
j−θ−
j)
n
r1θ+
r+1
n−2max
k#θ+
k−θ−
k$≥0.
The Proof is completed.
Based on the afore-mentioned discussion, we have the ranking method of IFPR
˜
P(˜pij)n×nμij,ν
ijn×n, which involves the following steps:
Step 1 For IFPR ˜
P(˜pij)n×nμij,ν
ijn×n, use Model (20) to obtain α∗
kk,
k1,2,···,n.
Step 2 Set α∗
kk in the DEA cross-efficiency Models (21)or(22), and solve the
Model (21) or Model (22) to obtain the optimal solutions u∗
1k,u∗
2k,...,u∗
nk,v∗
k,
k1,2,...,n.
Step 3 Use Formula (23) to compute the lower bound and upper bound of the
cross-efficiency, then get the interval cross-efficiency matrix Θ(˜
θjk)n×n, where
˜
θjk [θ−
jk,θ+
jk], j,k1,2,...,n.
Step 4 Calculate the average cross-efficiency value ˜
θj[θ−
j,θ+
j] of alternative xj
based on formula (24), j1,2,...,n.
Step 5 Calculate the normalized intuitionistic fuzzy priority weight vector ˜w
(˜w1,˜w2,..., ˜wn)Tby using formula (25), where ˜wiwμ
i,w
ν
i.
Step 6 Rank the normalized intuitionistic fuzzy priority weights ˜wjin descending
order, j1,2,...,n.
Step 7 Rank the alternatives xj(j1,2,...,n) according to the ranking of
˜wj,j1,2,···,n.
Step 8 End.
Example 1 Consider the alternative set X{x1,x2,x3,x4}, and the IFPR ˜
Pgiven by
the decision maker is as follows.
˜
P⎛
⎜
⎜
⎝
0.5,0.5
0.6,0.2
0.8,0.2
0.6,0.3
0.2,0.6
0.5,0.5
0.5,0.3
0.3,0.5
0.2,0.8
0.3,0.5
0.5,0.5
0.3,0.6
0.3,0.6
0.5,0.3
0.6,0.3
0.5,0.5
⎞
⎟
⎟
⎠
Step 1 According to Model (20), we can get the relative efficiency value of xk,
k1,2,3,4, i.e.,
α∗
11 1,α
∗
22 0.4082,α
∗
33 0.2857,α
∗
44 0.6236.
123
J. Liu et al.
Step 2 Set α∗
kk in the cross-efficiency DEA Model (21), and solve the Model (21)
to obtain the optimal solutions.
u∗
11 0.2500,u∗
21 0.0000,u∗
31 0.0000,u∗
41 0.0000,v
∗
10.2500;
u∗
12 0.0000,u∗
22 0.0328,u∗
32 0.0297,u∗
42 0.0000,v
∗
20.2500;
u∗
13 0.0000,u∗
23 0.0000,u∗
33 0.0000,u∗
43 0.1071,v
∗
30.2500;
u∗
14 0.1208,u∗
24 0.0000,u∗
34 0.0323,u∗
44 0.0000,v
∗
40.2500.
Step 3 Based on formula (23), we get the interval cross-efficiency matrix
Θ⎛
⎜
⎜
⎝
[1.0000,1.0000] [0.6717,1.0000] [0.6429,1.0000] [1.0000,1.0000]
[0.2500,0.6667] [0.2500,0.4082] [0.1837,0.4286] [0.2500,0.6236]
[0.2500,0.2500] [0.1750,0.2500] [0.1837,0.2857] [0.2500,0.2500]
[0.4286,0.6667] [0.3093,0.5833] [0.4286,0.4286] [0.4009,0.6236]
⎞
⎟
⎟
⎠
.
Step 4 From Θ, we can get the average cross-efficiency value of each alternative
respectively based on Formula (24), i.e.,
˜
θ1[0.8286,1.0000],˜
θ2[0.2334,0.5318],
˜
θ3[0.2147,0.2589],˜
θ4[0.3918,0.5755].
Step 5 According to Formula (25), we get the normalized intuitionistic fuzzy priority
weight vector, i.e.,
˜w(0.3294,0.5431,0.0928,0.7293,0.0853,0.8378,0.1558,0.7119)T.
Step 6 Based on Definition 6, we can get the ranking of normalized intuitionistic
fuzzy priority weight vector, which is as follows:
˜w1>˜w4>˜w2>˜w3.
Step 7 Thus, the rank of the alternatives xjaccording to the ranking of ˜wj(j
1,2,3,4) is as follows:
x1>x4>x2>x3.
Similarly, if we use Model (22) instead of Model (21) in Step 2, we can get
Θ⎛
⎜
⎜
⎝
[0.3750,1.0000] [0.6124,0.6124] [0.6429,1.0000] [0.6771,1.0000]
[0.2500,0.2500] [0.1531,0.4082] [0.1837,0.4286] [0.2500,0.4514]
[0.1071,0.2500] [0.1531,0.1531] [0.1837,0.2857] [0.1762,0.2500]
[0.2500,0.5833] [0.2624,0.4082] [0.4286,0.4286] [0.3363,0.6236]
⎞
⎟
⎟
⎠
,
123
Group decision making based on DEA cross-efficiency with…
Moreover, we have the average cross-efficiency value of four alternatives
˜
θ1[0.5768,0.9031],˜
θ2[0.2092,0.3846],
˜
θ3[0.1550,0.2347],˜
θ4[0.3193,0.5109].
Further, we can get
˜w(0.2626,0.5146,0.0952,0.7506,0.0706,0.8189,0.1454,0.6931)T.
It can be seen that we have the same ranking result of the derived intuitionistic
fuzzy priority weights, which is ˜w1>˜w4>˜w2>˜w3.
In a comparison of our method with some existing ranking methods of IFPR, our
proposed method has the following advantages.
First, the given IFPRs need an inconsistency repairing process to yield the accept-
able consistent IFPRs in Xu and Xia (2014), Liao et al. (2015) and Meng et al. (2017).
The inconsistency repairing have changed the original evaluation information given
by the decision maker, which can distort the given decision making information and
make the decision result less reliable. As to our ranking method, it is based entirely
on the original preference information given by decision maker and does not require
the consistency adjustment, so the result is more credible.
Second, in Wu et al. (2018), Liao and Xu (2014) and Wang (2013), it is assumed that
there exist specific relationships between the priority weight vector and the consistent
IFPRs. Then, the optimization models are established based on these assumptions to
get the priority weight vector. Since these assumptions are based on consistent IFPR,
the obtained priority weight vectors are always unreasonable for some inconsistent
IFPRs. Besides, these assumptions are not unique. By contrast, our approach does
not need to make any assumptions in advance, so it can avoid the risk of assumption
dependent.
Third, in our ranking method, the DEA cross-efficiency models combine the results
of self-evaluation and peer-evaluation to get the final efficiency value of each alterna-
tive, which makes the ranking result fairer.
4 Group decision making with intuitionistic fuzzy preference
relations
Let D{d1,d2,...,dm}be the set of decision makers and X{x1,x2,...,xn}
be the set of alternatives. Suppose that dlprovides his/her preference on X
{x1,x2,...,xn}using IFPR ˜
P(l)(˜p(l)
ij )n×n!μ(l)
ij ,ν(l)
ij "n×n,l1,2,...,m,
and λ(λ1,λ
2,...,λ
m)Tis the weight vector of decision makers satisfying λl≥0
and m
l1λl1, in which λlcan reflect the importance degree of dl.
In order to measure the similarity between two IFPRs, the definition of compatibility
degree was introduced by Xu (2013) as follows.
123
J. Liu et al.
Definition 11 Let ˜
P(l)!μ(l)
ij ,ν(l)
ij "n×nand ˜
P(k)!μ(k)
ij ,ν(k)
ij "n×nbe two
IFPRs, given by two decision makers dland dk,k,l1,2,...,m. Then, we call
c(˜
P(k),˜
P(l))n
i1n
j1μ(k)
ij μ(l)
ij +v(k)
ij v(l)
ij
max%n
i1n
j1[(μ(k)
ij )2+(ν(k)
ij )2],n
i1n
j1[(μ(l)
ij )2+(ν(l)
ij )2]&
the compatibility degree of ˜
P(l)and ˜
P(k).
It can be seen easily that the greater the value of c(˜
P(k),˜
P(l)), the nearer the
two IFPRs ˜
P(k)and ˜
P(l)will be. Moreover, we have 0 ≤c(˜
P(k),˜
P(l))≤1, and
c(˜
P(k),˜
P(l))1 if and only if ˜
P(k)˜
P(l), which means that ˜
P(l)and ˜
P(k)are
perfectly compatible.
Besides, based on the proposed interval transform function τ,˜
P(l)(˜p(l)
ij )n×n
!μ(l)
ij ,ν(l)
ij "n×ncan be transformed into IMPR ˜
H(l)[h(l),−
ij ,h(l),+
ij ]n×n, where
τ(˜p(l)
ij )h(l),−
ij ,h(l),+
ij μ(l)
ij
1−μ(l)
ij
,1−ν(l)
ij
ν(l)
ij ,l1,2,...,m,i,j1,2,...,n.
(26)
Then, the IFPRs provided by mdecision makers can be aggregated into a synthetic
interval preference relation Hs[hs,−
ij ,hs,+
ij ]n×n, where
hs,−
ij
m
'
l1h(l),−
ij λland hs,+
ij
m
'
l1h(l),+
ij λl,i,j1,2,...,n.(27)
It is evident that Hsis an IMPR. Note that the closer Hsand each of ˜
H(l)(l
1,2,...,m), the more representative the synthetic preference relation Hsis. Thus,
we develop the following goal programming to obtain the weight vector of decision
makers.
Min J
n
i1
n
ji+1
m
l1ε(l),+
ij +ε(l),−
ij +γ(l),+
ij +γ(l),−
ij
s.t.⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
ln h(l),−
ij −m
l1λlln h(l),−
ij −ε(l),+
ij +ε(l),−
ij 0,i<j,i,j1,2,...n,l1,2,...,m,
ln h(l),+
ij −m
l1λlln h(l),+
ij −γ(l),+
ij +γ(l),−
ij 0,i<j,i,j1,2,...n,l1,2,...,m,
m
l1λl1,
λl,ε
(l),+
ij ,ε
(l),−
ij ,γ(l),+
ij ,γ(l),−
ij ≥0,i<j,i,j1,2,...n,l1,2,...,m.
(28)
As can be seen, the objective function J, which reflects group consensus, can
measure the agreement between the individual preference relations and the synthetic
preference relation. Especially, when all ˜
P(l)(1,2,...,m) are perfectly compatible,
the following result holds:
123
Group decision making based on DEA cross-efficiency with…
Theorem 4 For any l,k∈{1,2,...,m},c(˜
P(l),˜
P(k))1if and only if J ∗0,
where J ∗is the optimal objective function value of Model (28).
Proof (Sufficiency) If c(˜
P(l),˜
P(k))1 for any l,k∈{1,2,...,m}, then ˜
P(1)
˜
P(2) ··· ˜
P(m). According to Eq. (26), we have h(1),−
ij h(2),−
ij ··· h(m),−
ij
and h(1),+
ij h(2),+
ij ··· h(m),+
ij ,i,j1,2,...n. Thus, all deviation variables
ε(l),+
ij ,ε(l),−
ij ,γ(l),+
ij and γ(l),−
ij are equal to 0 in Model (28), and therefore J∗0.
(Necessity) If J∗0, then it is evident that h(1),−
ij h(2),−
ij ··· h(m),−
ij and
h(1),+
ij h(2),+
ij ··· h(m),+
ij ,i,j1,2,···n. Based on Eq. (26), we have ˜
P(1)
˜
P(2) ··· ˜
P(m). Thus c(˜
P(l),˜
P(k))1 holds for any l,k∈{1,2,...,m}.
In conclusion, the process of group decision making method based on DEA cross-
efficiency with intuitionistic fuzzy preference relations can be summarized as follows:
Step 1 Use interval transform function τto translate ˜
P(l)(˜p(l)
ij )n×n
!μ(l)
ij ,ν(l)
ij "n×ninto ˜
H(l)[h(l),−
ij ,h(l),+
ij ]n×nbased on Eq. (26), l
1,2,...,m;
Step 2 Solve Model (28) to get the optimal weight vector λ(λ1,λ
2,...,λ
m)T
of decision makers;
Step 3 Aggregate ˜
H(l)[h(l),−
ij ,h(l),+
ij ]n×ninto the synthetic preference relation
Hs[hs,−
ij ,hs,+
ij ]n×nbased on Eq. (27);
Step 4 For multiplicative preference relation Hs[hs,−
ij ,hs,+
ij ], obtain the nor-
malized intuitionistic fuzzy priority weight vector ˜(˜1,˜2,..., ˜n)Tbased
on the ranking method of IFPR in Sect. 3.3, where ˜iμ
i,ν
i,i1,2,...,n;
Step 5 Rank the normalized intuitionistic fuzzy priority weights ˜iin the descend-
ing order based on Definition 6,i1,2,...,n.
Step 6 Rank the alternatives xi(i1,2,...,n) according to the ranking of ˜i,
i1,2,···,n.
Step 7 End.
Let ˜
P(l)(˜p(l)
ij )n×n!μ(l)
ij ,ν(l)
ij "n×n(l1,2,...,m)bemIFPRs, which are
given by mdecision makers respectively, over the alternative set X{x1,x2,...,xn},
where μij,ν
ij >0. Generally speaking, when all the decision makers think that
alternative xiis better than xj, for any k∈{1,2,...,n}, then μ(l)
ik ≥μ(l)
jk and ν(l)
ik ≤
ν(l)
jk,l1,2,...,m. Then, we have the following theorem.
Theorem 5 Let ˜
P(l)(˜p(l)
ij )n×n!μ(l)
ij ,ν(l)
ij "n×n(l1,2,...,m)bemIFPRs,
which are given by mdecision makers respectively. If for any k∈{1,2,...,n}and
l∈{1,2,...,m},μ(l)
ik ≥μ(l)
jk and ν(l)
ik ≤ν(l)
jk hold, then ˜i≥˜j,i,j1,2,...,n.
123
J. Liu et al.
Proof Let ˜
P(l)(˜p(l)
ij )n×n!μ(l)
ij ,ν(l)
ij "n×n(l1,2,...,m)bemIFPRs. From
Eqs. (26) and (27), we have τ(˜p(l)
ik )[h(l),−
ik ,h(l),+
ik ][μ(l)
ik
1−μ(l)
ik
,1−ν(l)
ik
ν(l)
ik
], τ(˜p(l)
jk)
[h(l),−
jk ,h(l),+
jk ][μ(l)
jk
1−μ(l)
jk
,1−ν(l)
jk
ν(l)
jk
], hs,−
ik
m
(
l1h(l),−
ik λland hs,+
jk
m
(
l1h(l),+
jk λl.
If μ(l)
ik ≥μ(l)
jk and ν(l)
ik ≤ν(l)
jk for all k∈{1,2,...,n}and l∈{1,2,...,m},it
follows that hs,−
ik ≥hs,−
jk and hs,+
ik ≥hs,+
jk . Assume that u∗(u∗
1k,u∗
2k,...,u∗
nk)T
and v∗
kare the optimal weights obtained by Model (21)or(22), since
θ−
jk
n
r1
u∗
rkhs,−
ik v∗
k,and θ+
jk
n
r1
u∗
rkhs,+
ik v∗
k,j1,2,...,n,
we have θ−
ik ≥θ−
jk and θ+
ik ≥θ+
jk. According to Eq. (24), it is evident that θ−
i≥
θ−
jand θ+
i≥θ+
j.FromEq.(25), we have μ
iθ−
i
ψ,μ
jθ−
j
ψ,ν
i1
ψn
r1,riθ+
r
and ν
j1
ψn
r1,r jθ+
r, where ψn
r1θ+
r+1
n−2max
k#θ+
k−θ−
k$. Thus, the score
function of ˜iand ˜jare S(˜i)1
ψ(θ−
i−n
riθ+
r) and S(˜j)1
ψ(θ−
j−
n
r jθ+
r), respectively. Consequently, we have
S(˜i)−S(˜j)1
ψ(θ−
i−θ−
j)+(θ+
i−θ+
j)≥0,
which implies that ˜i≥˜j.
Theorem 5shows that the obtained priority vector from our method is in accord
with the decision makers’ preferences. In other words, the ranking result can reflect
the preference of all the decision makers.
5 Numerical examples and comparison analysis
5.1 Numerical example
In the background of knowledge economy, the competition in the market is not just the
competition between enterprises, but the competition between supply chain alliances.
The supply chain is a complex system built by interdependent organizations and pro-
cesses, and the evaluation criteria of the supply chain involve customer service level,
financial status, innovation level, production flexibility, etc. The evaluation process is
somewhat subjective as the experts may not be very familiar with all the situation of
the supply chains they evaluated. Thus, it is suitable for the decision makers to use the
IFPRs to express their evaluation information. There are five decision makers d1,d2,
d3,d4and d5who provide their IFPRs on four supply chains {A1,A2,A3,A4}.After
doing some pairwise comparisons, the decision makers offer five IFPRs as follows:
123
Group decision making based on DEA cross-efficiency with…
˜
P(1) ⎛
⎜
⎜
⎝
0.50,0.50
0.50,0.20
0.70,0.10
0.50,0.30
0.20,0.50
0.50,0.50
0.60,0.20
0.30,0.60
0.10,0.70
0.20,0.60
0.50,0.50
0.30,0.60
0.30,0.50
0.60,0.30
0.60,0.30
0.50,0.50
⎞
⎟
⎟
⎠
,
˜
P(2) ⎛
⎜
⎜
⎝
0.50,0.50
0.60,0.20
0.60,0.20
0.60,0.30
0.20,0.60
0.50,0.50
0.50,0.20
0.30,0.50
0.20,0.60
0.20,0.50
0.50,0.50
0.20,0.60
0.30,0.60
0.50,0.30
0.60,0.20
0.50,0.50
⎞
⎟
⎟
⎠
,
˜
P(3) ⎛
⎜
⎜
⎝
0.50,0.50
0.55,0.25
0.65,0.20
0.35,0.55
0.25,0.55
0.50,0.50
0.40,0.25
0.55,0.30
0.20,0.65
0.25,0.40
0.50,0.50
0.60,0.20
0.55,0.35
0.30,0.55
0.20,0.60
0.50,0.50
⎞
⎟
⎟
⎠
,
˜
P(4) ⎛
⎜
⎜
⎝
0.50,0.50
0.60,0.30
0.70,0.20
0.60,0.30
0.30,0.60
0.50,0.50
0.50,0.40
0.30,0.40
0.20,0.70
0.40,0.50
0.50,0.50
0.30,0.60
0.30,0.60
0.40,0.30
0.60,0.30
0.50,0.50
⎞
⎟
⎟
⎠
,
˜
P(5) ⎛
⎜
⎜
⎝
0.50,0.50
0.60,0.30
0.75,0.15
0.60,0.20
0.30,0.60
0.50,0.50
0.45,0.20
0.60,0.20
0.15,0.75
0.20,0.45
0.50,0.50
0.40,0.40
0.20,0.60
0.20,0.60
0.40,0.40
0.50,0.50
⎞
⎟
⎟
⎠
.
Step 1 BasedonEq.(26), ˜
P(1),˜
P(2),˜
P(3),˜
P(4) and ˜
P(5) are transformed into ˜
H(l)
(l1,2,3,4,5) respectively, where
˜
H(1) ⎛
⎜
⎜
⎝
[1.0000,1.0000] [1.0000,4.0000] [2.3333,9.0000] [1.0000,2.3333]
[0.2500,1.0000] [1.0000,1.0000] [1.5000,4.0000] [0.4286,0.6667]
[0.1111,0.4286] [0.2500,0.6667] [1.0000,1.0000] [0.4286,0.6667]
[0.4286,1.0000] [1.5000,2.3333] [1.5000,2.3333] [1.0000,1.0000]
⎞
⎟
⎟
⎠
,
˜
H(2) ⎛
⎜
⎜
⎝
[1.0000,1.0000] [1.5000,4.0000] [1.5000,4.0000] [1.5000,2.3333]
[0.2500,0.6667] [1.0000,1.0000] [1.0000,4.0000] [0.4286,1.0000]
[0.2500,0.6667] [0.2500,1.0000] [1.0000,1.0000] [0.2500,0.6667]
[0.4286,0.6667] [1.0000,2.3333] [1.5000,4.0000] [1.0000,1.0000]
⎞
⎟
⎟
⎠
,
˜
H(3) ⎛
⎜
⎜
⎝
[1.0000,1.0000] [1.2222,3.0000] [1.8571,4.0000] [0.5385,0.8182]
[0.3333,0.8182] [1.0000,1.0000] [0.6667,3.0000] [1.2222,2.3333]
[0.2500,0.5385] [0.3333,1.5000] [1.0000,1.0000] [1.5000,4.0000]
[1.2222,1.8571] [0.4286,0.8182] [0.2500,0.6667] [1.0000,1.0000]
⎞
⎟
⎟
⎠
,
123
J. Liu et al.
˜
H(4) ⎛
⎜
⎜
⎝
[1.0000,1.0000] [1.5000,2.3333] [2.3333,4.0000] [1.5000,2.3333]
[0.4286,0.6667] [1.0000,1.0000] [1.0000,1.5000] [0.4286,1.5000]
[0.2500,0.4286] [0.6667,1.0000] [1.0000,1.0000] [0.4286,0.6667]
[0.4286,0.6667] [0.6667,2.3333] [1.5000,2.3333] [1.0000,1.0000]
⎞
⎟
⎟
⎠
,
˜
H(5) ⎛
⎜
⎜
⎝
[1.0000,1.0000] [1.5000,2.3333] [3.0000,5.6667] [1.5000,4.0000]
[0.4286,0.6667] [1.0000,1.0000] [0.8182,4.0000] [1.5000,4.0000]
[0.1765,0.3333] [0.2500,1.2222] [1.0000,1.0000] [0.6667,1.5000]
[0.2500,0.6667] [0.2500,0.6667] [0.6667,1.5000] [1.0000,1.0000]
⎞
⎟
⎟
⎠
.
Step 2 By solving Model (28), we get the optimal weight vector of five decision
makers, i.e., λ(0.0883,0.3689,0.0941,0.2657,0.1830)T;
Step 3 Using Eq. (27), ˜
H(l)(l1,2,3,4,5) are aggregated into the synthetic
preference relation
Hs⎛
⎜
⎜
⎝
[1.0000,1.0000] [1.4196,3.0567] [2.0316,4.5796] [1.3142,2.3333]
[0.3271,0.7044] [1.0000,1.0000] [0.9616,3.0000] [0.5949,1.5000]
[0.2184,0.4922] [0.3333,1.0399] [1.0000,1.0000] [0.4286,0.9154]
[0.4286,0.7609] [0.6667,1.6809] [1.0924,2.3333] [1.0000,1.0000]
⎞
⎟
⎟
⎠
.
Step 4 According to the ranking method for IFPR in Sect. 3.3 with Model (21), we
obtain the normalized intuitionistic fuzzy priority weight vector
˜(0.2798,0.5975,0.1036,0.7164,0.0694,0.7929,0.1318,0.7133)T.
Meanwhile, if we use Model (22) instead of Model (21), we have
(0.1866,0.5155,0.1043,0.7059,0.0646,0.7889,0.0923,0.6918)T.
Step 5 Based on Definition 6, the ranking of the normalized intuitionistic fuzzy
priority weight vector ˜in the descending order is as follows:
˜10.2798,0.5975>˜40.1318,0.7133>˜
20.1036,0.7164>˜30.1318,0.7133
Additionally, the ranking of is
˜
10.1866,0.5155>˜
40.0923,0.6918>˜
20.1043,0.7059>˜
30.0646,0.7889
As can be seen, ˜and have the same ranking results.
Step 6 The final ranking of the four alternative supply chains is A1>A4>A2>
A3.
Step 7 Thus, A1is the best supply chain.
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Group decision making based on DEA cross-efficiency with…
5.2 Comparison analysis
This subsection further compares our method with several existing representative
methods, which were developed in Wang (2013), Meng et al. (2017), Xu (2012) and Wu
et al. (2018). First of all, we use these methods to solve the above problem respectively.
(1) Using Wang (2013)’s Method. Using the linear goal programming model (4.18)
in Wang (2013), we obtain the normalized intuitionistic fuzzy weight vector:
˜w(0.4701,0.4709,0.0749,0.6250,0,0.9034,0.1549,0.6626)T.
Moreover, the score functions of the four weights are obtained as follows:
S(˜w1)−0.0008,S(˜w2)−0.5501,S(˜w3)−0.9034,S(˜w4)−0.5077.
So, the final ranking is A1>A4>A2>A3, which is the same as that derived by
our approach.
(2) Using Meng et al. (2017)’s Method. The intuitionistic fuzzy prior-
ity weight vector is derived using Meng et al. (2017)’s method, i.e.,
˜w(0.6146,0.2560,0.3686,0.3861,0.2469,0.5385,0.4080,0.4286).
From Zhang and Xu (2012)forIFVs,wehave
L(˜w1)0.6588,L(˜w2)0.4930,L(˜w3)0.3800,L(˜w4)0.4911.
Hence, the ranking is A1>A2>A4>A3, which is different from the results
derived by our approach, as well as the Wang (2013)’s method as the positions of the
supply chain A2and A4are changed.
Furthermore, we will use the method in Wu et al. (2018) and Xu (2012) to obtain
the priority vector of synthetic IFPR, which is yielded by aggregate the five IFPRs,
i.e., ˜
P(l),l1,2,...,5.
(3) Using Xu (2012)’s Method. We can get the priority weight vector,
˜w([0.2845,0.3059],[0.2290,0.2597],[0.1971,0.2251],[0.2369,0.2617])T.
The following possibility degree matrix is construct to rank the priority weights.
P⎛
⎜
⎜
⎝
0.51 1 1
00.510.4111
000.50
00.5889 1 0.5
⎞
⎟
⎟
⎠
.
By summing all elements in each line of P,wehave p13.5000,p2
1.9111,p30.5000,p42.0889. Hence, the ranking of the four alternative supply
chains is A1>A4>A2>A3, which is the same as that derived by our approach.
123
J. Liu et al.
(4) Using Wu et al. (2018)’s Method. The crisp priority vector is obtained by using
the arithmetic mean operator:
w10.8750,w
20.4430,w
30.1250,w
40.5570.
And thus the final ranking of the four alternative supply chains is A1>A4>A2>
A3, which is the same as that derived by our approach.
The above numerical example shows that our approach produces the same ranking
order, i.e., A1>A4>A2>A3, with Xu’s, Wu et al.’s and Wang’s methods, while
Meng et al.’s method derives a slightly different order in which the positions of A2
and A4are changed, which is a further proof of the credibility and reliability of our
approach.
We further illustrate the advantages of our method by the Example 2 compared with
the four existing representative methods.
Example 2 Let ˜
Pbe an IFPR on alternative set X{x1,x2,x3,x4}, and
˜
P⎛
⎜
⎜
⎝
0.50,0.50
0.35,0.55
0.40,0.40
0.35,0.55
0.55,0.35
0.50,0.50
0.70,0.10
0.60,0.20
0.40,0.40
0.10,0.70
0.50,0.50
0.70,0.20
0.55,0.35
0.20,0.60
0.20,0.70
0.50,0.50
⎞
⎟
⎟
⎠
.
Using our approach, we have
˜w(0.1145,0.7653,0.2034,0.5490,0.1106,0.6983,0.1101,0.7397)T,
then the ranking order is x2>x3>x4>x1.
Using Wang (2013)’s Method, we have
S(˜w1)−0.6237,S(˜w2)0.1239,S(˜w3)−0.6237,S(˜w4)−0.6761,
by which the ranking order is x2>x1x3>x4.
UsingXu(2012)’s method, we have
p11.5309,p23.5000,p32.2753,p40.6938,
which means that the ranking order is x2>x3>x1>x4.
UsingWuetal.(2018)’s Method, we have
w10.2250,w
20.8750,w
30.2750,w
40.6250,
by which the ranking order is x2>x4>x3>x1.
Using Meng et al. (2017)’s Method, since ˜
Phas poor consistency, and it can be
adjusted to a multiplicative consistent IFPR ˜
T.
123
Group decision making based on DEA cross-efficiency with…
˜
T⎛
⎜
⎜
⎝
0.5000,0.5000
0.2799,0.7066
0.3747,0.4243
0.4372,0.3443
0.7066,0.2799
0.5000,0.5000
0.5906,0.2227
0.6517,0.1695
0.4243,0.3747
0.2227,0.5906
0.5000,0.5000
0.5646,0.4161
0.3443,0.4372
0.1695,0.6517
0.4161,0.5646
0.5000,0.5000
⎞
⎟
⎟
⎠
.
Then, from ˜
T,wehave
L(˜w1)0.4526,L(˜w2)0.6891,L(˜w3)0.4680,L(˜w4)0.3910.
Thus, the ranking order is x2>x3>x1>x4.
Comparing the ranking of xi(i1,2,...,4) derived by our approach with those
of Wang (2013), Xu (2012), Wu et al. (2018) and Meng et al. (2017), we find that
these methods generate a different ranking, especially on x1and x4. From the IFPR
matrix ˜
P(˜pij)4×4,wehave ˜p41 0.55,0.35, which means the certainty degree
that x4is preferred to x1is 0.55, while the certainty degree of x4is non-preferred to
x1is 0.35. It is obvious that x1should not be superior to x4. Therefore, the ranking
result of our method is more reasonable than that of Xu (2012), Wu et al. (2018),
Wang (2013) and Meng et al. (2017) in this case. What’s more, from the multiplicative
consistent IFPR ˜
T(˜
tij)4×4adjusted from ˜
P,wehave˜
t41 0.3443,0.4372,
which is quite different from ˜p41 0.55,0.35. The distortion of the given decision
making information makes the decision result less reliable.
As can be seen, compared with Xu (2012), Wu et al. (2018), Wang (2013) and
Meng et al. (2017), our proposed method has the following advantages:
(1) In Meng et al. (2017)’s method, the inconsistent IFPRs need to be repaired to yield
the acceptable consistent IFPRs, which can distort the given decision making
information and make the decision result less reliable. Besides, the inconsistency
repairing process is complex and tedious. As to our method, the decision making
process is based entirely on the original preference information given by decision
makers and does not require consistent improving. Thus, our method can avoid
information distortion and the result is more credible.
(2) In Wu et al. (2018)’s method and Xu (2012)’s method, the priority vector of IFPR
is given in the form of interval, which is not normalized. From Theorem 3,we
find that our method can yield the normalized intuitionistic fuzzy weight vector.
(3) InWuetal.(2018)’s method and Wang (2013)’s method, it is assumed that there
exist some relationships between priority weight vector and consistent IFPRs,
then the optimization models are established based on these assumptions to get the
priority weight vector. Since these assumptions are based on consistent IFPR, the
obtained priority weight vectors are always unreasonable for some inconsistent
IFPRs. Besides, these assumptions are not unique. By contrast, our approach
does not need to make any assumptions in advance, so it can avoid the risk of
assumption dependent.
In summary, our method does not need consistency adjustment, making any assump-
tions and can yield a normalized intuitionistic fuzzy weight vector. In particular, when
the IFPR given by the decision maker has a poor consistency, our method has better
applicability and can derive more reasonable ranking result than some known methods.
123
J. Liu et al.
6 Conclusions
This paper has been studying group decision making based on DEA cross-efficiency
with intuitionistic fuzzy preference relations.
Firstly, the defined interval transform function has established the substantial rela-
tionship between IFPRs into IMPRs, and it has been proved that the transform function
can ensure the consistency of the relationship. Then, an interval transform function
based DEA model is developed to obtain the ranking vector of consistent IFPR, in
which each of the alternatives is viewed as a DMU. Moreover, for any IFPR, we pro-
posed two DEA cross-efficiency models to get the peer-evaluation cross-efficiency
values of all alternatives, and we can calculate the normalized intuitionistic fuzzy
weight vector of the IFPR based on the cross-efficiency values. Because the proposed
normalized intuitionistic fuzzy weight vector derivation method from IFPR does not
need repairing the given IFPR, it can avoid information distortion and obtain more cred-
ible decision making results. A goal programming model is investigated to derive the
weights of decision makers. Meanwhile, a step-by-step procedure for group decision
making approach based on DEA cross-efficiency with intuitionistic fuzzy preference
relations is presented. Finally, the proposed approach is verified by numerical exam-
ples and the comparison analysis, by which one can find the advantages of the new
method.
To the best of our knowledge, employing the theory of DEA cross-efficiency to the
group decision making with IFPRs are completely new to the literature and have not
been studied elsewhere before. Probably, we also can use the DEA cross-efficiency
model to any other decision making environments, such as triangular fuzzy preference
relations, trapezoidal fuzzy preference relation.
Acknowledgements The work was supported by National Natural Science Foundation of China (Nos.
71501002, 61502003, 71871001, 71771001, and 71701001), Anhui Provincial Natural Science Foundation
(Nos. 1608085QF133, 1508085QG149), Anhui Provincial Philosophy and Social Science Planning Youth
Foundation (No. AHSKQ2016D13).
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Affiliations
Jinpei Liu1,2 ·Jingmiao Song1·Qin Xu3·Zhifu Tao4·Huayou Chen5
Jingmiao Song
1253732255@qq.com
Qin Xu
xuqin2013@aliyun.com
Zhifu Tao
zhifut_0514@163.com
Huayou Chen
huayouc@126.com
1School of Business, Anhui University, Hefei 230601, Anhui, China
123
J. Liu et al.
2Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State
University, Raleigh, NC 27695-7906, USA
3School of Computer Science and Technology, Anhui University, Hefei 230601, Anhui, China
4School of Economics, Anhui University, Hefei 230601, Anhui, China
5School of Mathematical Sciences, Anhui University, Hefei 230601, Anhui, China
123
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