A Novel Approach to Supplier Selection based on Vague Sets Group Decision (vol 36, pg 9557, 2009)

Abstract

In practice, in purchasing decision-making, many quantitative and qualitative factors, with vagueness and imprecision, have to be considered. This makes the decision process very complicated and unstructured. Besides the fuzzy sets theory, vague sets theory is one of the methods used to deal with uncertain information. Since vague sets can provide more information than fuzzy sets, it is considered superior in mathematical analysis of uncertain information. In this paper, a new approach based on vague sets group decision is proposed to deal with the supplier selection problem in supply chain systems. The work procedure is shown briefly, as follows: First, linguistic values are used to assess the ratings and weights for quantitative or qualitative factors. Second, degree of similarity and probability of vague sets are used to determine the ranking order of all alternatives. Finally, a numerical example of the selection problem of suppliers is shown, to highlight the procedure of the proposed approach, at the end of this paper.

Full text

An novel approach to supplier selection based on vague sets group decision
Dongfeng Zhang
a,*
, Jinlong Zhang
a
, Kin-Keung Lai
b
, Yaobin Lu
a
a
School of Management, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan, Hubei 430074, China
b
Department of Management Science, City University of Hong Kong, Hong Kong
article info
Keywords:
Supplier selection
Group decision
Vague sets
Linguistic variables
abstract
In practice, in purchasing decision-making, many quantitative and qualitative factors, with vagueness
and imprecision, have to be considered. This makes the decision process very complicated and unstruc-
tured. Besides the fuzzy sets theory, vague sets theory is one of the methods used to deal with uncertain
information. Since vague sets can provide more information than fuzzy sets, it is considered superior in
mathematical analysis of uncertain information. In this paper, a new approach based on vague sets group
decision is proposed to deal with the supplier selection problem in supply chain systems. The work pro-
cedure is shown briefly, as follows: First, linguistic values are used to assess the ratings and weights for
quantitative or qualitative factors. Second, degree of similarity and probability of vague sets are used to
determine the ranking order of all alternatives. Finally, a numerical example of the selection problem of
suppliers is shown, to highlight the procedure of the proposed approach, at the end of this paper.
Ó2008 Elsevier Ltd. All rights reserved.
1. Introduction
With the globalization of economic markets and the develop-
ment of information technology, a well-designed and implemented
supply chain management (SCM) system is now regarded as an
important tool to increase competitive advantage (Choi, Bai, Geun-
es, & Romeijn, 2007; Li, Yamaguchi, & Nagai, 2007). In upstream
echelons of supply chain, vendor selection or supplier evaluation
continues to be a key element in the industrial purchasing process,
and appears to be one of the major activities of the professionals in
the industry (Patton, 1997; Michaels, Kumar, & Samu, 1995). Sup-
plier selection and evaluation is the process of finding the appro-
priate suppliers who are able to provide the buyer with the right
quality products and/or services at the right price, in the right
quantities and at the right time (Mandal & Deshmukh, 1994; Sarkis
& Talluri, 2002).
Evaluation and selection of suppliers is a typical multiple crite-
ria decision-making (MCDM) problem involving multiple criteria
that can be both qualitative and quantitative. The MCDM provides
an effective framework for vendor comparison, based on evalua-
tion of multiple and often conflicting criteria. There is an abun-
dance of supplier evaluation and selection models proposed in
supply chain literature. The main methods are linear weighting
methods (LW) (Thompson, 1990; Timmerman, 1986), analytic
hierarchy process (AHP) (Barbarosoglu & Yazgac, 1997; Narasim-
han, 1983), analytic network process (Sarkis & Talluri, 2000), total
cost approaches (Monezka & Trecha, 1998; Smytka & Clemens,
1993) and mathematical programming (MP) techniques (Buffa &
Jackson, 1983; Chaudhry, Forst, & Zydiak, 1993).
Although linear weighting is a very simple method, it depends
heavily on human judgment and also weighs the attributes equally,
which is rarely true, in practice. While MP techniques cause signif-
icant problems in considering qualitative factors, AHP cannot effec-
tively take into account risk and uncertainty in estimating the
suppliers’ performance because it presumes that the relative
importance of attributes affecting the suppliers’ performance is
known with certainty (Dyer, Fishburn, & Steuer, 1992). The draw-
back of MP is that it requires arbitrary aspiration levels and cannot
accommodate subjective attributes (Khorramshahgol, Azani, &
Gousty, 1988).
Most of these methods do not seem to address the complex and
unstructured nature and context of many present day purchasing
decisions (de Boer, van der Wegen, & Telgen, 1998). In fact, in many
existing decision models, only quantitative criteria are considered
for supplier selection. Several important factors are often not taken
into account in the decision-making process, such as incomplete
information, additional qualitative criteria and imprecise prefer-
ences. Based on the vast literature on supplier selection (Choi &
Hartley, 1996; Weber, Current, & Benton, 1991), we can conclude
that supplier selection may involve several and different types of
criteria, combinations of different decision models, group deci-
sion-making and various forms of uncertainty. It is difficult to find
the best way to evaluate and select supplier, companies use a
0957-4174/$ - see front matter Ó2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.eswa.2008.07.053
*Corresponding author. Tel.: +86 13476153808.
E-mail address: hustzdf@gmail.com (D. Zhang).
Expert Systems with Applications 36 (2009) 9557–9563
Contents lists available at ScienceDirect
Expert Systems with Applications
journal homepage: www.elsevier.com/locate/eswa

variety of different methods to deal with it. Therefore, the most
important issue in the process of supplier selection is to develop
a suitable method to select the right supplier.
In essence, the supplier selection problem in supply chain is a
group decision-making problem, under multiple criteria. The de-
gree of uncertainty, the number of decision-makers (DMs) and
the nature of the criteria have to be taken into account while solv-
ing this problem. The decision-makers always express their prefer-
ences on alternatives or on attributes of suppliers, which can be
used to rank the suppliers or in selecting the most desirable ones.
The preferences on different suppliers and on attributes are DMs’
subjective judgments. In conventional MCDM methods, ratings
and weights of the attributes are known precisely (Delgado, Verde-
gay, & Vila, 1992; Hwang & Yong, 1981; Kaufmann & Gupta, 1991).
Generally, DMs’ judgments are uncertain and cannot be estimated
by exact numerical values. Under many conditions, crisp data are
inadequate to model real-life situations; human judgments,
including preferences, are often vague and preferences cannot be
estimated in exact numerical values.
In recent years, fuzzy set approaches have been proposed to deal
with the supplier selection problem under uncertainty (Wang,
2005). A more realistic approach may be to use linguistic assess-
ments, instead of numerical values. In other words, ratings and
weights of the criteria in the problem are assessed by means of lin-
guistic variables (Bellman & Zadeh, 1970; Chen, 2000; Herrera, Her-
rera-Viedma, & Verdegay, 1996; Herrera & Herrera-Viedma, 2000).
The fuzzy set theory offers a possibility of handing data and infor-
mation involving subjective characteristics of human nature in the
decision-making process. Kickert (1978) has discussed fuzzy mul-
ti-criteria decision-making. Zimmermann (1987) illustrated a fuzzy
set approach to multi-objective decision-making; he has compared
some approaches to solve multi-attribute decision-making prob-
lems based on the fuzzy set theory. Yager (1978) presented a fuzzy
multi-attribute decision-making method that uses crisp weights,
and in 1988, he introduced an ordered weighted aggregation oper-
ator and investigated its properties (Yager, 1988). Laarhoven and
Pedrycz (1983) presented a method for multi-attribute decision-
making, using fuzzy numbers as weights. But these methods have
a shortcoming that the fuzzy set is not fuzzy completely; for exam-
ple, their fuzzy values are also exact. To improve the fuzzy set, Gau
and Buehrer (1993) proposed the vague set theory. Then, based on
the vague set theory, Chen and Tan (1994) presented some new
techniques for handling multi-criteria fuzzy decision-making prob-
lems. Hong and Choi (2000), Jun (2007) extended the study on vague
sets-based MCDM method.
In this paper, we have proposed a new approach based on vague
sets group decisions to deal with the problem of supplier selection
under uncertain environments. In supplier selection process, the
degree of uncertainty of the attributes has to be taken into account
(Chen, Lin, & Huang, 2006). Considering the fuzziness in the deci-
sion data, in the group decision-making process, linguistic vari-
ables that can be expressed in vague values are used, to assess
the weights of all criteria and the ratings of each alternative with
respect to each criterion. Linguistic variables are also used to deter-
mine weights of the importance of different decision-makers; then
we adjust the weights by considering the similarities and differ-
ences among them. After that, we integrate the judgments of all
decision-makers into a final decision matrix. Using probability de-
gree to compare the vague sets of the evaluation object, we obtain
the ranking order of every alternative.
The paper is organized as follows. The next section introduces
the basic definitions and notations of the vague sets. In Section 3,
we present a fuzzy decision-making method based on vague sets
to cope with the supplier selection problem. And then, the pro-
posed method is illustrated with an example. Finally, some conclu-
sions are drawn at the end of this paper.
2. The basic theory
2.1. Vague sets
Let Ube the universe of discourse, with a generic element of U
denoted by u. A vague set Ais characterized by a truth-member-
ship function t
A
and a false-membership function f
A
, where t
A
(u)
is a lower bound on the grade of membership of u, derived from
the evidence for u;f
A
(u) is a lower bound on the negation of u, de-
rived from the evidence against u; and t
A
(u)+f
A
(u)61. The grade
of membership of uin the vague set Ais bound to a subinterval
[t
A
(u),1 f
A
(u)] of [0,1]. The vague value [t
A
(u),1 f
A
(u)] indicates
that the exact grade of membership
l
A
(u)ofumaybe unknown,
but it is bound by t
A
(u)6
l
A
(u)61f
A
(u), where t
A
(u)+f
A
(u)61.
For example, Fig. 1 shows a vague set in the universe of
discourseU.
When the universe of discourse Uis continuous, a vague set A
can be written as
A¼Z
U
½t
A
ðuÞ;1f
A
ðuÞ=uðu2UÞ:
When the universe of discourse Uis discrete, a vague set Acan be
written as
A¼X
n
i¼1
½t
A
ðu
i
Þ;1f
A
ðu
i
Þ=u
i
ðu
i
2UÞ:
2.2. Operation between vague sets
Let x,ybe two vague values in the universe of discourse U,
x=[t
x
,1f
x
], y=[t
y
,1f
y
], where t
x
,f
x
,t
y
,f
y
2[0,1] and
t
x
+f
x
61,t
y
+f
y
61; the operation and relationship between vague
values is defined as follows:
Definition 1. The minimum operation of vague values xand yis
defined by
x^y¼½minðt
x
;t
y
Þ;minð1f
x
;1f
y
Þ
¼½minðt
x
;t
y
Þ;1maxðf
x
;f
y
Þ:
Definition 2. The maximum operation of vague values xand yis
defined by
x_y¼½maxðt
x
;t
y
Þ;maxð1f
x
;1f
y
Þ
¼½maxðt
x
;t
y
Þ;1minðf
x
;f
y
Þ:
Definition 3. The complement of vague value xis defined by

x¼½f
x
;1t
x
:
Fig. 1. Vague set.
9558 D. Zhang et al. / Expert Systems with Applications 36 (2009) 9557–9563

Let A,Bbe two vague sets in the universe of discourse
U={u
1
,u
2
,...,u
n
}, A¼P
n
i¼1
½t
A
ðu
i
Þ;1f
A
ðu
i
Þ=u
i
, and B¼P
n
i¼1
½t
B
ðu
i
Þ;
1f
B
ðu
i
Þ=u
i
, then the operations between vague sets are defined
as follows.
Definition 4. The intersection of vague sets Aand Bis defined by
A\B¼X
n
i¼1
f½t
A
ðu
i
Þ;1f
A
ðu
i
Þ ^ ½t
B
ðu
i
Þ;1f
B
ðu
i
Þg=u
i
:
Definition 5. The union of vague sets Aand Bis defined by
A[B¼X
n
i¼1
f½t
A
ðu
i
Þ;1f
A
ðu
i
Þ _ ½t
B
ðu
i
Þ;1f
B
ðu
i
Þg=u
i
:
Definition 6. The complement of vague set Ais defined by
A¼X
n
i¼1
½f
A
ðu
i
Þ;1t
A
ðu
i
Þ=u
i
:
2.3. Similarity measure between vague sets
Zhang, Huang, and Li (2004) proposed a method to calculate
the similarity measure between vague values x=[t
x
,1f
x
], y=
[t
y
,1f
y
]:
Sðx;yÞ¼1dðx;yÞ
ffiffiffi
2
p;ð1Þ
where dðx;yÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðt
x
t
y
Þ
2
þð1f
x
ð1f
y
ÞÞ
2
q¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðt
x
t
y
Þ
2
þðf
x
f
y
Þ
2
q
is the distance between xand y.
Also, we define the similarity measure between vague sets as
follows:
Definition 7. Let A,Bbe two vague sets in the universe of
discourse U={u
1
,u
2
,...,u
n
}, A¼P
n
i¼1
½t
A
ðu
i
Þ;1f
A
ðu
i
Þ=u
i
,B¼
P
n
i¼1
½t
B
ðu
i
Þ;1f
B
ðu
i
Þ=u
i
; the similarity measure between Aand B
is defined by
SðA;BÞ¼1
nX
n
i¼1
Sð
l
A
ðu
i
Þ;
l
B
ðu
i
ÞÞ:ð2Þ
2.4. Comparison between vague sets
Substantively, vague sets-based multiple criteria fuzzy deci-
sion-making is to compare the vague sets of the evaluation object.
Formally, a vague value is also an interval-value. So, according to
interval-value, we give the definition of comparison between va-
gue sets.
Definition 8. For vague value x=[t
x
,1 f
x
], y=[t
y
,1f
y
], the
probability of xPyis defined by
PðxPyÞ¼Maxð0;LðxÞþLðyÞMaxð0;1f
y
t
x
ÞÞ
LðxÞþLðyÞ;ð3Þ
where L(x)=1f
x
t
x
,L(y)=1f
y
t
y
is the length of vague value
x,y.
According to Definition 8, we can easily get the properties as
follows:
Property 1. 06P(x Py)61.
Property 2. If P(x Py)=P(yPx), then P(xPy)=P(yPx) = 0.5.
Property 3. P(x Py)+P(yPx)=1.
Property 4. For any three vague values x,y,z,ifP(xPy)P0.5,
P(y Pz)P0.5, then P(x Pz)P0.5.
Definition 9. Let A,Bbe two vague sets in the universe of
discourse U={u
1
,u
2
,...,u
n
}, A¼P
n
i¼1
½t
A
ðu
i
Þ;1f
A
ðu
i
Þ=u
i
,B¼P
n
i¼1
½t
B
ðu
i
Þ;1f
B
ðu
i
Þ=u
i
, the probability of APBis defined by
PðAPBÞ¼1
nX
n
i¼1
Pð
l
A
ðu
i
ÞP
l
B
ðu
i
ÞÞ:ð4Þ
2.5. Defuzzification of vague values
Definition 10. For vague value x=[t
x
,1f
x
], define the defuzzifi-
cation function to get the precise value as follows:
DfzzðxÞ¼t
x
=ðt
x
þf
x
Þ:ð5Þ
2.6. Weighted sum of vague values
Definition 11. For nVague values x
i
¼½t
x
i
;1f
x
i
, whose weights
vector w={w
1
,w
2
,...,w
n
} are nprecise values, the weighted sum of
x
i
(i=1,...,n) is defined as follows:

x¼X
n
i¼1
w
i
x
i
¼X
n
i¼1
w
i
t
i
;1X
n
i¼1
w
i
f
i
"#
;ð6Þ
where P
n
i¼1
w
i
¼1.
3. Model formulation
A new approach based on vague sets group decision is proposed
to rank the order of preference of suppliers. This method is very
suitable for solving the group decision-making problem under an
uncertain environment. It not only considers the relative impor-
tance of different decision-makers, but also includes the accor-
dance and difference in the decision group. After all, it integrates
the judgments of all the decision-makers into a decision matrix,
from which we can get the rank vector of all supplier alternatives.
Assume that A={A
1
,A
2
,...,A
m
} is a discrete set of mpossible
supplier alternatives, and C={C
1
,C
2
,...,C
n
} is a set of nattributes
of suppliers. The attributes are additively independent.
W={W
1
,W
2
,...,W
n
} is the vector of attribute weights. In this pa-
per, the attribute weights and ratings of suppliers are linguistic
variables. Here these linguistic variables can be expressed in vague
values on a 1–7 scale, as shown in Table 1.
The procedures are summarized as follows:
Step 1: Form a committee of decision-maker and identify the
importance of weights vector of decision-makers.
Assume that a decision group has Kperson, weights vector
D={D
1
,D
2
,...,D
K
} can be obtained by professional knowledge
and experience of experts, which is the subjective weights vector
Table 1
The scale of linguistic variables
Attribute ratings Weights Vague value
Very poor, VP Very low, VL [0.0,0.1]
Poor, P Low, L [0.1,0.3]
Medium poor, MP Medium low, ML [0.3,0.4]
Fair, F Medium, M [0.4,0.5]
Medium good Medium high, MH [0.5,0.6]
Good, G High, H [0.6,0.9]
Very good, VG Very high, VH [0.9,1.0]
D. Zhang et al. / Expert Systems with Applications 36 (2009) 9557–9563 9559

of decision-makers. D
k
(k=1,...,K) is the importance degree of the
kth DM, and D
k
¼½t
D
k
;1f
D
k
is linguistic variable in Table 1.
Step 2: Use linguistic variables to identify the attribute weights
and attribute ratings of supplier alternatives.
For every DM in the decision group, we can get a vector of attri-
bute weights and a preference matrix of supplier alternatives.
Namely, W
k
¼fW
k
1
;W
k
2
;...;W
k
n
gðk¼1;...;KÞis the vector of
attribute weights given by kth DM, where W
k
j
¼ðt
W
k
j
;1f
W
k
j
Þ
ðj¼1;...;nÞis a linguistic variable in Table 1. The preference ma-
trix given by kth DM is written as
R
k
¼
R
k
11
R
k
12
 R
k
1n
R
k
21
R
k
22
 R
k
2n
.
.
..
.
...
..
.
.
R
k
m1
R
k
m2
 R
k
mn
2
6
6
6
6
6
4
3
7
7
7
7
7
5
;
where R
k
ij
ði¼1;...;m;j¼1;...;nÞis the attribute rating of supplier
alternative A
i
on attribute C
j
given by kth DM, and R
k
ij
¼½t
R
k
ij
;1f
R
k
ij

is a linguistic variable in Table 1.
Step 3: Calculate the weighted decision matrix of kth DM.
Considering the different importance of each attribute, the
weighted decision matrix can be expressed as
M
k
¼
M
k
11
M
k
12
 M
k
1n
M
k
21
M
k
22
 M
k
2n
.
.
..
.
...
..
.
.
M
k
m1
M
k
m2
 M
k
mn
2
6
6
6
6
6
4
3
7
7
7
7
7
5
;
where
M
k
ij
¼W
k
j
^R
k
ij
ði¼1;...;m;j¼1;...;nÞ:ð7Þ
Each line M
k
i
¼P
n
j¼1
M
k
ij
=C
i
represents the evaluation of kth DM vis-
a-vis alternative A
i
on attributes set C={C
1
,C
2
,...,C
n
}. It is also a va-
gue set.
Step 4: Adjust the importance degree of decision-makers
according to the preference accordance in the decision group.
Since the final decision must be close to the preference of most
DMs, it is reasonable for us to increase the weight of DMs whose
preference is close to the group preference. According to Definition
7, calculate the similarity between the pth DM and qth DM as
follows:
S
pq
¼SðM
p
;M
q
Þ¼1
mX
m
i¼1
SðM
p
i
;M
q
i
Þ:ð8Þ
Thus, we can get the preference accordance matrix of all DMs:
S¼½S
pq
¼
S
11
S
12
 S
1K
S
21
S
22
 S
2K
.
.
..
.
...
..
.
.
S
K1
S
K2
 S
KK
2
6
6
6
6
4
3
7
7
7
7
5
:
Obviously, Sis a symmetry matrix. Using the line sum of Sto get the
similarity weights vector h={h
1
,h
2
,...,h
K
}, where
h
k
¼P
K
q¼1;q–k
S
kq
P
K
p¼1
P
K
q¼1;q–p
S
pq
¼P
K
q¼1
S
kq
1
P
K
p¼1
P
K
q¼1
S
pq
K:ð9Þ
Since his derived form the preference matrix given by all DMs, it is
called the objective weights vector here.
Step 5: Adjust the weights vectors of all DMs by both subjective
and objective weights vectors.
Use formula (5) to get the precise value w={w
1
,w
2
,...,w
K
}of
the subjective weights vector D={D
1
,D
2
,...,D
K
}, where
w
k
¼t
D
k
=ðt
D
k
þf
D
k
Þ:ð10Þ
Normalize wto get the final subjective weight vector, which is still
said wwith no confusion in the case.
So, there is one h
k
and one w
k
corresponding to kth DM. Calcu-
late the adjusted weights vector d={d
1
,d
2
,...,d
K
} as follows:
d
k
¼aw
k
þð1aÞh
k
;k¼1;2;...;K;ð11Þ
where a2[0,1] represents the preference to subjective weights
against objective weights. The larger ais, the more is the attention
of DMs to subjective weights. Contrarily, the more is the attention
of DMs to objective weights?
Step 6: Integrate all DMs’ preference matrix to generate the
whole decision matrix
G¼
G
11
G
12
 G
1n
G
21
G
22
 G
2n
.
.
..
.
...
..
.
.
G
m1
G
m2
 G
mn
2
6
6
6
6
4
3
7
7
7
7
5
;
where
G
ij
¼X
K
k¼1
d
k
M
k
ij
¼X
K
k¼1
d
k
t
M
k
ij
;1X
K
k¼1
d
k
f
M
k
ij
"#
;ð12Þ
which is obtained by Definition 11.
Each line G
i
in matrix Grepresents the evaluation of alternative
A
i
, by the whole decision group. Obviously, G
i
is a vague set.
Step 7: Calculate the probability matrix of all supplier
alternatives
P¼
p
11
p
12
 p
1m
p
21
p
22
 p
2m
.
.
..
.
...
..
.
.
p
m1
p
m2
 p
mm
2
6
6
6
6
4
3
7
7
7
7
5
;
where
p
il
¼PðG
i
PG
l
Þ¼1
nX
n
j¼1
PðG
ij
PG
lj
Þ;ð13Þ
which is obtained by Definition 9.
Step 8: Calculate the order vector of all supplier alternatives.
By Definition 8, we have p
ii
= 0.5 p
il
+p
li
= 1. So, Pis a fuzzy
complementary judgment matrix. According to the algorithm pro-
posed by Xu (2001), order vector e={e
1
,e
2
,...,e
m
} for all supplier
alternatives can be obtained by
e
i
¼P
m
l¼1
p
il
þ
m
2
1
mðm1Þ:ð14Þ
When e
i
is bigger, the ranking order of A
i
is better. Otherwise, the
ranking order is worse.
According to the above procedure, we can determine the rank-
ing order of all supplier alternatives and select the best one from
among a set of feasible suppliers.
4. Numerical experiments
There are six suppliers A
1
,A
2
,...,A
6
selected as alternatives,
against four attributes C
1
,C
2
,...,C
4
. The four attributes are product
quality, service quality, delivery time and price, respectively (Wang
& Hu, 2005). C
1
,C
2
and C
3
are benefit attributes; larger values are
better. C
4
is cost attribute; smaller values are better. The selection
structure is shown in Fig. 2.
The numerical calculation process is shown as follows:
Step 1:
A committee of four DMs (DM1, DM2, DM3 and DM4) has been
formed to make the selection decision. The four DMs’ importance
weights are shown in Table 2.
9560 D. Zhang et al. / Expert Systems with Applications 36 (2009) 9557–9563

Step 2:
Identify the attribute weights and attribute ratings of supplier
alternatives. The attribute weights given by four DMs are shown
in Table 3.
The attribute ratings of supplier alternatives given by four DMs
are shown in Table 4.
Step 3:
Calculate the weighted decision matrix for all suppliers. Accord-
ing to formula (7), we can obtain the weighted decision matrix
shown in Table 5.
Step 4:
Adjust the importance degree of decision-makers. According to
formula (8), we can get the preference accordance matrix of four
DMs as follows:
S¼
1:0000 0:9184 0:8889 0:9432
0:9184 1:0000 0:9152 0:8802
0:8889 0:9152 1:0000 0:9191
0:9432 0:8802 0:9191 1:0000
2
6
6
6
4
3
7
7
7
5
:
According to formula (9), the similarity weights vector of four DMs
can be obtained as follows:
h¼f0:2516 0:2483 0:2491 0:2509 g;
which is also called objective weights vector. In this numerical
example, it seems that the evaluation of weights of all four DMs
are close to each other.
Step 5:
Adjust the weights vector of four DMs. According to formula
(10), the precise value of the subjective weights vector Dis ob-
tained as follows:
w¼f1:0000 0:8571 0:5556 0:8571 g:
Normalize winto:
w¼f0:3058 0:2621 0:1699 0:2621 g:
Fig. 2. Suppliers selection structure.
Table 2
DMs’ importance weights
DM1 DM2 DM3 DM4
Weights VH H MH H
Table 3
Attribute weights
Ci DMi
DM1 DM2 DM3 DM4
C1 VH H H H
C2H VHVHH
C3 MH H H MH
C4MMMHMH
Table 4
Group decision table for all suppliers alternatives
Ci/Ai DMi
DM1 DM2 DM3 DM4
C1 A1 G MG G G
A2 MG G F MG
A3FFMGG
A4 F MG MG F
A5 MG F F MG
A6 G MG MG MG
C2 A1 G G MG MG
A2 G MG MG G
A3FFPF
A4 P MP MP P
A5 MP MP MP MP
A6 MP P P MP
C3 A1 G MG MG G
A2 MG G G G
A3 G G F MG
A4 G MG MG G
A5 MG F F MG
A6FFMGF
C4 A1 F G G G
A2 G G F MG
A3 VG VG G G
A4 G MG G G
A5 MG MG G MG
A6 G VG VG G
D. Zhang et al. / Expert Systems with Applications 36 (2009) 9557–9563 9561

Assume a= 0.5, that is, subjective weights were assumed to have
the same importance as objective weights. According to formula
(11), we can get adjusted weights vector as follows:
d¼f0:2787 0:2552 0:2095 0:2565 g:
Step 6:
Generate the whole decision matrix. According to formula (12),
the integrated decision matrix of four DMs is obtained as follows:
Step 7:
Calculate the probability matrix. According to formula (13), the
probability matrix of six supplier alternatives is obtained as
follows:
P¼
0:5000 0:5210 0:6894 0:7500 0:8081 0:7814
0:4790 0:5000 0:6711 0:7614 0:7940 0:7029
0:3106 0:3289 0:5000 0:6976 0:6743 0:6713
0:2500 0:2386 0:3024 0:5000 0:4701 0:4832
0:1919 0:2060 0:3257 0:5299 0:5000 0:5115
0:2186 0:2971 0:3287 0:5168 0:4885 0:5000
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
:
Step 8:
Calculate the order vector of six suppliers. According to formula
(14), we can obtain the order vector as follows:
e¼f0:2017 0:1969 0:1728 0:1415 0:1422 0:1450 g:
Therefore, the ranking order of six suppliers will be as follows:
A
1
>A
2
>A
3
>A
6
>A
5
>A
4
:
We can say that supplier A
1
is the best supplier among the six sup-
pliers. A
1
should be an important alternative for the company. The
next important alternatives are A
2
and A
3
,A
6
and A
5
are better sup-
pliers and A
4
is the worst supplier.
5. Conclusion
Many practitioners and researchers have presented the
advantages of supply chain management. In order to increase
the competitive advantage, many companies consider a well-
designed and implemented supply chain system to be an
important tool. Under this condition, building on the closeness
and long-term relationships between buyers and suppliers is a
critical success factor to establish the whole supply chain sys-
tem. Therefore, supplier selection problem becomes the most
important issue to implement a successful supply chain system.
In this paper, a new vague sets-based approach was proposed
to deal with supplier selection problem under uncertainty envi-
ronments. An example of the selection problem of suppliers is
used to illustrate the proposed approach. The experimental re-
sults show that the proposed approach is reliable and reason-
able. This proposed approach can help in more accurate
selections in other spheres also, such as management and
economics.
Acknowledgements
This research is supported by the grant from the NSFC of China
(Project Nos. 70571025 and 70731001). At the same time, we
thank the editors and the anonymous reviewer(s) for their sugges-
tions that have helped us to improve this paper.
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C4 A1 [0.4 0.5] [0.4 0.5] [0.5 0.6] [0.5 0.6]
A2 [0.4 0.5] [0.4 0.5] [0.4 0.5] [0.5 0.6]
A3 [0.4 0.5] [0.4 0.5] [0.5 0.6] [0.5 0.6]
A4 [0.4 0.5] [0.4 0.5] [0.5 0.6] [0.5 0.6]
A5 [0.4 0.5] [0.4 0.5] [0.5 0.6] [0.5 0.6]
A6 [0.4 0.5] [0.4 0.5] [0.5 0.6] [0.5 0.6]
G¼
½0:5745 0:8234½0:5534 0:7602½0:5000 0:6000½0:4466 0:5466
½0:5046 0:6556½0:5535 0:7606½0:5465 0:7394½0:4257 0:5257
½0:4723 0:6236½0:3371 0:4581½0:5046 0:6556½0:4466 0:5466
½0:4465 0:5465½0:1929 0:3465½0:5000 0:6000½0:4466 0:5466
½0:4535 0:5535½0:3000 0:4000½0:4535 0:5535½0:4466 0:5466
½0:5279 0:6836½0:2071 0:3535½0:4210 0:5210½0:4466 0:5466
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
:
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