like to have a decision which not only makes as much profit as possible, but also avoids as much risk as possible. Specially,
Chamodrakas et al. [4] addressed this issue through the introduction of a fuzzy set representation of the closeness to the
ideal and to the negative ideal solution and presented an innovative fuzzy approach for ranking alternatives in multiple cri-
teria decision making problems based on TOPSIS. This model enables a parameterization of the method according to the risk
attitude of the decision maker. Besides, computing the optimal point in the VIKOR is based on the particular measure of
‘‘closeness’’ to the PIS. Therefore, it is suitable for those situations in which the decision maker wants to have maximum prof-
it and the risk of the decisions is less important for him [1,5–10]. Sayadi et al. [6] extended the VIKOR method for decision
making problems with interval number. The extended VIKOR method’s ranking is obtained through comparison of interval
numbers and for doing the comparisons between intervals, we introduce
v
as optimism level of decision maker in this paper.
Sanayei et al. [9] proposed a hierarchy MCDM model based on fuzzy sets theory and VIKOR method to deal with the supplier
selection problems in the supply chain system. Chen and Wang [7] provided a rational and systematic process for developing
the best alternative and compromise solution under each of the selection criteria using the fuzzy VIKOR method. The study’s
finding offers an important reference for resolving fuzzy multi-criteria decision-making problems. Vahdani et al. [8] pre-
sented the interval-valued fuzzy VIKOR method, aiming at solving MCDM problems in which the weights of criteria are un-
equal, using interval-valued fuzzy set concepts. San Cristóbal [10] applied the VIKOR method in the selection of a Renewable
Energy project corresponding to the Renewable Energy Plan launched by the Spanish Government. The method is combined
with the AHP (Analytical Hierarchy Process) method for weighting the importance of the different criteria, which allows
decision-makers to assign these values based on their preferences.
As a generalization of fuzzy set [11–20], Torra and Narukawa [21] and Torra [22] proposed the hesitant fuzzy set which
permitted the membership having a set of possible values and discussed the relationship between hesitant fuzzy set and
intuitionistic fuzzy set. Xia and Xu [23] gave an intensive study on hesitant fuzzy information aggregation techniques and
proposed a series of operators under various situations to solve the decision making problems with anonymity. Xu and
Xia [24,25] proposed a variety of distance measures for hesitant fuzzy sets, based on which the corresponding similarity
measures can be obtained, and defined the distance and correlation measures for hesitant fuzzy information and discussed
their properties in detail. Xia et al. [26] developed several series of aggregation operators for hesitant fuzzy information with
the aid of quasi-arithmetic means. Gu et al. [27] investigated the evaluation model for risk investment with hesitant fuzzy
information. They utilized the hesitant fuzzy weighted averaging (HFWA) operator to aggregate the hesitant fuzzy informa-
tion corresponding to each alternative, and then rank the alternatives according to the score function.
From above analysis, we can see that hesitant fuzzy set is a very useful tool to deal with uncertainty. Moreover, when
giving the membership degree of an element, the difficulty of establishing the membership degree is not because we have
a margin of error, or some possibility distribution on the possibility values, but because we have several possible values. For
such cases, the hesitant fuzzy set is very useful in avoiding such issues in which each criterion can be described as a hesitant
fuzzy set defined in terms of the opinions of decision makers. More and more multiple criteria decision making theories and
methods under hesitant fuzzy environment have been developed. Therefore, we extend the concept of VIKOR method and
TOPSIS method to develop a methodology for solving MADM problems with hesitant fuzzy element. In this paper, we devel-
op the E-VIKOR method and TOPSIS method to solve the MCDM problems with hesitant fuzzy set information. In Sections 2
and 3, the information in the format of hesitant fuzzy element and corresponding concepts are described, and the basic
essential of the VIKOR method is introduced. In Section 4, the problem on multiple criteria decision marking is described,
and the principles and steps of the proposed E-VIKOR method and TOPSIS method are presented. In Section 5, a numerical
example illustrates an application of the E-VIKOR method, and the result by the TOPSIS method is compared. In Section 6,we
conclude the paper.
2. Preliminaries
Here we give a brief review of some preliminaries.
Definition 1. [1,2]. Let X be a fixed set, a hesitant fuzzy set (HFS) on X is in terms of a function that when applied to X returns
a subset of [0, 1].
To be easily understood, we express the HFS by a mathematical symbol:
E ¼f< x; h
E
ðxÞ >
j
x 2 Xg; ð1Þ
where h
E
(x) is a set of some values in [0,1], denoting the possible membership degrees of the element x2X to the set E. For
convenience, we call h = h
E
(x) a hesitant fuzzy element (HFE) and H the set of all HFEs.
Let h, h
1
and h
2
be three HFEs, then
(1) h
k
¼[
c
2h
f
c
k
g;
(2) kh ¼[
c
2h
f1 ð1
c
Þ
k
g;
(3)
~
h
1
[
~
h
2
¼[
c
1
2
~
h
1
;
c
2
2
~
h
2
maxf
c
1
;
c
2
g;
(4)
~
h
1
\
~
h
2
¼[
c
1
2
~
h
1
;
c
2
2
~
h
2
minf
c
1
;
c
2
g;
(5) h
1
h
2
¼[
c
1
2h
1
;
c
2
2h
2
f
c
1
þ
c
2
c
1
c
2
g;
N. Zhang, G. Wei / Applied Mathematical Modelling 37 (2013) 4938–4947
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