Abstract—The Dual Hesitant Fuzzy Sets (DHFSs) is a useful tool to deal with vagueness and ambiguity in the multiple attribute decision making (MADM) problems. The distance and similarity measures analysis are important research topics. In this paper, we propose some new distance measures for dual hesitant fuzzy sets, and study the properties of the measures. In the end, we develop an approach for multi-criteria decision making under dual hesitant fuzzy environment, and illustrate an example to show the behavior of the proposed distance measures. Index Terms—dual hesitant fuzzy set, distance measures, similarity measures, multi-criteria decision making I. INTRODUCTION hu and Xu [1] introduced the definition of dual hesitant fuzzy set, which is a new extension of fuzzy sets (FSs) [2]. Zhu and Xu’s DHFSs used the membership hesitancy function and the non-membership hesitancy function to support a more exemplary and flexible access to assign values for each element in the domain. DHFS can be regarded as a more comprehensive set, which supports a more flexible approach when the decision makers provide their judgments. The existing sets, including FSs [2], IFSs [3] and HFSs [4] can be regarded as special cases of DHFSs. When people make a decision, they are usually hesitant and irresolute for one thing or another which makes it difficult to reach a final agreement. They further indicated that DHFSs can better deal with the situations that permit both the membership and the no-membership of an element to a given set having a few different values, which can arise in a group decision making problem. For example, in the organization, some decision makers discuss the membership degree 0.6 and the non-membership 0.1 of an alternative A satisfies a criterion x . Some possibly assign (0.8, 0.2), while the others assign (0.7, 0.2). No consistency is reached among these decision makers. Accordingly, the difficulty of establishing a common membership degree and a non-membership degree is not because we have a margin of error (intuitionistic fuzzy set), or some possibility distribution values (type-2 fuzzy set), but because we have a set of possible values (hesitant fuzzy set). For such a case, the satisfactory degrees can be represented by Manuscript received February 27, 2015; revised July 4, 2016. Lei Wang is with College of Communications Engineering, PLA University of Science and Technology, Nanjing, China. (e-mail: [email protected]). Xiang Zheng is with College of Communications Engineering, PLA University of Science and Technology, Nanjing, China. (e-mail: [email protected]) a dual hesitant fuzzy element 0.6, 0.8, 0.7 , 0.1, 0.2 , which is obviously different from intuitionistic fuzzy number (0.8, 0.2) or (0.7, 0.2) and hesitant fuzzy number 0.6, 0.8, 0.7 . Distance measures of FSs are an important research topic in the FS theory, which has received much attention from researchers [6-9]. Among them, the most widely used distance measures [10-12] are the Hamming distance, Euclidean distance, and Hausdorff metric. Later on, the distance measures about other extensions of fuzzy sets have also been developed. Later on, the distance measures about other extensions of fuzzy sets have also been developed. For example, Xu [13] introduced the concepts of deviation degrees and similarity degrees between two linguistic values, and between two linguistic preference relations, respectively. Li and Cheng [14] generalized the Hamming distance and the Euclidean distance by adding a parameter and gave a similarity formula for IFSs only based on the membership degrees and non-membership degrees. Hung and Yang [15] and Grzegorzewski [16] suggested a lot of similarity measures for IFSs and interval-valued fuzzy sets based on the Hausdorff metric. Xu and Xia [17] proposed a variety of distance measures for hesitant fuzzy sets. They investigated the connections of the distance measures and further developed a number of hesitant ordered weighted distance measures. Xu and Xia [18] define the distance for hesitant fuzzy information and then discuss their properties in detail. The aforementioned measures, however, cannot be used to deal with the distance measures of dual hesitant fuzzy information. However, little has been done about this issue. Thus it is very necessary to develop some theories about dual hesitant fuzzy sets. To do this, the remainder of the paper is organized as follows. Section 2 presents some basic concepts related to IFSs, HFSs and DHFSs. Section 3 aims to present the axioms for distance measures, gives some new distance measures for DHFSs. In Section 4, proposes an approach to multi-criteria decision making. Section 5 gives some conclusions. II. PRELIMINARIES A. IFSs and HFSs Definition 1 [3]. Let X be a fixed set, an intuitionistic fuzzy set (IFS) A on X is an object having the form: { , ( ), () } A A A x x x x X Notes on Distance and Similarity Measures of Dual Hesitant Fuzzy Sets Lei Wang, Xiang Zheng*, Li Zhang, Qiang Yue, Member, IAENG Z IAENG International Journal of Applied Mathematics, 46:4, IJAM_46_4_11 (Advance online publication: 26 November 2016) ______________________________________________________________________________________
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Abstract—The Dual Hesitant Fuzzy Sets (DHFSs) is a useful
tool to deal with vagueness and ambiguity in the multiple
attribute decision making (MADM) problems. The distance and
similarity measures analysis are important research topics. In
this paper, we propose some new distance measures for dual
hesitant fuzzy sets, and study the properties of the measures. In
the end, we develop an approach for multi-criteria decision
making under dual hesitant fuzzy environment, and illustrate an
example to show the behavior of the proposed distance
measures.
Index Terms—dual hesitant fuzzy set, distance measures,
similarity measures, multi-criteria decision making
I. INTRODUCTION
hu and Xu [1] introduced the definition of dual hesitant
fuzzy set, which is a new extension of fuzzy sets (FSs) [2].
Zhu and Xu’s DHFSs used the membership hesitancy
function and the non-membership hesitancy function to
support a more exemplary and flexible access to assign values
for each element in the domain. DHFS can be regarded as a
more comprehensive set, which supports a more flexible
approach when the decision makers provide their judgments.
The existing sets, including FSs [2], IFSs [3] and HFSs [4]
can be regarded as special cases of DHFSs. When people
make a decision, they are usually hesitant and irresolute for
one thing or another which makes it difficult to reach a final
agreement. They further indicated that DHFSs can better deal
with the situations that permit both the membership and the
no-membership of an element to a given set having a few
different values, which can arise in a group decision making
problem. For example, in the organization, some decision
makers discuss the membership degree 0.6 and the
non-membership 0.1 of an alternative A satisfies a criterion
x . Some possibly assign (0.8, 0.2), while the others assign
(0.7, 0.2). No consistency is reached among these decision
makers. Accordingly, the difficulty of establishing a common
membership degree and a non-membership degree is not
because we have a margin of error (intuitionistic fuzzy set), or
some possibility distribution values (type-2 fuzzy set), but
because we have a set of possible values (hesitant fuzzy set).
For such a case, the satisfactory degrees can be represented by
Manuscript received February 27, 2015; revised July 4, 2016.
Lei Wang is with College of Communications Engineering, PLA
University of Science and Technology, Nanjing, China. (e-mail:
TableⅣ SCORE VALUES OBTAINED BY ATS-WGDHFPA OPERATOR BASED ON THE
GENERALIZED DUAL HESITANT WEIGHTED HAUSDORFF DISTANCE AND THE
RANKING OF ALTERNATIVES.
d1 λ=0.05 λ=0.1 λ=1 λ=10 λ=100
A1 0.2104 0.2107 0.2157 0.2675 0.1032
A2 0.4831 0.4833 0.4864 0.5207 0.5530
A3 0.6528 0.6529 0.6551 0.6784 0.7081
A4 -0.0661 -0.0658 -0.0604 -0.0115 -0.5224
We now present a figure to clearly demonstrate how the
score values vary as the parameter λ increases and the
aggregation arguments are kept fixed (see Fig. 1).
0 20 40 60 80 100 120 140 160 180 2000.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
Lambda
Sco
re v
alu
se
S(x1)
S(x2)
S(x3)
S(x4)
S(x2)
S(x4)
S(x3)
S(x1)
Fig. 1.Variation of the score valuse with respect to the
parameter λ.
V. CONCLUSION
In this paper, we have given a further study about the
distance measures for DHFSs. Based on ideas of the
well-known Hamming distance, the Euclidean distance,
the Hausdorff metric and their generalizations, we have
developed a class of dual hesitant distance measures, and
discussed their properties and relations as their parameters
change. We have also given a variety of ordered weighted
distance measures for DHFSs in which the distances are
rearranged in decreasing order, and given three ways to
determine the associated weighting vectors. With the
relationship between distance measures and similarity
measures, the corresponding similarity measures for DHFSs
have been obtained. It should be pointed out that all of the
above measures are based on the assumption that if the
corresponding DHFEs in DHFSs do not have the same
length, then the shorter one should be extended by adding the
minimum value in it until both the DHFEs have the same
length. In fact, we can extend the shorter DHFE by adding any
value in it until it has the same length of the longer one
according to the decision makers’ preferences and actual
situations. Finally, an approach for multi-criteria decision
making has been developed based on the proposed distance
measures under dual hesitant fuzzy environments.
ACKNOWLEDGMENTS
The authors are very grateful to the anonymous reviewers
for their insightful and constructive comments and
suggestions that have led to an improved version of this paper.
The work was supported by the National Natural Science
Foundation of China (No.71501186) and the Natural Science
Foundation for Young Scholars of Jiangsu Province
(No.bk20140065).
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