Similarity and entropy measures for hesitant fuzzy sets

• Date: 01 May 2018
• Published date: 01 May 2018
• DOI: http://doi.org/10.1111/itor.12477

Intl. Trans. in Op. Res. 25 (2018) 857–886

DOI: 10.1111/itor.12477

INTERNATIONAL

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IN OPERATIONAL

RESEARCH

Similarity and entropy measures for hesitant fuzzy sets

Junhua Hu, Yan Yang, Xiaolong Zhang and Xiaohong Chen

School of Business, Central South University, Lu Shan Nan Lu 932, Yue Lu District,

Changsha 410083, Hunan Province, P. R. China

E-mail: hujunhua@csu.edu.cn [Hu]

Received 14 September 2016; receivedin revised form 26 September 2017; accepted 28 September 2017

Abstract

Hesitant fuzzy sets (HFSs) are beneficial tools for expressing the hesitancy of decision makers (DMs) to

access alternatives in daily life,thereby enabling the membership of an element to a set that is represented by

several possible values. This study proposes an intervalbound footprint (IBF), which describes the fluctuation

range of the values of hesitant fuzzy elements (HFEs) arranged in order. In addition, a few similarity and

entropy measures for HFSs are deduced. First, the interval bound footprint, upper bound footprint, and

lower bound footprint for HFEs are defined and their corresponding properties are discussed. Subsequently,

several similarity and entropy measures for HFSs are presented based on IBF. Lastly, a hesitant fuzzy multi-

criteria decision-making method based on the proposed similarity and entropy measures is introduced. We

use a numerical example to discuss the differencesamong the proposed similarity measures and the applicable

environment based on the risk preferences of the different DMs.

Keywords:hesitant fuzzy sets; similarity measure; entropy measure; multi-criteria decision making

1. Introduction

Hesitant fuzzy sets (HFSs), which are among the important extensions of fuzzy sets (FSs) (Bustince

et al., 2016), were first introduced by Torra (Torra and Narukawa, 2009; Torra, 2010) and had

also been extensively studied and applied to various fields (Joshi and Kumar, 2016; Zhang and Xu,

2016; Gou et al., 2017a; Li and Wang, 2017). Decision makers (DMs) may have varying opinions

when evaluating alternatives due to different backgrounds or experiences. Consequently, consistent

evaluation information is difficult to obtain. Therefore, evaluating the alternatives is represented

by several possible values instead of the consistent information. HFSs, which allow possible values

for the membership of an element to a given set, are sufficiently suitable and powerful to handle

similar situations. To date, studies on HFSs have mainly focused on three aspects, namely, hesitant

fuzzy aggregation operators, multi-criteria decision making (MCDM) methods, and hesitant fuzzy

information measures.

For hesitant fuzzy aggregation operators, Xia and Xu (2011) defined new operational laws for

HFSs based on the algebraic t-norms and t-conorms and developed a few hesitant fuzzy aggregation

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2017 The Authors.

International Transactionsin Operational Research C

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858 J. Hu et al. / Intl. Trans. in Op. Res.25 (2018) 857–886

operators for HFSs. Furthermore, Liao and Xu (2014, 2015b) proposeda family of hybrid weighted

aggregation operators to fuse hesitant fuzzy information and introduced extended hesitant fuzzy

hybrid weighted aggregation operators thereafter to solve decision-making problems. Many other

hesitant fuzzy aggregation operators were also presented (Wei et al., 2016; Zhao et al., 2016;

Liao and Xu, 2017a). Moreover, numerous aggregation operators for certain extensions of HFSs

were developed. Chen et al. (2013) defined the concept of an interval-valued hesitant fuzzy set

(IVHFS) and presented several corresponding interval-valued hesitant aggregation operators. Wei

and Liao (2016) defined two types of aggregation operators for hesitant 2-tuple sets. Research

was likewise conducted on certain aggregation operators for other extensions of HFSs, such as

Bonferroni means for hesitant fuzzy linguistic term sets (HFLTSs) (Gou et al., 2017b), dual hesitant

fuzzy power aggregation operators (Wang et al., 2016b), and interval-valued hesitant fuzzy rough

approximation operators (Zhang et al., 2016b).

For the MCDM methods, HFSs were integrated into several classic techniques to solve decision

making problems under a hesitant environment. These methods include TOPSIS with incomplete

hesitant information (Xu and Zhang, 2013), a VIKOR-based method that accommodates hesitant

fuzzy circumstances (Liao and Xu, 2013), a hesitant fuzzy QUALIFLEX approach (Zhang and

Xu, 2015a), and a hesitant ELECTRE II method (Chen and Xu, 2015). In addition, Liao et al.

(2014) presented a beneficial tool called hesitant fuzzy preference relation (HFPR), which had

elicited considerable attentionsince its introduction. Liu et al. (2016) developed a novel approach to

determine the value of the consistency index for HFPR. He and Xu (2016) proposed error analysis

methods to solve group decision methods with HFPRs. Xu et al. (2016) introduced two models to

derive the priority weights from an incomplete HFPR. Furthermore, various extensions of HFSs

are gradually maturing. For example, HFLTSs (Rodriguez et al., 2012), which is among the most

advantageous tools for decision making, have been studied by various scholars. Gou et al. (2017b)

proposed an MCDM method that combined Bonferronimeans and HFLTSs.Gou et al. (2017c) also

presented another method based on hesitant fuzzy linguistic entropy and cross-entropy measures.

Zhang et al. (2016a) also applied the hesitant fuzzy linguistic VIKOR approach to solvethe inpatient

admission problem. Moreover, many MCDM approaches that used other extensions of HFSs have

been presented, such as a dynamic decision-making method based on hesitant probabilistic FSs

(Gao et al., 2017), TOPSIS method based on interval-valued intuitionistic hesitant fuzzy Choquet

integral (Joshi and Kumar, 2016), and a novel three-way decision model that combined dual HFSs

and decision theoretic rough sets (Liang et al., 2017).

Information measures, including distance, similarity, and entropy measures, play indispensable

roles in fuzzy MCDM problems. Many studies have focused on similarity and entropy measures in

different fuzzy environments, such as FSs (Liu, 1992; Mitchell, 2003) and intuitionistic fuzzy sets

(IFSs) (Hung and Yang, 2006; Hu et al., 2014). Hesitant fuzzy information measures have been

studied from several different perspectives since the introduction of HFSs. On the one hand, many

researchers have proposed various distance,similarity, and entropy measurements forHFSs. Xu and

Xia (2011a, 2011b) presented a variety of distance measurements among HFSs. Peng et al. (2013)

developed a generalized hesitant fuzzy synergetic weighted distance measure. Zeng et al. (2016)

proposed several distance and similarity measurements for HFSs. Liao and Xu (2017b) intro-

duced novel entropy measuresfor HFSs. In addition, various scholars developed many information

measurements based on HFSs extensions,such as distance measurements for hesitant fuzzy linguistic

numbers (Wang et al., 2016a) and dual HFSs (Ren et al., 2017), similarity measurementsfor IVHFSs

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2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation of OperationalResearch Societies