Multicriteria decision‐making approach based on gray linguistic weighted Bonferroni mean operator

• DOI: http://doi.org/10.1111/itor.12220
• Date: 01 September 2018
• Author: Jian‐Qiang Wang,Xiao‐Hong Chen,Zhang‐Peng Tian,Jing Wang
• Published date: 01 September 2018

Intl. Trans. in Op. Res. 25 (2018) 1635–1658

DOI: 10.1111/itor.12220

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RESEARCH

Multicriteria decision-making approach based on gray

linguistic weighted Bonferroni mean operator

Zhang-Peng Tian, Jing Wang, Jian-Qiang Wang and Xiao-Hong Chen

School of Business, Central South University, Changsha 410083, PR China

E-mail: 137467762@qq.com [Tian]; 30422815@qq.com [J. Wang]; jqwang@csu.edu.cn [J.-Q. Wang];

375104630@qq.com [Chen]

Received 26 August2014; received in revised form 5 October 2015; accepted 6 October 2015

Abstract

The main purpose of this paper is to provide a multicriteriadecision-making (MCDM) approach that applies

the gray linguistic Bonferroni mean (BM) operator to address the situations where the criterion values

take the form of gray linguistic numbers (GLNs) and the criterion weights are known. First, the related

operations and comparison method for GLNs are provided. Subsequently, a BM operator and weighted BM

operator of GLNs are developed. Then, based on the gray linguistic weighted BM operator, an MCDM

approach is proposed. Finally, an illustrative example is given and a comparison analysis is conducted

between the proposed approach and other existing methods to demonstrate the effectiveness and feasibility

of the developed approach.

Keywords:multicriteria decision making; gray linguistic sets; Bonferroni mean operator

1. Introduction

In practice, multicriteria decision-making (MCDM) methods are widely used for ranking alterna-

tives or selecting the optimal alternative with respect to several concerned criteria (Liu et al., 2015).

However, in some cases, it is difficult for decision makers to explicitly express preference in solving

MCDM problems with uncertain or incomplete information. Under these circumstances, fuzzy sets

(FSs), proposed by Zadeh (1965), where each element has a membership degree represented by a

real number in [0, 1], are regarded as a valuable tool for solving MCDM problems (Bellman and

Zadeh, 1970; Yager, 1977). Sometimes, FSs cannot handle the cases where the membership degree

is uncertain and hard to define by a crisp value. Therefore, the concept of interval-valued FSs

(IVFSs) was proposed to capture the uncertainty of membership degree(Turksen, 1986). Generally,

if the membership degree is given, then the nonmembership degree can be determined by default.

In order to deal with the uncertainty of nonmembership degree, Atanassov (1986) introduced the

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International Transactionsin Operational Research C

2015 International Federation of OperationalResearch Societies

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1636 Z.-P. Tian et al. / Intl. Trans. in Op. Res. 25 (2018) 1635–1658

intuitionistic FSs (IFSs) which are an extension of Zadeh’s FSs. IFSs consider both membership

degree and nonmembership degree simultaneously, and thereforeIFSs are more flexible in handling

information containing uncertainty and incompleteness than traditional FSs. Currently, IFSs have

been widely applied in solving MCDM problems (Zeng and Su, 2011; Xu, 2012; Cao et al., 2015;

Wu et al., 2015; Yu, 2015). Moreover, intuitionistic interval fuzzy numbers (Wang et al., 2014a),

triangular intuitionistic fuzzy numbers (Wan, 2013; Wang et al., 2013; Wan and Dong, 2014a), and

intuitionistic trapezoidal fuzzy numbers (Liu and Yu, 2013; Liu and Liu, 2014), which were derived

from IFSs, are also useful tools to cope with fuzzy and uncertain information. In fact, the degrees

of membership and nonmembership in IFSs may be expressed using interval numbers instead of

specific numbers. Interval-valued intuitionistic FSs (IVIFSs) (Atanassov and Gargov, 1989) were

hence proposed as an extension of FSs and IVFSs. In recent years, MCDM problems with IVFSs

have attracted much attention from researchers (Xu and Chen, 2011; Huang et al., 2013; Meng

et al., 2013; Tan et al., 2014; Wan and Dong, 2014b), in which the aggregation operators, prospect

score function, and possibility degree were involved.

Although FSs and IFSs have been developed and generalized, they cannot deal with all sorts of

fuzziness in real decision-making problems, such as problems that are too complex or ill-defined

to be solved by quantitative expressions. Zadeh (1975) introduced the linguistic variable, which is

an effective tool because using linguistic information can enhance the reliability and flexibility of

classical decision models (Yang and Wang, 2013; Merig´

o et al., 2014; Yang, 2014; Xu et al., 2015).

Recently, linguistic variables have been studied in depth and numerous MCDM methods integrated

with other theories havebeen developed. Intuitionistic linguistic sets (ILSs), which combine IFSs and

linguistic variables, are applied to solve multicriteria group decision-making (MCGDM) problems

(Liu and Jin, 2012). ILSs and their extensions (Liu, 2013; Wang et al., 2015d) can describe both a

linguistic variable and an intuitionistic fuzzy number, in which the former can provide a qualitative

assessment value, while the latter can define the confidence degree for the given evaluation value.

Hesitant fuzzy linguistic sets (HFLSs), which are based on linguistic term sets and hesitant FSs,

are used for expressing decision makers’ hesitance that exists in giving the associated membership

degrees of one linguistic term (Lin et al., 2014; Wang et al., 2014c). Hesitant fuzzy linguistic

term sets (HFLTSs) contain several consecutive linguistic terms rather than a single term, which are

particularly helpful forconditions where hesitance in evaluation occurs due to uncertain information

and/or incomplete knowledge (Rodr´

ıguez et al., 2012; Wang et al., 2015b). Besides, a method based

on the cloud model, which can depict the uncertainty of qualitative concept, has been successfully

utilized to deal with linguistic information (Wang et al., 2014b, 2015c). A method based on the

2-tuple linguistic information model (Herrera and Mart´

ınez, 2000; Merig´

o and Gil-Lafuente, 2013;

Wang et al., 2015a), which can effectively avoid information distortion, has hitherto occurred in

linguistic information processing (Rodr´

ıguez and Mart´

ınez, 2013). It is clear that all those proposals

of linguistic variables, promising as they are, still need to be refined. As they can only express the

uncertain information but not the incomplete one, some new theories are required.

Chen (1994) introduced the grayFS (GFS), which is one of the most significant theories in dealing

with MCDM problems with incomplete informationcaused by poor data (Yin, 2013). A GFS could

be specified for IVFSs or rough sets (RSs) under special conditions (Yang and John, 2012). Luo and

Liu (2004) developed an MCDM method based on the linear combination of fuzzy information

and gray information, in which the maximum entropy was defined to determine criterion weights.

Chithambaranathan et al. (2015) developed a hybrid framework for evaluating the environmental

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2015 The Authors.

International Transactionsin Operational Research C

2015 International Federation of OperationalResearch Societies