An approach to multiplicative linguistic group decision making based on possibility degrees

• Author: Zeshui Xu,Meimei Xia
• Date: 01 September 2018
• DOI: http://doi.org/10.1111/itor.12222
• Published date: 01 September 2018

Intl. Trans. in Op. Res. 25 (2018) 1611–1634

DOI: 10.1111/itor.12222

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RESEARCH

An approach to multiplicative linguistic group decision making

based on possibility degrees

Meimei Xiaaand Zeshui Xub

aSchool of Economics and Management, Beijing Jiaotong University, Beijing, China

bBusiness School, Sichuan University, Chengdu, China

E-mail: meimxia@163.com [Xia]; xuzeshui@263.net [Xu]

Received 10 February2015; received in revised form 3 October 2015; accepted 6 October 2015

Abstract

In order to compare two uncertain multiplicative linguistic variables, a possibility degree formula has been

developed and its properties are also studied. Then a possibility degree matrix(also considered as a reciprocal

preference relation) is constructed to compare a collection of uncertain multiplicative linguistic variables.

Two models are further provided to derive the priority vector from the possibility degree matrix based on

the additive consistency and multiplicative consistency. Especially, if the parameters are assigned specific

values, then the models reduce to the existing ones. A group decision making method has been developed

to deal with the situations where the preferences on alternatives are expressed by uncertain multiplicative

linguistic variables. In this method, the possibility degree matrix is first constructed, from which the priority

of alternatives is obtained using the developed models. At the end, an example is given to illustrate the

proposed method.

Keywords:group decision making; possibility degree; uncertain multiplicative linguistic variable

1. Introduction

In practical decision making problems,group decision making is very popular because it can achieve

more objective results. In group decision making, the preference information over alternatives can

be represented by different forms according to the decision makers’ preferences, such as fuzzy set

(Zadeh, 1965), interval-valued fuzzy set (Zadeh, 1975), linguistic fuzzy set (Herrera et al., 1996),

intuitionisitic fuzzy set (Atanassov, 1986), and so on. Among these representative forms, a linguistic

variable (the element of a linguistic fuzzy set) is an important qualitative tool to express human

language, and thus can better describe the decision makers’ preferences. For example, to evaluate

the “comfort” of a car, linguistic terms such as “good,” “fair,” or “poor” could be preferred by the

decision makers instead of numerical values.

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2015 The Authors.

International Transactionsin Operational Research C

2015 International Federation of OperationalResearch Societies

Published by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148,

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1612 M. Xia and Z. Xu / Intl. Trans. in Op. Res. 25 (2018) 1611–1634

In the last decades, a lot of workhas been conducted on group decision making based on linguistic

variables (Herrera and Mart´

ınez, 2000, 2001; Herrera et al., 2005; Dong et al., 2008), such as the

consistency and the consensus problems (Herrera-Viedma et al., 2005; Chiclana et al., 2009; Dong

et al., 2010; Wu et al., 2015; Ure˜

na et al., 2015a, 2015b) and the multicriteria decision making

problems (Xu, 2005, 2012; Wei, 2010a, 2010b, 2011a, 2011b). In real life, there exist problems

that need to assess the linguistic variables with linguistic term sets that are not uniformly and

symmetrically distributed (Wang and Hao, 2006; Herrera et al., 2008; Dong et al., 2009a, 2013b;

Dong and Herrera-Viedma, 2015b). Herrera et al. (2008) defined the unbalanced linguistic term set

and developed a methodology to deal with unbalanced linguistic information based on the concept

of linguistic hierarchy and the 2-tuple fuzzy linguistic representation model. But the unbalanced

linguistic variables areirregularly distributed, which may not reflect some situations.Take the law of

diminishing marginal utility in economics as an example, to invest the same resources in a company

with bad performance and in a company with good performance, the former enhances more quickly

than the latter. In other words, the gap between the grades expressing bad information should be

smaller than the one between the grades expressing good information.In addition, the computing of

the unbalanced linguistic variable is a little complex. Dong et al. (2008, 2011, 2013a, 2015) proposed

the concept of scale function based on the Analytic Hierarchy Process (AHP) linguistic term set

and developed a 2-tuple fuzzy linguistic multicriteria approach to select the individual numerical

scale and prioritization method for AHP. Dong et al.’s method (2008, 2011, 2013a, 2015) focused

on the generation of numerical scales for each user based on the linguistic set, whose subscripts

are uniformly and symmetrically distributed. Xu (2004a, 2009) defined the multiplicative linguistic

variables based on the Saaty Scale (Saaty, 1977) and the Ma-Zheng Scale (Ma and Zheng, 1991),

whose distribution can describe the law of diminishing marginal utility.

Sometimes, in the group decision making problems, the decision makers would prefer to pro-

vide uncertain linguistic information (Xu, 2004b) because of time pressure, lack of knowledge,

or data, and their limited expertise related to the problem domain (Xu, 2004b; Wei, 2009; Fan

and Liu, 2010a). How to compare the uncertain linguistic variables is an important step to ob-

tain the final result in group decision making problems. Xu (2004b) proposed a possibility degree

formula to compare two uncertain linguistic variables, Xu (2006) also proposed another possi-

bility degree formula for uncertain linguistic variables, but these two formulas are only suitable

for some specific linguistic variables such as the uncertain additive linguistic variables (Herrera

et al., 1996), and are invalid for others such as the uncertain multiplicative linguistic variables (Xu,

2004a, 2009).

In this paper, we mainly study group decision making problems under uncertain multiplicative

linguistic environments. To compare the linguistic preferences provided by the decision makers, a

new possibility degree formula is developed, based on which the possibility degree matrix (also

considered as the reciprocal preference relation) is constructed. We also develop several models to

derive the priority of alternatives from a possibility degree matrix. To do this, the remainder of

this paper is constructed as follows: Section 2 gives some basic concepts. Section 3 proposes a new

possibility degree formula for comparing uncertain multiplicative linguistic variables and discusses

its properties. Section 4 constructs some models to derive the priority vectors from reciprocal pref-

erence relations. Section 5 develops a group decision making method under uncertain multiplicative

linguistic environments based on the defined possibility degree formula and models.

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2015 The Authors.

International Transactionsin Operational Research C

2015 International Federation of OperationalResearch Societies