Decision making with intuitionistic linguistic preference relations

• Author: Qingxian An,Fanyong Meng,Jie Tang,Xiaohong Chen
• Date: 01 September 2019
• DOI: http://doi.org/10.1111/itor.12383
• Published date: 01 September 2019

Intl. Trans. in Op. Res. 26 (2019) 2004–2031

DOI: 10.1111/itor.12383

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RESEARCH

Decision making with intuitionistic linguistic preference

relations

Fanyong Menga,b, Jie Tanga, Qingxian Anaand Xiaohong Chena,c

aSchool of Business, Central South University, Changsha 410083, China

bCollaborative Innovation Center on Forecast and Nanjing University of Information Science and Technology,

Nanjing 210044, China

cSchool of Accounting, Hunan University of Commerce,Changsha 410205, China

E-mail: mengfanyongtjie@163.com [Meng]; tjie411@126.com [Tang]; anqingxian@csu.edu.cn [An];

cxh@csu.edu.cn [Chen]

Received 17 December 2015; receivedin revised form 21 October 2016; accepted 24 November 2016

Abstract

To address the preferred and nonpreferred degrees of linguistic variables, this paper introduces intuitionis-

tic linguistic preference relations (ILPRs) that apply intuitionistic linguistic variables (ILVs) to denote the

decision makers’ preferences. To judge the consistency of ILPRs, a consistent concept is introduced, and a

consistency index is defined. When ILPRs are unacceptably inconsistent, a method to improve the consis-

tency is introduced. Then, an approach to rank ILVs is introduced.In some situations where ILPRs might be

incomplete, the consistency-based linear programming model is constructed to evaluate the missing values.

Considering group decision making, a group consensus index is defined, and its several desirable properties

are discussed. Meanwhile, an acceptability-based consistency and consensus approach is developed, and the

associated example is offered to show the efficiency of the proposed procedure.

Keywords: intuitionistic linguistic preference relation; consistency index; group consensus index; aggregation operator;

programming model

1. Introduction

A decision-making problem usually requires at least one decision maker to evaluate and rank a

heterogeneous set of objects. The analytic hierarchy process (AHP) (see Saaty, 1980) is an efficient

and powerful tool to cope with decision-making problems that need the decision maker to compare

each pair of objects according to every criterion and give their judgments. Since Saaty (1980)

first introduced the AHP to us, it has been largely developed (see Orlovsky, 1978; Buckley, 1985;

Choudhury et al., 2006; Xu, 2006, 2007; Srdjevic, 2007; Xia et al., 2013). According to the preference

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F. Meng et al. / Intl. Trans.in Op. Res. 26 (2019) 2004–2031 2005

value, the preferencerelation can be classified into two types. One type is the quantitative preference

relation (see Orlovsky, 1978; Buckley, 1985; Singh et al., 1989; Xu, 2007, Xu and Yager, 2009; Xia

et al., 2013; Meng et al., 2016c), and the other type is the qualitativepreference relation (see Xu, 2006;

Tapia-Garc´

ıa et al., 2012; Zhang et al., 2012; Xia et al., 2014; Meng et al., 2016a, 2016b). Because

reciprocal preference relations (see Saaty, 1980) and fuzzy preference relations (see Orlovsky, 1978)

need the decision maker to offer his/her exact preference degree or intensity of one object over

another, this might be difficult or even impossible in some situations. Thus, researchers defined

several families of extended preference relations to denote the decision makers’ vague and uncertain

judgments (see Buckley, 1985; Xu, 2007; Xu and Yager, 2009; Tapia-Garc´

ıa et al., 2012; Wu and

Chiclana, 2014; Das and Guha, 2015; Das et al., 2016; Meng and Chen, 2016).

All above-mentioned preference relations need the decision maker to offer the quantitative judg-

ments. However, in some practical situations, the problems might be too complex or ill-defined to

use quantitative preferences. To cope with this issue, linguistic variables (see Zadeh, 1975) are a good

choice, as they clearly addressthe decision makers’ qualitative judgments. For instance, Herreraand

Mart´

ınez (2000) introduced a simple and novel linguistic representation model called the 2-tuple

linguistic representation model that consists of a pair of values: a linguistic term and a symbolic

translation value in [−0.5, 0.5). Dong et al. (2008) proved that the extended linguistic aggrega-

tion (ELA) operator (see Xu, 2004) is equivalent to the associated 2-tuple linguistic aggregation

operator (Zhang et al., 2004). This means that there is no information loss by using the ELA oper-

ator to aggregate linguistic variables. Dong and Zhang (2014), Dong and Herrera-Viedma (2015),

and Dong et al. (2010) researched several methods for decision making with linguistic preference

relations.

Although linguistic preference relations are a powerful tool to express the decision makers’

qualitative preferences, they are based on the assumption that the preferred degree of one object

over another for a linguistic term is 1. However, as Xu (2007) noted, in some real-life situations

the decision maker might be not so confident for his/her preferences for objects because of various

reasons, such as imprecise or insufficientlevel of knowledge of the problem and the inter complexity

of objects. In such cases, it might be more suitable to express his/her preferences for objects with

a certain degree. To address these situations, intuitionistic linguistic variables (ILVs; see Wang and

Li, 2009) might be a good choice, which are expressedby a linguistic ter m, membership degree, and

nonmembership degree. In this paper, we pay our attention to decision making with intuitionistic

linguistic preference relations (ILPRs). To do this, the remainder of this paper is organized as

follows.

In Section 2, some basic and relative concepts are briefly reviewed. In Section 3, the concepts

of ILPRs and consistent ILPRs are introduced. Then, a consistency index (CI) is presented, and a

method to improvethe consistency is introduced. After that, the extended intuitionistic linguistic or-

dered weighted averaging (EILOWA)operator is defined. When the weights of the ordered positions

are not known, a model for determining the optimal symmetric weighting vector is constructed.

In Section 4, considering incomplete ILPRs, a consistency-based linear programming model is

constructed. In Section 5, a group consensus index (GCOI) is defined to measure the consensus of

ILPRs. When the consensus of individual ILPRs is unacceptable, an adjusted method to improve

the group consensus is presented. Then, an approach to group decision making with ILPRs is

developed. Concluding comments are given in the last section.

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