Elsevier

Information Sciences

Volume 369, 10 November 2016, Pages 128-143
Information Sciences

Probabilistic linguistic term sets in multi-attribute group decision making

https://doi.org/10.1016/j.ins.2016.06.021Get rights and content

Abstract

When expressing preferences in qualitative setting, several possible linguistic terms with different weights (represented by probabilities) may be considered at the same time. The probabilistic distribution is usually hard to be provided completely and ignorance may exist. In this paper, we first propose a novel concept called probabilistic linguistic term set (PLTS) to serve as an extension of the existing tools. Then we put forward some basic operational laws and aggregation operators for PLTSs. After that, we develop an extended TOPSIS method and an aggregation-based method respectively for multi-attribute group decision making (MAGDM) with probabilistic linguistic information, and apply them to a practical case concerning strategy initiatives. Finally, the strengths and weaknesses of our methods are clarified by comparing them with some similar techniques.

Introduction

Decision making is a common activity in every aspect of our daily life. People usually have to provide their opinions linguistically, due to the nature of qualitative criteria or the high cost of deriving accurate numerical values [21], [39]. Linguistic decision making, which considers the linguistic information as the values of linguistic variables, has achieved great development and played an important role in various fields, such as enterprise strategy planning [20], quality assessment [3], the selection of investment strategy [24], etc. In practical decision making problems, the decision makers (DMs) can use linguistic terms like ``good'', ``fair'' or ``poor'' to express their preferences over the considered alternatives, and make the optimal decision by using some proper decision making methods.

Up to now, a lot of research has been done on linguistic decision making [33]. The complexity of the real decision making problems needs multiple DMs to participate in decision making processes, and thus, group decision making (GDM) under linguistic environment has been widely studied over the last decades. For example, Herrera and Verdegay [13] put forward the linguistic assessments in GDM in 1993. Then Herrera et al. [11], [12] proposed a series of GDM models under linguistic setting. Later, Xu [31] established the goal programming model for multi-attribute decision making (MADM) with linguistic information. Ben–Arieh and Chen [2] studied the opinion aggregation and measure of consensus in linguistic GDM. Xu [30] gave an approach based on the uncertain linguistic ordered weighted geometric (LOWG) and the induced uncertain LOWG operators to GDM with uncertain multiplicative linguistic preference relations.

In practical applications, uncertainties may exist when expressing preferences by means of linguistic information. For instance, it is difficult for a DM to express his/her preferences by using just one linguistic term due to the complexity and uncertainty of the real problems. The DMs may be hesitant between several possible linguistic terms. Considering this fact, Rodriguez et al. [22] came up with hesitant fuzzy linguistic term sets (HFLTSs) based on hesitant fuzzy sets [25] and linguistic term sets (LTSs) [38], which allow a DM to propose several possible values for a linguistic variable. Wei et al. [28] defined the operations on HFLTSs, and presented some new linguistic aggregation operators for fusing HFLTSs. Beg and Rashid et al. [1] proposed a method to aggregate the opinions provided by the DMs on different criteria, where the opinions take the form of HFLTSs. To make a better use of HFLTSs in MADM, Zhu and Xu [42] put forward hesitant fuzzy linguistic preference relations (HFLPRs) and defined some consistency measures for HFLPRs. Lots of research has been done on the GDM problems based on HFLTSs and amounts of reliable and interesting results has been provided. Beg and Rashid [1] extended the TOPSIS method for HFLTSs. Meanwhile, Rodríguez et al. [23] developed a GDM model which deals with comparative linguistic expressions based on HFLTSs. Liu and Rodríguez [19] further studied the fuzzy envelopes of HFLTSs and their application in MADM. Dong et al. [4] put forward an optimal model for GDM with HFLTSs by minimizing the adjusted simple terms. In addition, to ease the computation, Wang [26], and Zhang and Wu [41] extended the HFLTSs to suit the non-continuous linguistic terms, namely, the extended HFLTSs (EHFLTSs).

However, in most of the current studies about HFLTSs, all possible values provided by the DMs have equal importance or weight. Obviously, it is not in accordance with the reality. In both individual decision making and GDM problems, the DMs may prefer some of the possible linguistic terms so that the set of possible values may have different importance degrees (e.g., taking the form of probability distributions).

Example 1

[14]. When evaluating the quietness of an engine of a certain brand (e.g., Honda engine), an expert may express that he/she is 50% sure that it is good, and 30% sure that it is excellent. Then the value of the linguistic variable (named Quietness) can be denoted by Quietness(Honda)={good(0.5),excellent(0.3)}

Example 2

A hundred consumers are invited to express how they feel about the overall comfortable degree of a vehicle (e.g., Honda XR-V). Suppose that sixty-five consumers state that it is high, twenty consumers state that it is very high, ten consumers insist that it is slightly high, and others do not express any opinions. In this case, the obtained information can be summarized as follows: Comfort(HondaXRV)={slightlyhigh(0.1),high(0.65),veryhigh(0.2)}

Therefore, according to the above examples, the assessment information includes not only several possible linguistic terms but also the associated probabilistic information. This information can be interpreted as probabilistic distribution [7], [29], [40], importance [19], degree of belief [36], [37] and so on. The ignorance of this information may lead to erroneous decision results. Moreover, it is usually hard to obtain the complete information of probabilistic distribution [25], [36], as seen in Examples 1 and 2. Therefore, in this paper, we introduce a more general concept, referred to as probabilistic linguistic term sets (PLTSs), to extend HFLTSs through adding probabilities without loss of any original linguistic information provided by the DMs. With the PLTSs, the DMs can not only provide several possible linguistic values over an object (alternative or attribute), but also reflect the probabilistic information of the set of values. By this way, we can get comprehensive and accurate preference information of the DMs.

There are some similar techniques which consider distinguishing the positions of possible linguistic terms. But there are some differences which we summarize in Table 1. Wang and Hao [27] considered two proportional linguistic terms at first. Then the model has been improved by Zhang et al. [40], Dong et al. [7], and Wu and Xu [29] respectively, by a general form of probabilistic distributions. Based on this idea, the numerical scale models have also been investigated in Refs. [5], [6], [8]. If the probabilistic information is complete, then both the existing techniques and this proposal can be seen as the similar tools. However, we enable the DMs to express even incomplete probabilistic distributions, i.e., partial ignorance is acceptable. Thus, the proposed PLTS can be seen as an extension of the existing techniques. It is more convenient for the DMs to provide their preferences. Moreover, Yang and Xu [36], [37] considered partial ignorance as well (in which the probabilistic information is interpreted as the degrees of belief) under the framework of evidential reasoning. But the theory of evidential reasoning is generally not an approach of computing with words (CWW) because the order relation defined on linguistic terms is completely ignored and a linguistic term set is only considered as a set of linguistic grades.

The rest of this paper is organized as follows: Section 2 reviews the background regarding the HFLTSs and some basic operations. In Section 3, we propose the concept of PLTSs, and put forward some basic operations and aggregation operators for PLTSs. Section 4 focuses on multi- attribute group decision making (MAGDM) under linguistic environment. We develop an extended TOPSIS method and an aggregation-based method respectively for MAGDM with probabilistic linguistic information. Finally, a case study is provided in Section 5 to illustrate the usefulness of our methods in strategy initiatives, and the conclusions are included in Section 6.

Section snippets

Preliminaries

In this section, we review some concepts and operations related to LTSs and HFLTSs.

Probabilistic linguistic term sets

In order to overcome the abovementioned issue of HFLTSs, in this section, we will propose a novel concept called PLTSs, and investigate the comparison method, the basic operation laws and the aggregation operators.

Problem description

In the following, we consider the MAGDM problem with probabilistic linguistic information: Let x={x1,x2,...,xm} be a finite set of m alternatives, and a={a1,a2,...,an} be a set of n attributes, whose weight vector is w=(w1,w2,...,wn)T, where wj ≥ 0 (j=1,2,...,n) and j=1mwj2=1. Due to the complexity and uncertainty of the practical decision making problems, the information about the attributes’ weights is completely unknown or partly known.

The DMs provide their evaluation values respectively,

A case study

In the following, we further illustrate the practicality of PLTSs by utilizing a practical example adapted from Parreiras et al. [20]:

The board of directors in a company, which includes five members, is to plan the development of large projects (strategy initiatives) for the following five years. Suppose that there are three possible projects xi (i=1,2,3) to be evaluated. It is necessary to compare these projects so as to select the most important of them as well as order them from the point of

Conclusions

It is not rational to completely ignore the importance and probability of possible linguistic terms in GDM with linguistic information. When the probabilistic information of the linguistic terms is incomplete, we have extended the existing techniques and come up with PLTSs in this paper. Some basic operations, comparison laws and aggregation operators have been studied. To ease the use of PLTSs, the extended TOPSIS method and the aggregation-based method have been developed after the

Acknowledgements

The authors thank the Editor-in-Chief, the Associate Editor and the anonymous reviewers for their helpful comments and suggestions, which have led to an improved version of this paper. The work was supported by the National Natural Science Foundation of China (Nos. 61273209, 71571123), and the Central University Basic Scientific Research Business Expenses Project (No. skgt201501).

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