Managing the consensus in group decision making in an unbalanced fuzzy linguistic context with incomplete information

Abstract

To solve group decision-making problems we have to take in account different aspects. On the one hand, depending on the problem, we can deal with different types of information. In this way, most group decision-making problems based on linguistic approaches use symmetrically and uniformly distributed linguistic term sets to express experts’ opinions. However, there exist problems whose assessments need to be represented by means of unbalanced linguistic term sets, i.e., using term sets which are not uniformly and symmetrically distributed. On the other hand, there may be cases in which experts do not have an in-depth knowledge of the problem to be solved. In such cases, experts may not put their opinion forward about certain aspects of the problem and, as a result, they may present incomplete information. The aim of this paper is to present a consensus model to help experts in all phases of the consensus reaching process in group decision-making problems in an unbalanced fuzzy linguistic context with incomplete information. As part of this consensus model, we propose an iterative procedure using consistency measures to estimate the incomplete information. In addition, the consistency measures are used together with consensus measures to guided the consensus model. The main novelty of this consensus model is that it supports the management of incomplete unbalanced fuzzy linguistic information and it allows to achieve consistent solutions with a great level of agreement.

Introduction

The increasing complexity of the social–economic environment nowadays has caused that the decision-making processes are being widely studied [15], [18]. Many organizations have moved from a single decision maker or expert to a group of experts to accomplish this task successfully. A group decision making (GDM) problem is usually understood as a decision problem which consists in finding the best alternative(s) from a set of feasible alternatives, $X=\{x_1\text{,}\dots \text{,}{x}_n\}$, according to the preferences provided by a group of experts, $E=\{e_1\text{,}\dots \text{,}{e}_m\}$, characterized by their experience and knowledge. To do this, experts have to express their preferences by means of a set of evaluations over the set of alternatives.

In this paper, we assume that experts use preference relations [8], [31], [47], [48], amongst other reasons, because they are a useful tool in the aggregation of experts preferences into group preference [8], [9], [10], [31], [32], [35], [45], [48] and focuses exclusively on two alternatives at a time, which facilitates experts when expressing their preferences. However, this way of providing preferences limits experts in their global perception of the alternatives and, as a consequence, the provided preferences could be not rational. Usually, rationality is related to consistency, which is associated with the transitivity property. Many properties have been suggested to model transitivity of a fuzzy preference relation [32]. One of these properties is the additive consistency, which, as shown in [32], can be seen as the parallel concept of Saaty’s consistency property in the case of multiplicative preference relations [47]. Obviously, the consistent information, i.e., information which does not imply any kind of contradiction, is more relevant or important than information containing some contradictions. Thus, it would be of great importance to measure the level of consistency of each expert in the GDM problem.

In these problems, a difficulty that has to be addressed is the lack of information. As aforementioned, each expert has his/her own experience concerning the problem being studied, which also may imply a major drawback, that of an expert not having a perfect knowledge of the problem to be solved. Indeed, there may be cases where an expert would not be able to efficiently express any kind of preference degree between two or more of the available options. This may be due to an expert not possessing a precise or sufficient level of knowledge of part of the problem, or because that expert is unable to discriminate the degree to which some options are better than others. Experts in these situations would rather not guess those preference degrees and, as a consequence, they might provide incomplete information [1], [2], [4], [29], [30], [38], [39], [45], [49], [50]. Therefore, it would be of great importance to provide the experts with tools that allow them to express this lack of knowledge in their opinions.

Another important issue to bear in mind is the different types of information used by the experts to provide their opinions. Usually, many problems present quantitative aspects which can be assessed by means of precise numerical values [8], [30], [29], [37]. However, some problems present also qualitative aspects that are complex to assess by means of precise and exact values. In these cases, the fuzzy linguistic approach [19], [23], [34], [41], [49], [50], [55], [56], [57] can be used to obtain a better solution. This is the case, for instance, when experts try to evaluate the “comfort” of a car, where linguistic terms like “good”, “fair”, “poor” are used [40]. Many of these problems use linguistic variables assessed in linguistic term sets whose terms are uniformly and symmetrically distributed, i.e., assuming the same discrimination levels on both sides of mid linguistic term. However, there exist problems that need to assess their variables with linguistic term sets that are not uniformly and symmetrically distributed [21], [33]. This type of linguistic term sets are called unbalanced linguistic term sets (see Fig. 1).

To solve GDM problems, the experts are faced by applying two processes before obtaining a final solution [22], [25], [31], [36], [37]: the consensus process and the selection process (see Fig. 2). The former consists in obtaining the maximum degree of consensus or agreement between the set of experts on the solution set of alternatives. Normally, the consensus process is guided by a human figure called moderator [7], [22], [25], [36], who is a person that does not participate in the discussion but monitors the agreement in each moment of the consensus process and is in charge of supervising and addressing the consensus process toward success, i.e., to achieve the maximum possible agreement and to reduce the number of experts outside of the consensus in each new consensus round. The latter refers to obtaining the solution set of alternatives from the opinions on the alternatives given by the experts. It involves two different steps [26], [46]: aggregation of individual opinions and exploitation of the collective opinion. Clearly, it is preferable that the set of experts achieves a great agreement amongst their opinions before applying the selection process and, therefore, in this paper we focus on the consensus process.

A consensus process is defined as a dynamic and iterative group discussion process, coordinated by a moderator helping experts bring their opinions closer. If the consensus level is lower than a specified threshold, the moderator would urge experts to discuss their opinions further in an effort to bring them closer. On the contrary, when the consensus level is higher than the threshold, the moderator would apply the selection process in order to obtain the final consensus solution to the GDM problem. In this framework, an important question is how to substitute the actions of the moderator in the group discussion process in order to automatically model the whole consensus process. Some automatic consensus approaches have been proposed in [6], [29], [31], [34], [42]. Most of these consensus models use only consensus measures to control and guide the consensus process. However, if a consensus process is seen as a type of persuasion model [16], other criteria could be used to guide consensus reaching processes as, for example, the cooperation or consistency criterion. Some fuzzy consensus approaches based on both consistency and consensus measures can be found in [14], [17], [24], [29].

The aim of this paper is to present a consensus model to deal with GDM problems in which experts use incomplete unbalanced fuzzy linguistic preference relations to provide their preferences. This consensus model will not only be based on consensus measures but also on consistency measures. We use two kinds of consensus measures to guide the consensus reaching process, consensus degrees, which evaluate the agreement of all the experts, and proximity measures, which evaluate the agreement between the experts’ individual opinions and the group opinion. To compute them, first, all missing values are estimated using a consistency-based estimation procedure. This estimation procedure is based on the Tanino’s consistency principle and makes use of all the estimation possibilities that derive from it. In this approach, the computation of missing values in an expert’s incomplete unbalanced fuzzy linguistic preference relation is done using only the preference values provided by that particular expert. By doing this, it is assured that the reconstruction of the incomplete unbalanced fuzzy linguistic preference relation is compatible with the rest of the information provided by that expert. Also, the main aim in the design of these approaches is to maintain or maximise the expert’s global consistency, as it has been shown in [11]. Afterwards, some consistency measures for each expert are computed. Both consistency and consensus measures are used to design a feedback mechanism, and, in such a way, we substitute the actions of the moderator and give advice to the experts on how they should change and complete their opinions to obtain a solution with a high consensus degree (making experts’ opinions closer).

The rest of the paper is set out as follows. Section 2 deals with the preliminaries necessary to develop our consensus model. In Section 3, the consensus model for GDM problems with incomplete unbalanced fuzzy linguistic information is presented. Section 4 shows a practical example to illustrate the application of the consensus model. Finally, some concluding remarks are pointed out in Section 5.