Managing consensus and self-confidence in multiplicative preference relations in group decision making

Abstract

Preference relations have been widely used in Group Decision Making (GDM) to represent decision makers’ preferences over alternatives. Recently, a new kind of preference relation called the self-confident multiplicative preference relation has been presented, which is formed considering multiple self-confidence levels into the multiplicative preference relation. This paper proposes an iteration-based consensus building framework for GDM problems with self-confident multiplicative preference relations. In this consensus building framework, an extended logarithmic least squares method is presented to derive the individual and collective priority vectors from the self-confident multiplicative preference relations. Then, a two-step feedback adjustment mechanism is used to assist the decision makers to improve the consensus level, which adjusts both the preference values and the self-confidence levels. The simulation experiments are devised to testify the efficiency of the proposed consensus building framework. Simulation results show that compared with only adjusting the preference values in the iteration-based consensus model, adjusting both the preference values and the self-confidence levels can accelerate the consensus success ratio and improve the consensus success ratio.

Introduction

Group Decision Making (GDM) can be defined as a situation where a group of decision makers participate in a decision process, provide their opinions regarding the given alternatives, and select the best alternative(s) by the aggregation of their opinions. Preference relation is one of the most widely used preference representation structures in GDM [52]. Several different types of preference relations have been proposed in the literature, such as additive preference relations (also called fuzzy preference relations) (e.g., [15], [24], [44], [45], [50]), multiplicative preference relations (e.g., [47], [48], [49]), linguistic preference relations (e.g., [17], [37], [55]).

Recently, a new kind of preference relation has been proposed by considering self-confidence levels in the preference relation, which is named by self-confident preference relation [40]. In Ureña et al. [51], the concept of self-confidence level was introduced based on the hesitancy degree of the reciprocal intuitionistic fuzzy preference relation, and a new aggregation operator was proposed by taking into account both the consistency level and the confidence level. Formally, Liu et al. [40] proposed self-confident multiplicative, additive, and linguistic preference relations through investigation of the self-confident preference relations under the heterogeneous context. Each element of the self-confident preference relations consists of two components: the first part is the preference evaluation between pairs of alternatives, and the second part represents the decision maker's self-confidence level of the first part (i.e., preference evaluation) and it is often defined on a rating scale (numerical scale or linguistic scale).

Although some preliminary works have been made regarding GDM with self-confident preference relations, the consensus issue is not considered in the literature. In practical GDM problems, two processes are often carried out to obtain a final collective solution: 1) the consensus process and 2) the selection process. The consensus process is adopted to improve the consensus level among a group of decision makers, and the selection process is applied to obtain a collective ranking of alternatives. Traditionally, consensus is defined as the full and unanimous agreement of all the decision makers regarding all the feasible alternatives. However, in real life, a complete agreement may be difficult to achieve, and it may not always necessary in practice. This belief has led to the use of the “soft” consensus (e.g., [31], [34]). So far, a large number of studies based on ‘‘soft’ ’ consensus have been reported, which can be classified into the following categories [21]:

• Consensus models with different preference representation structures (e.g., [2], [3], [9], [10], [30], [32], [33], [35], [41]). For instance, Kacprzyk et al. [35] proposed a consensus building model for GDM with additive preference relations. Altuzarra et al. [3] proposed a consensus building model for GDM with incomplete preference relations. Alonso et al. [2] presented a novel consensus building model which uses linguistic preference relations to support consensus in new Web 2.0 Communities.

• Consensus models based on consistency and consensus measures (e.g., [12], [17], [58], [62]). For example, Chiclana et al. [12] proposed a consensus building model for GDM problems that proceeds from consistency to consensus. Zhang et al. [62] and Wu and Xu [58] developed consensus building models that simultaneously manage individual the consistency and consensus.

• Consensus models considering the behaviors/attitudes of decision makers (e.g., [22], [42], [46], [53], [54], [55], [56], [60]). Palomares et al. [42], Quesada et al. [46] and Xu et al. [60] all have studied the consensus building models by addressing non-cooperative behaviors of the decision makers. Meanwhile, Wu and Chiclana [53] and Wu et al. [55] have proposed trust-based consensus building models.

• Consensus models under dynamic/Web contexts (e.g., [1], [16], [36], [44]). Pérez et al. [44] proposed a dynamic consensus building model to manage decision situations in which the set of alternatives changes dynamically. Alonso et al. [1] and Kacprzyk and Zadrozny [36] investigated web-based consensus support systems. Dong et al. [16] proposed a consensus building model for the dynamic GDM problem.

• Consensus models with minimum adjustments or cost (e.g., [6], [7], [19], [26], [63]). Ben-Arieh and Easton [6] presented a consensus building model with minimum cost, which was subsequently extended to a maximum number of decision makers consensus model in Ben-Arieh et al. [7]. Meanwhile, Dong et al. [19] proposed a consensus building models with minimum adjustments under the linguistic context. Recently, the consensus building model with minimum cost has been pursued by Gong et al. [26] and Zhang et al. [63].

• Consensus models for GDM with multiple attributes (e.g., [21], [59], [61], [63]). For example, Xu et al. [61] proposed a consensus approach for eliminating conflicts in emergency multiple attribute GDM problems. Recently, Dong et al. [21] reported a consensus building model for a complex and dynamic multiple attribute GDM problem.

The above literature review shows that the existing consensus models have made considerable progress. However, they are incapable of handling GDM with self-confident preference relations due to the complexity of this kind of decision problem. Motived by the challenge to deal with the consensus issue in the GDM with self-confident preference relations, this study continues the research line of Ureña et al. [51] and Liu et al. [40] and proposes a novel iteration-based consensus building framework to manage consensus and self-confidence in the GDM with multiplicative preference relations. In this framework, an extended logarithmic least squares method is used in the selection process to derive individual and collective priority vectors from self-confident multiplicative preference relations. Then, a two-step feedback adjustment mechanism in the consensus process is proposed, which adjusts both the preference values and the self-confidence levels to help the decision makers to improve the consensus level. Following this, a simulation experiment is put forward for measuring the consensus efficiency of the novel consensus building model.

The remainder of the paper is organized as follows: Section 2 introduces preliminaries regarding the 2-tuple linguistic model, multiplicative preference relations, logarithmic least squares method and self-confident multiplicative preference relations. Then, Section 3 develops a novel consensus building framework for GDM with self-confident multiplicative preference relations. In Section 4, a consensus process is presented to improve the consensus level among the decision makers. An illustrative example is given in Section 5. Next, Section 6 proposes the simulation method to measure the consensus efficiency of the novel consensus building model. Finally, concluding remarks and future research directions are discussed in Section 7.