Reaching a consensus with minimum adjustment in MAGDM with hesitant fuzzy linguistic term sets

Highlights

• We propose a distance-based consensus measure for the MAGDM with HFLTSs.

• We develop a minimum adjustment distance consensus rule for the MAGDM with HFLTSs.

• A minimum distance aggregation model is presented to derive the collective opinion.

• We present a consensus reaching process for the MAGDM with HFLTSs.

• The convergence proof of the consensus reaching process is provided.

Abstract

In real decision making, decision makers tend to express their opinions with uncertainty when facing complicated decision problem and environment. This paper develops a novel consensus reaching process for multiple attribute group decision making (MAGDM) with hesitant fuzzy linguistic term sets (HFLTSs). Firstly, we define a new distance measure for two HFLTSs and propose a distance-based consensus measure for the MAGDM with HFLTSs. Then, based on this consensus measure, we develop a minimum adjustment distance consensus rule for the MAGDM with HFLTSs, which can minimize the adjustment distance between the original and adjusted opinions in the process of reaching consensus. Moreover, to obtain the collective opinion with maximum consensus, we develop a minimum distance aggregation model, which is to minimize the maximum of the distance between each decision maker's individual opinion and the collective opinion. Furthermore, based on the proposed consensus rule and aggregation model, we present a consensus reaching process for MAGDM with HFLTSs. Finally, we provide the convergence proof of the consensus reaching process, and a numerical example is used to demonstrate the validity of the consensus reaching process.

Introduction

In real group decision making (GDM), decision makers usually express their opinions with qualitative information that are represented by means of linguistic variables. In the past few decades, a wide range of linguistic computational models, such as the semantic model [1], [2], [3], the symbolic model [4], [5] and the model based on linguistic 2-tuples [6], [7], [8], [9], have been developed and applied in GDM with linguistic information. Herrera et al. [10] gave a systematic survey of the linguistic computational models in GDM.

The good performance of linguistic computing dealing with uncertainty has caused a spread use of it in different types of decision based applications [11], [12]. However, in the above linguistic computational models, decision makers are restricted to express their opinions with a single and simple linguistic term. In most situations, it is hard for decision maker to use a single linguistic term to express his/her opinion when facing with a complex decision problem under uncertainty. Decision makers may hesitate between different linguistic terms and require richer expressions to express their knowledge more accurately. For this reason, Rodríguez et al. [13] proposed the concept of hesitant fuzzy linguistic term set (HFLTS) to provide a linguistic and computational basis to increase the richness of linguistic elicitation based on the fuzzy linguistic approach and the use of context-free grammars by using comparative terms. Following this, Rodríguez et al. [14] further developed a new linguistic GDM model dealing with linguistic information based on the HFLTSs and context-free grammars. To facilitate the process of computation with words (CWW), Liu and Rodríguez [15] presented a fuzzy envelope for the HFLTS to carry out the CWW process. Meanwhile, Beg and Rashid [16] proposed a new aggregation method for multiple attribute group decision making (MAGDM) with HFLTSs. Moreover, Wei et al. [17] developed novel comparison methods based on possibility degree formulas and defined two aggregation operators for the HFLTSs. Liao et al. [18]proposed a family of distance and similarity measure for the HFLTSs. Additionally, several other multiple attribute decision making models with HFLTSs were investigated and developed in [19], [20].

In GDM, decision makers’ opinions may differ substantially. A consensus process is considered as a dynamic and interactive group discussion process to help decision makers improve the consensus level [21], [22]. Generally, it is hard to reach a complete agreement among decision makers in real GDM, so soft consensus measure is widely used in consensus models [23], [24], [25], [26]. A large number of researches focus on the consensus reaching process in GDM problems [21], [22], [27], [28], [29], [30], [31], [32], [33], [34]. Herrera et al. [35] proposed the first soft consensus model in GDM with fuzzy linguistic preferences. Later, Herrera-Viedma et al. [36] presented a basic consensus framework in GDM problem, in which a feedback mechanism is designed to aid decision makers to improve the consensus level. Furthermore, Herrera-Viedma et al. [37] presented a systematic overview of consensus models based on soft consensus measures. In the consensus reaching process, it is significant to design an effective feedback mechanism to guide decision makers reach consensus with minimum adjustments. Ben-Arieh and Easton [38] proposed the minimum cost consensus models and maximum expert consensus models, which can obtain the optimal adjusted individual opinions with minimum consensus cost or maximum experts with consensus under a given cost budget. Inspired by Ben-Arieh and Easton's consensus models, a mass of studies focus on minimizing the adjustment between decision makers’ original and adjusted opinions in the consensus reaching process [32], [39], [40], [41].

However, to date, there exist few studies on the consensus models in the MAGDM with HFLTSs. Wu and Xu [42] proposed a new approach to deal with the consensus reaching process for MAGDM with hesitant fuzzy linguistic information. Following this, Wu and Xu [43] further developed separate consistency and consensus processes to deal with hesitant fuzzy linguistic preference relation individual rationality and group rationality. Moreover, Dong et al. [40] defined a novel distance-based approach to measure the difference between two HFLTSs, and proposed a two-stage consensus model with minimum adjusted simple terms in the hesitant linguistic GDM. As it was pointed out in [44], different aggregation methods, consensus measures and feedback mechanisms categorize the different consensus models in GDM. Thus, we focus on the enrichment and perfection of the study on the consensus model in the hesitant linguistic MAGDM context. In order to do this, we mainly tackle the following three issues in this paper:

• How to define the consensus measure in the hesitant linguistic MAGDM context.

• How to aggregate the individual opinions into the collective opinion with hesitant linguistic information.

• How to design an effective feedback mechanism to guide decision makers to reach consensus with minimum adjustment.

Motivated by these three issues, we firstly define a new distance-based consensus measure and present a minimum adjustment distance consensus rule in the MAGDM problem with HFLTSs. Then, we develop a minimum distance aggregation model, which is to minimize the maximum of the distance between each decision maker's individual opinion and the collective opinion. Based on the proposed consensus rule and aggregation model, we design a consensus reaching process to guide decision makers to reach consensus with minimum adjustment distance.

The rest paper is organized as follows. Section 2 provides the basic notations and operation laws regarding the HFLTSs and proposes the MAGDM problems with HFLTSs. Section 3 defines a new distance measure between two HFLTSs and the consensus measure in this study. A consensus rule with minimum adjustment distance is presented in Section 4. Following this, Section 5 develops a minimum distance aggregation model and presents a consensus reaching process based on the proposed consensus rule for the MAGDM problems with HFLTSs. Finally, an illustrative example is provided in Section 6, and concluding remarks are included in Section 7.