Decision Support
Adaptive consensus reaching process with hybrid strategies for large-scale group decision making

https://doi.org/10.1016/j.ejor.2019.10.006Get rights and content

Highlights

  • We propose an approach to determine sub-groups’ weights.

  • We give two hierarchies’ consensus measures for large-scale group decision making.

  • We introduce an adaptive consensus model for large-scale group decision making.

  • We give hybrid feedback strategies to improve the consensus degree.

  • We validate the method with legislative amendment in United Nations.

Abstract

Large-scale group decision making, which involves dozens to hundreds of experts, is attracting increasing attention and has become an important topic in the field of decision making. Because of the clustering process, a large-scale group decision making problem can be divided into two levels: inter sub-group and intra sub-group. In existing consensus models under the large-scale group decision making environment, the degree of consensus within the intra sub-group is not truly taken into account. To deal with this issue, this work develops an adaptive consensus model for the sub-groups composed of hybrid strategies, with or without a feedback mechanism, according to the different levels of inter and intra degrees of consensus. These different levels of consensus are divided into four scenarios (high–high, high–low, low–high, low–low), and different feedback suggestions are generated corresponding to different cases. This hybrid mechanism can reduce the cost of supervision for the moderator. The fuzzy c-means clustering algorithm is used to classify experts. A weight-determining method combining the degree of cohesion and the size of a sub-group is introduced. Finally, an illustrative example is offered to verify the practicability of the proposed model. Some discussions and comparisons are provided to reveal the advantages and features of the proposed model.

Introduction

With the rapid expansion of technological paradigms such as e-democracy (Efremov, Insua & Lotov, 2009; Kim, 2008) and social networks (Liang, Liao & Liu, 2017), a decision-making problem may include a large number of experts. Consequently, large-scale group decision making (LSGDM), which includes dozens to hundreds of experts, is attracting increasing attention and has become an important topic in the decision-making field. Unlike the traditional group decision making (GDM) problems in which few experts are involved, there are two mainly used standards to identify the LSGDM problems regarding the number of experts: one is 11 experts (Xu, Du & Chen, 2015; Zhang, Dong & Herrera-Viedma, 2018) and the other is 20 experts (Wu & Xu, 2018). In this study, we use the latter standard. That is to say, a GDM problem can be considered as an LSGDM problem if at least 20 experts participate.

Consensus is an important issue in GDM. Strictly speaking, consensus means a unanimous agreement among all participants concerning all alternatives. However, this concept is often unrealistic. Many scholars have softened this strict definition (Wu & Xu, 2016, 2018; Xu, Du et al., 2015; Zhang et al., 2018). Kacprzyk and Fedrizzi (1986, 1988, 1989) conducted a series of work on soft consensus. A review on soft consensus can be found in Herrera-Viedma, Cabrerizo, Kacprzyk and Pedrycz (2014). Nevertheless, most existing consensus models (Chen, Lee, Yang & Sheu, 2012; Dong & Cooper, 2016; Gupta, 2018; Labella, Liu, Rodríguez & Martínez, 2018) only considered small-scale experts, and thus, may not be appropriate for handling LSGDM problems because of the high adjustment cost or time limitation. A comprehensive analysis of the limitations of classical consensus models can be found in Labella et al. (2018).

In practice, it is a challenge to reach an agreement with all experts owing to their different attitudes, motivations, and perceptions. Thus, the consensus reaching process (CRP) is essential to obtain the best solution agreed by all experts in GDM. Scholars have developed different CRP methods and structures in different contexts, such as the CRP in a dynamic/web context (Pérez, Cabrerizo & Herrera-Viedma, 2010, 2011), CRP in a social network (Capuano, Chiclana, Fujita & Herrera-Viedma, 2018), consensus with minimum cost (Cheng, Zhou, Cheng, Zhou & Xie, 2018), and adaptive CRP (Chen et al., 2012; Dong & Cooper, 2016; Gupta, 2018; Mata, Martínez & Herrera-Viedma, 2009; Pang, Liang & Song, 2017; Pérez, Cabrerizo, Alonso & Herrera-Viedma, 2014; Rodríguez, Labella, Tré & Martínez, 2018; Zhang, Zhu, Liu, Chen & Ma, 2017). Among these models, the adaptive CRP can generate recommendations adaptively by considering dynamic parameters such as levels of group consensus (Mata et al., 2009) and the weights of experts (Pérez et al., 2014). For instance, Mata et al. (2009) proposed a consensus model in which the amount of recommendations and the number of experts involved in each iteration were adapted to different levels of group consensus. At the initial iteration stage, given that the level of consensus is very low, all the experts are advised to modify their evaluation information. With the level of consensus increases, both the number of changes and the number of experts who need to modify their evaluations decrease. This adaptive CRP was based on four categories of consensus levels: very low, low, medium, and high. Pérez et al. (2014) introduced a CPR that adapted to the weights of experts. The logic of this model is that the experts with higher weights have greater expertise, and thus, they can be asked to perform a smaller amount of modifications in the CRP.

The existing methods can be used to handle GDM problems under different circumstances. However, there are still some limitations:

  • (1)

    The clustering process plays a vital role in solving LSGDM problems, and this is the main difference between general GDM and LSGDM. Determining a cluster's weight is critical in LSGDM. Most of the existing studies (Wu & Xu, 2018; Xu, Du et al., 2015; Zhang et al., 2018) simply used the number of experts in a cluster as the standard to determine the weights of clusters. In this way, the clusters with the same number of experts but possessing different inner characteristics in terms of cohesiveness or diversity would still have the same weights. Thus, we need to develop a method to determine the weights of different clusters that considers both the degree of cohesion and the size of a cluster.

  • (2)

    The use of the adaptive consensus model in an LSGDM context is still at an initial exploratory stage, and thus, it faces big challenges. In LSGDM, owing to the various backgrounds and expertise of experts, it is more difficult to reach a consensus result than in conventional GDM. Thus, it is necessary to research the adaptive consensus model within a large-scale context. Although Rodríguez et al. (2018) introduced an adaptive consensus model under the LSGDM context similar to the method of Mata et al. (2009), their model classified the global consensus into three levels: low, high, and high enough. Nevertheless, this model did not take into account the intra consensus of a sub-group. In LSGDM, the clustering process should be used to cluster the experts into several groups based on their preference information to simplify the decision-making process. As a result, two levels of consensus are generated, i.e., the level of consensus within a sub-group and the level of consensus of a sub-group to the global group. These two kinds of consensus are respectively named as the intra consensus and inter consensus in this study. The existing studies about LSGDM did not generate feedback suggestions considering these two kinds of consensus at the same time.

To address these shortcomings, in this study, we develop a novel adaptive consensus method under the LSGDM context. This method first uses the fuzzy c-means (FCM) clustering algorithm to classify the experts into several clusters. Then, a weight-determining method for clusters is proposed. Afterwards, the consensus measures corresponding to the intra consensus and inter consensus are defined. Based on these consensus measures, an adaptive feedback mechanism, composed of hybrid consensus strategies with or without a feedback mechanism according to the different degrees of inter and intra agreements, is presented in detail. Finally, a case study is provided to better understand the theory of this study.

The primary contributions of this work are summarized as follows:

  • (1)

    A weight-determining method for clusters is developed. This method considers both the size and the cohesion degree of a cluster, and thus improves the weight-determining method that only considers the size of a cluster. Furthermore, a cohesive cluster will exert much influence in the global group, and thus this method can accelerate the convergence speed of the CRP objectively since the aim of the CRP is to make the group information central. An aggregation formula is proposed to integrate these two parameters. The degree of cohesion of a cluster includes the silhouette coefficient.

  • (2)

    According to the different levels of inter and intra consensus of a sub-group, this study develops an adaptive consensus model with hybrid strategies for multiple groups. This model can generate different recommendations according to the degrees of consensus of two levels. These different strategies can reduce the numbers of involved experts and the recommendations for modifications in the CRP within the LSDGM context. Thus, the proposed method can reduce the supervision and adjustment costs of the CRP.

  • (3)

    An illustrative example about the legislative amendment of the International Trade Law within the United Nations system is provided to explore and verify the applicability and feasibility of the proposed model. Then, comparisons regarding different parameters including the number of clusters and the consensus threshold are provided. We also give a comparison with Rodríguez et al. (2018)'s method to validate the advantage of the adaptive mechanism.

To achieve the above goals, this paper is organized as follows: Section 2 describes some concepts used in this study, including the LSGDM, CRP, and FCM. Section 3 presents a novel adaptive consensus model under the LSGDM context. In Section 4, an illustrative example is included to verify the applicability and feasibility of this model. Some comparisons and discussions are also provided. Concluding remarks are presented in Section 5.

To facilitate the comprehension of the paper, Table 1 summarizes some of the used notations.

Section snippets

Preliminaries

In this section, we review the tools employed to build the proposed model, that is, the concepts of LSGDM, CRP, and FCM.

An adaptive CRP within the LSGDM context

In this section, an adaptive consensus model for LSGDM problems with hybrid strategies for multiple groups is proposed. The main innovation of this model is that it generates feedback recommendations according to the different levels of intra consensus (consensus in a sub-group) and inter consensus (consensus of a sub-group to the global group). The consensus measure is divided into two levels in LSGDM because of the clustering algorithm. The intra consensus level can reflect the degree of

Illustrative example

This section presents an illustrative example to verify the applicability of the proposed adaptive CRP with hybrid strategies for LSGDM problems with RCMs.

Discussions and comparative analyses

In this section, we provide some discussions about setting parameters for the proposed model and the comparisons with related LSGDM methods.

Conclusions

This study developed an adaptive consensus model with hybrid strategies within the LSGDM context. A key characteristic of this model is the design of a hybrid feedback mechanism to improve the consensus performance in LSGDM problems. This model is composed of four parts: classifying experts, determining the weights of sub-groups, computing degrees of consensus, and improving levels of consensus. An illustrative example about the legislative amendment in UNCITRAL was implemented to illustrate

Acknowledgments

The work was supported by the National Natural Science Foundation of China (71571156 and 71971145), and the 2019 Sichuan Planning Project of Social Science (No. SC18A007).

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