Abstract: Empathy relationship, as a social relationship, objectively exists in some realistic group decision-making problems, so group size becomes an important factor affecting group consensus. It increases the uncertainty of group decision-making problems and leads to a costly and time-consuming consensus reaching process. Therefore, this paper aims to investigate a large-scale group consensus decision-making method in a probabilistic linguistic empathy network environment. A robust cost consensus model is then developed using the chance-constrained robust optimization method during the feedback adjustment process to account for the effects caused by the uncertainty of the unit adjustment cost.
   Considering that it is difficult for people to clearly evaluate their empathy relationship with others in real life, it is more expressed in natural language. We first define a probabilistic linguistic empathy function to evaluate the empathy relationship among decision makers, and thus build a probabilistic linguistic empathy network. Based on such a network, the decision makers’ preferences can be decomposed into their intrinsic preference, representing their true opinion, and their empathetic preference for other decision makers in the network.
   Secondly, a fuzzy C-means clustering method based on preference relationship is used to divide decision makers with high similarity into several subgroups. Since the empathy relationship among decision makers should be used as a reliable indicator to assign weights to each cluster, this paper considers the size, cohesion, and overall empathy degree of each cluster to determine the importance of clusters. In addition, since decision makers in the same cluster have highly similar preferences, the degree of attitudinal empathy based on the empathy relationship is utilized to determine the weight of decision makers within a class.
   When the group consensus degree cannot reach a predefined consensus threshold, to improve the consensus quality, opinion adjustment becomes a natural phenomenon. At this point, an efficient feedback adjustment process needs to be implemented. This paper designs an optimization-based feedback mechanism incorporating a minimum cost consensus model, which is guided by empathetic relationship. Affected by factors such as social experience and educational background, the unit adjustment cost of decision makers from different organizations may present uncertainty. Robust optimization, as a powerful tool for dealing with uncertainty, is often combined into minimum cost consensus models to deal with uncertain unit adjustment costs. However, most of these robust consensus models pre-set uncertainty level parameters to control the fluctuation range of uncertain parameters, resulting in results that may be too conservative. In order to make full use of the stochastic nature of fluctuating data when establishing the uncertainty immune solution, this paper uses the chance-constrained robust optimization method to develop a robust cost consensus model. Then the appropriate confidence level can be determined to deal with uncertainty according to the actual situation, so as to guarantee the stability of the model while reducing its conservatism.
   Finally, since decision makers tend to accept the preferences of decision makers with whom they have an empathetic relationship, they are important reference information for generating preference adjustment recommendations. This paper combines empathetic evolutionary preferences with optimal adjustment preferences to generate adjustment recommendations for decision makers.
   This paper abstracts the government (the moderator) and 20 emergency experts (decision makers) from the realistic group decision-making problems in the formulation of the epidemic prevention and control plan to conduct consultations. Then a concrete implementation of the proposed framework is demonstrated, in which decision makers’ preferences and empathy relationships are randomly generated by computer. Through computer simulation experiments and comparison with other consensus models, the practicability and effectiveness of the chance-constrained robust optimization method in large-scale group consensus decision-making problems are verified. The research results show that considering that the empathy relationship can promote the group consensus, the chance-constrained robust consensus model can better balance economy and conservatism.

Key words: group decision making, chance-constrained robust optimization, minimum cost consensus, probabilistic linguistic empathy network, feedback mechanism

摘要： 本文旨在面向概率语言移情网络建立大规模群体共识决策方法,利用机会约束鲁棒优化方法处理单位调整成本的不确定性。首先,建立概率语言移情网络评估决策者间的移情关系。其次,利用模糊C-均值聚类对决策者分类,并基于聚类的规模、内聚度和总体移情度确定聚类权重。在反馈调整过程中,利用置信水平控制不确定参数的波动范围,建立机会约束鲁棒成本共识模型为决策者调整意见提供参考。最后,通过疫情防控方案选择的应用和对比分析,证明考虑移情关系能促进共识达成,且机会约束鲁棒共识模型能更好的平衡经济性和保守性。

关键词: 大规模群体决策, 机会约束鲁棒优化, 最小成本共识, 概率语言移情网络, 反馈机制