关键词: 群体决策, 最小成本共识模型, 聚合算子, 数据驱动鲁棒优化, 置信水平

Abstract: The quality and reliability of the composite indicator are directly influenced by the aggregation of individual preference information. Even slight perturbations in the aggregation weights may result in the selection of unreasonable solutions, leading to economic and social losses. Therefore, it is crucial to determine the aggregation weights of decision-makers (DMs) appropriately. Robust optimization (RO) methods have gained significant attention due to their ability to generate uncertainty-immune solutions. However, these methods typically construct uncertainty sets based on experience, which introduces certain conservatism to the model results. In contrast, data-driven RO methods construct uncertainty sets based on uncertain observations, allowing for a reasonable balance between conservatism and robustness in decision outcomes. Consequently, it is necessary to develop a model that can effectively manage the uncertainty associated with aggregation weights using the data-driven RO methods.
To tackle the uncertainty and randomness of DMs' aggregation weights, a series of data-driven robust minimum-cost consensus models is proposed in this paper. Firstly, a minimum-cost consensus model with consensus constraints is introduced as the foundation of the study. This model ensures that DMs achieve an acceptable level of consistency. Secondly, a kernel density estimation (KDE) method is used to derive probability density functions of uncertain weights based on historical data. These functions are utilized to construct uncertainty intervals with confidence levels, enabling control over the perturbation range of uncertain weights in the aggregation operator. Subsequently, two types of flexible uncertainty sets, namely flexible uncertainty set I and flexible uncertainty set II, are defined. These sets correspond to the set of three different shapes, including the box set, ellipsoidal set, and polyhedral set. By employing these uncertainty sets, the data-driven robust minimum-cost consensus models are developed to address six different uncertain environments.
Finally, this paper abstracts a group decision-making problem from the carbon quota allocation problem. By estimating the probability density function for the government's allocation of quotas to each enterprise, the confidence-based uncertainty intervals and corresponding data-driven robust models can be established to deal with the uncertainty caused by deviations, which demonstrates the applicability of the proposed models. To analyze the performance of the model, three experiments, including all uncertainties with the same confidence level, different uncertainties with different confidence levels and flexible uncertainty sets versus classical uncertainty sets, are conducted in this paper, and the following meaningful results are obtained: (1)The data-driven method can effectively improve the quality of the aggregation operator by traversing the historical data of weights; (2)The government can choose the appropriate uncertainty set according to the degree of risk preference to make decisions; (3)The proposed model contributes to reducing the price of robustness in data analysis results to a certain extent.
In summary, the propose data-driven robust consensus models provides novel insights and approaches for solving the uncertainty in the consensus reaching process of group decision-making problems. As decision-making environments grow more complex, involving a greater number of DMs, reaching consensus becomes increasingly challenging. Additionally, DMs' weights will be influenced by more factors such as social relationships. Therefore, in future research, the proposed model can be considered to be extended to large-scale group decision-making problems.

Key words: group decision making, minimum cost consensus model, aggregation operator, data-driven robust optimization, confidence level