Maximum satisfaction consensus with budget constraints considering individual tolerance and compromise limit behaviors

In the consensus reaching process (CRP), decision-makers (DMs) often present behaviors of tolerance and limited compromise when adjusting their opinions, which may lead to the actual decision deviating from the solution recommendation obtained by applying the current minimum cost consensus theory. In this paper, we propose a consensus-reaching approach based on maximum satisfaction under budget constraints for DMs with these two behaviors. To obtain a feasible budget constraint of the CRP, a reference budget consensus model (RBCM) incorporating both of these behaviors is first developed. Under the constraint of the limited budget, we propose a maximum satisfaction consensus model (MSCM) to maximize the satisfaction level of all DMs based on the goal programming theory. To further achieve different levels of consensus, we propose a soft MSCM under a certain threshold using soft consensus measures. Moreover, the proposed consensus models are illustrated by the data of China’s Taihu Lake pollution control. The results suggest that: First, the individual tolerance behavior has a higher impact on the consensus outcome than the compromise limit, whereas the latter is more likely to affect the satisfaction of each DM. Second, as the budget increases, the consensus opinion and group satisfaction level will increase toward a higher average cost until they remain constant. Last, a higher consensus threshold will reduce DMs’ satisfaction levels, and an appropriate threshold should be selected as the control baseline of the CRP in the real application.