Multi-stage optimization models for individual consistency and group consensus with preference relations

In this paper, a systematic optimization framework is developed to address the individual consistency and group consensus issues in decision making problems that involve human judgment for which pairwise comparisons are frequently adopted. In existing optimization approaches, the modified preferences have been limited to continuous numerical terms, and the uniqueness of these models has not been explicitly addressed. To resolve these issues, in this paper, two frameworks are developed; one to improve individual level consistency and the other to achieve group level consensus. Using discrete scales, the proposed models are proven to have equivalent integer linear programming forms that can be solved using a sequential optimization strategy in which the size of the change, the number of modifications, and the number of individuals who need to revise their preferences are sequentially optimized. To enhance the acceptability of the suggested preferences, an interactive consistency process and interactive consensus process based on the multi-stage models are also designed. Numerical examples are presented to illustrate the developed approaches.