Group Decision Making with Transitive Preferences Under Ordinal and Cardinal Consistencies: An Optimization Approach

Abstract

With the ubiquity of preference relations used in group decision making (GDM), controlling the transitivity of preferences characterized by individual consistency has attracted much attention. However, few previous studies consider ordinal and cardinal consistencies simultaneously and the ordinal consistency is often ignored in consensus reaching process in existing studies. In this study, the conditions of individual ordinal consistency are proved to be equivalent to a series of inequalities, based on which, an optimization model is developed to deal with the ordinal inconsistency problem. In addition, a second optimization model is proposed to address the coexistence of both ordinal and cardinal inconsistencies. A framework is designed to provide a complete strategy for controlling consistency. Such a framework is also generalized to accommodate the consensus problem in GDM. Comparing to the existing consistency improvement approaches, the proposed approach explicitly solves the ordinal consistency problem by an optimization approach. Furthermore, the proposed group consensus model guarantees both ordinal consistency and acceptable cardinal consistency when consensus is achieved. Finally, classical examples with extensive comparisons are conducted to show the effectiveness of the proposed approaches.