Managing transitivity and consistency of preferences in AHP group decision making based on minimum modifications

Preference transitivity characterized by ordinal consistency is a fundamental principle for decision making models based on pairwise comparison matrices (PCMs). However, little previous research has addressed ordinal consistency in an optimal way. Further, because ordinal consistency is not considered in the consensus reaching process, non-transitive preferences may still exist in the revised PCMs. In this paper, optimization models are proposed to obtain transitive preferences for solving individual consistency and group consensus problems. First, the conditions satisfying the ordinal consistency of PCMs are analysed and a system of constraints is derived to allow for the ordinal consistency to be explicitly controlled in the optimization model. A mixed integer linear optimization model is then proposed to assist decision makers satisfy both the ordinal and cardinal consistencies. A second mixed integer linear optimization model is then designed to ensure that the consensus level in group decision making problems can be achieved when both the group as a whole and all individuals have acceptably cardinal and ordinal consistencies. Optimization models considering ordinal consistency and classical cardinal consistency indices are open problems needing to be managed in future. Compared with existing methods, the proposed models provide an optimal way to minimize modifications in deriving transitive preferences. Finally, the feasibility and validity of the models are verified through comparisons with classic models.