An approach to incomplete multiplicative preference relations and its application in group decision making

To measure the consistency of multiplicative preference relations, a reasonable concept of consistent multiplicative preference relations is defined, and the multiplicative geometric consistent index (MGCI) is presented. In a simulation method, the average values of the MGCI are studied. Considering incomplete multiplicative preference relations, a consistency-based linear programming model is constructed, which is optimal from the point of view of consistency with respect to the known judgments. In a group decision making context, the group consensus index (GCOI) is given to measure the consensus of individual multiplicative preference relations. Then, the hybrid weighted geometric mean (HWGM) operator is defined to calculate the collective multiplicative preference relation. Based on the consistency and consensus analysis, an approach to group decision making with incomplete multiplicative preference relations is developed. Furthermore, we briefly research the application of the new method to incomplete interval multiplicative preference relations and incomplete fuzzy preference relations. Meanwhile, the associated examples are offered to show the practicability and efficiency of the proposed procedure.