A Belief Comparison Matrix and its Integration Method for AHP with Incomplete Information

In this paper, a belief comparison matrix and its integration method are proposed for the analytic hierarchy process (AHP) with incomplete information. Different from most of current methods, the belief comparison matrices can distinguish the “uncertainty” from “ignorance” for AHP under the incomplete information circumstance. On the base of Dempster–Shafer theory of evidence, the pair wise comparison values with respect to preference first are given as belief values. Then the belief judgment matrices are integrated by the evidence reasoning (ER) combination rules. After the integration comparison values matrix being transformed into the reciprocal judgment matrix by a weighted geometric averaging operator, the weights vector can be worked out by using the original approach of AHP. A numerical example is studied to illustrate the detailed implementation process of the approach and demonstrate its potential applications in MADM problems with incomplete information.