Acceptable consistency of aggregated comparison matrices in analytic hierarchy process

Abstract

The analytic hierarchy process (AHP) is a method for solving multiple criteria decision problems, as well as group decision making. The weighted geometric mean method (WGMM) is appropriate when aggregation of individual judgements (AIJ) is used. This paper presents a new proof which confirms the theorem that if the comparison matrices of all decision makers are of acceptable consistency, then the weighted geometric mean complex judgement matrix (WGMCJM) also is of acceptable consistency. This theorem was presented and first proved by Xu (2000), but Lin et al. ( 2008) rejected the proof. We also discuss under what conditions the WGMCJM is of acceptable consistency when not all comparison matrices of decision makers are of acceptable consistency. For this case we determine the upper bound for the consistency ratio of WGMCJM and provide numerical examples.

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