Inconsistency indices in pairwise comparisons: an improvement of the Consistency Index

Abstract

The Consistency Index and the Consistency Ratio of the analytic hierarchy process (AHP) were designed to measure the ratio of inconsistent judgments among pairwise comparisons (PCs), which have been the principal indices for the past four decades. Definitions of inconsistency measures for PCs have yet to be established, however, because of the difficulty in quantifying subjectivity in judgments. Therefore, an empirical review that can take such subjective factors into account is essential. In this paper, the Consistency Ratio is thus reviewed using subjective data, and then a new inconsistency index for PCs is proposed based on the review. The review is based on subjective data obtained from two opinion surveys, which focuses on the relationship between the Consistency Ratio and two indicators: (1) the conformity of the results of the AHP and that of the ranking method, and (2) the goodness-of-fit of weight elicited by the AHP to human perception. A new inconsistency index is then proposed based on the mathematical property of a pairwise comparison matrix and further validated based on the conformity and the goodness-of-fit of weight. The results show that the proposed index detects inconsistency among real-world PCs more sensitively than could the Consistency Ratio; the index might suggest the reliability of the output of a pairwise comparison matrix.