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

Sato, Yuji; Tan, Kim Hua

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.

• Journal Article Type: Article
• Acceptance Date: Nov 10, 2021
• Online Publication Date: Jan 25, 2022
• Publication Date: Jan 25, 2022
• Deposit Date: Feb 11, 2022
• Publicly Available Date: Jan 26, 2023
• Journal: Annals of Operations Research
• Print ISSN: 0254-5330
• Electronic ISSN: 1572-9338
• Peer Reviewed: Peer Reviewed
• DOI: https://doi.org/10.1007/s10479-021-04431-3
• Keywords: Decision support systems・Pairwise comparison・Inconsistency・Human perception
• Public URL: https://nottingham-repository.worktribe.com/output/7416259
• Publisher URL: https://link.springer.com/article/10.1007/s10479-021-04431-3