# Classifying Inconsistency in AHP Pairwise Comparison Matrices Using Machine Learning

arXiv: https://arxiv.org/abs/2505.06293 · HTML: https://mlsanity.com/paper/2505.06293/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 34 references that are in the map.
- First posted on arXiv 7 May 2025.

## Builds on (most cited first, 2 of 2)

- [On Axiomatization of Inconsistency Indicators for Pairwise Comparisons](https://mlsanity.com/paper/1307.6272/) (2013)
- [Axiomatizations of inconsistency indices for triads](https://mlsanity.com/paper/1801.03355/) (2018)

## Abstract (arXiv)

Assessing consistency in Pairwise Comparison Matrices (PCMs) within the Analytical Hierarchy Process (AHP) poses significant challenges when using the traditional Consistency Ratio (CR) method. This study introduces a novel alternative that leverages triadic preference reversals (PR) to provide a more robust and interpretable assessment of consistency. Triadic preference reversals capture inconsistencies between a pair of elements by comparing the direction of preference derived from the global eigenvector with that from a 3x3 submatrix (triad) containing the same pair, highlighting local-global preference conflicts. This method detects a reversal when one eigen ratio exceeds one while another falls below one, signaling inconsistency. We identify two key features: the proportion of preference reversals and the maximum reversal, which mediate the impact of a PCM's order on its consistency. Using these features simulated PCMs are clustered into consistent and inconsistent classes through k-means clustering, followed by training a logistic classifier for consistency evaluation. The PR method achieves 97% accuracy, significantly surpassing the Consistency Ratio (CR) method's 50%, with a false negative rate of only 2.6% compared to 5.5%. These findings demonstrate the PR method's superior accuracy in assessing AHP consistency, thereby enabling more reliable decision-making. The proposed triadic preference reversal (PR) approach is implemented in the R package AHPtools publicly available on the Comprehensive R Archive Network (CRAN).