New consistency index for AHP works across any matrix size

Pairwise comparisons are the backbone of the Analytic Hierarchy Process (AHP), a structured technique for complex decision-making. Since Thomas Saaty introduced his consistency index (CI) and consistency ratio (CR) in the 1970s, the field has grappled with whether a single threshold (typically 0.1) fairly assesses consistency across matrices of different sizes. A larger matrix (more criteria) naturally tends to produce higher inconsistency values, potentially penalizing decisions with many factors. This paper by Tsuneshi Obata and Shunsuke Shiraishi (arXiv:2607.07950) directly addresses that gap.

The authors build on their earlier work linking consistency to the coefficients of a matrix's characteristic polynomial. They define a fundamental property any size-independent index must satisfy, then refine their previous index to meet this property. Through rigorous analysis and simulations with randomly generated matrices, they demonstrate that their refined index coincides with the existing consistency index when properly normalized—but now remains invariant to matrix size. The study explicitly visualizes how matrix size distorts raw CI values, making the case for a size-independent alternative. For practitioners in AI, operations research, and decision support systems, this means more reliable consistency checks when comparing dozens of alternatives under multiple criteria, especially in automated decision pipelines where matrix size varies dynamically.