Size Independence of Consistency Index for Pairwise Comparison Matrices in Analytic Hierarchy Process

Jul 16, 2026

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Pairwise comparisons are central to the analytic hierarchy process (AHP), where inconsistencies in judgment must be quantified using consistency indices such as Saaty’s CI and CR. A fixed threshold (e.g., 0.1) may not correspond to equivalent inconsistency across different matrix sizes. The study analyzes how consistency relates to matrix size by linking a super-PCM with sub-PCMs formed by removing one item, then proposes a size-independence property. A refined index satisfying the property is identified and verified via random-matrix experiments and visualization.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages16

Reading Time40 min

Introduction

Pairwise Comparison and Consistency

Pairwise Comparison Matrix

arXiv :2607 .07950v1 [ stat .ME] 8 Jul 2026 Size Independence of Consistency Index for Pairwise Comparison Matrices in Analytic Hierarchy Process Tsuneshi OBATA‗ Shunsuke SHIRAISHI† Abstract Pairwise comparisons are fundamental in the analytic hierarchy process. Various consistency indices have been proposed to assess inconsistencies in these comparisons. Since Saaty ﬁrst proposed his consistency index, the assessment of the degree of consistency in pairwise comparison matrices has remained an open and hot topic in the study of the analytic hierarchy process. The consistency indices CI and CR proposed by Saaty are deﬁned using the principal eigenvalue of the pairwise comparison matrix. In our previous study, we introduced an alternative index derived from the relationship between the coeﬃcient of the characteristic polynomial and the consistency of comparisons. Saaty proposed a ﬁxed threshold of 0.1 for CI or CR as a guideline for an acceptable level of consistency, regardless of the matrix size. However, whether this threshold represents an equivalent level of consistency across diﬀerent matrix sizes, that is, across diﬀerent numbers of evaluation items, remains unclear. This study analysed the relationship between consistency and matrix size by examining pairwise comparison matrices constructed from subsets of evaluation items. Based on this analysis, we propose the fundamental property to be satisﬁed by a size-independent consistency index. Furthermore, we reﬁne our previously proposed index to ensure that it satisﬁes this property, demonstrating that it coincides with the existing consistency index. Finally, we visualise the relationship between the matrix size and consistency index values using randomly generated pairwise comparison matrices, thereby providing insights into the impact of matrix size on consistency evaluation. 1 Introduction Pairwise comparison matrices (PCMs) are fundamental tools in the analytic hierarchy process (AHP) and other decision making methods, which enable decision makers to express their relative preferences between items1. As human judgements are rarely perfectly consistent, various consistency indices have been proposed to quantify deviations from perfect consistency. The assessment of the degree of consistency in PCMs has remained an open and hot topic [2, 5] . The consistency index (CI) and consistency ratio (CR) introduced by Saaty [8, 9] are widely employed in practice. ‗Faculty of Science and Engineering, Otemon Gakuin University, 2-1-15 Nishiai, Ibaraki, Osaka, 567-8502 Japan e-mail: [t-obata@haruka.otemon.ac.jp ORCID:](t-obata@haruka.otemon.ac.jp ORCID:) 0000-0002-5924-112X †Faculty of Applied Information Science, Hiroshima Institute of Technology, 2-1-1 Miyake, Saeki-ku, Hiroshima 731-5193 Japan e-mail: [s.shiraishi.wx@it-hiroshima.ac.jp ORCID:](s.shiraishi.wx@it-hiroshima.ac.jp ORCID:) 0000-0003-2401-1938 1In this paper, we employ the term items to refer to the objects being compared in AHP, such as criteria, sub-criteria, and alternatives. The interpretation of the consistency values across PCMs of varying sizes remains a longstanding question [1, 7] . For example, in Saaty’s framework, the thresholds CI ≤ 0. 1 and CR ≤ 0. 1 are applied uniformly regardless of the matrix size 􀀽 . However, the same numerical value for a particular index does not necessarily represent the degree of inconsistency as 􀀽 changes. This raises the question of the properties that a size-independent consistency index should possess. In this study, we address the size-independence problem by analysing the relationship between a PCM of size 􀀽 (referred to as a super-PCM) and 􀀽 distinct PCMs of size 􀀽 −1 obtained by eliminating one item (referred to as sub-PCMs) . The consistency of a super-PCM should be comparable with that of its sub-PCMs. We formalise this intuition by introducing a property that links the index values of a super-PCM and its sub-PCMs, and discuss its implications

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What is the outcome of refining the previously proposed consistency index?

The refined index is shown to satisfy the size-independence property and it coincides with an existing consistency index, while random-matrix experiments illustrate how indices respond to matrix size changes.

Jul 16, 2026

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