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<p>Pairwise comparisons (PCs) form a crucial part of many popular multiple-criteria decision-making methods designed to solve complex real-world problems. The aim of the paper is to investigate the relationship between cardinal and ordinal inconsistency in the multiplicative pairwise comparisons framework, both for discrete and continuous scales. In particular, the study focuses on thresholds of cardinal inconsistency (expressed by suitable triad based inconsistency indices: Koczkodaj’s index <i>KI</i>, the Peláez-Lamata index <i>PLI</i>, the triads geometric consistency index <i>T-GCI</i> and the Cavallo-d’Apuzzo <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( I_{CD} \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mrow> <mi mathvariant="italic">CD</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> index) such that below these thresholds a PC matrix has to be ordinally consistent. The thresholds above which a PC matrix has to be ordinally inconsistent are provided as well. In addition, estimates for Saaty’s index <i>CR</i> are also included. The derived thresholds enable a decision maker to quickly assess ordinal consistency without checking the transitivity of all triads contained in the matrix, thus enabling a fast check of rationality of judgments contained in a PC matrix.</p>

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Cardinal-ordinal inconsistency thresholds in discrete and continuous pairwise comparisons

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月17日

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  • Jiri Mazurek

摘要

Pairwise comparisons (PCs) form a crucial part of many popular multiple-criteria decision-making methods designed to solve complex real-world problems. The aim of the paper is to investigate the relationship between cardinal and ordinal inconsistency in the multiplicative pairwise comparisons framework, both for discrete and continuous scales. In particular, the study focuses on thresholds of cardinal inconsistency (expressed by suitable triad based inconsistency indices: Koczkodaj’s index KI, the Peláez-Lamata index PLI, the triads geometric consistency index T-GCI and the Cavallo-d’Apuzzo \( I_{CD} \) I CD index) such that below these thresholds a PC matrix has to be ordinally consistent. The thresholds above which a PC matrix has to be ordinally inconsistent are provided as well. In addition, estimates for Saaty’s index CR are also included. The derived thresholds enable a decision maker to quickly assess ordinal consistency without checking the transitivity of all triads contained in the matrix, thus enabling a fast check of rationality of judgments contained in a PC matrix.