Cardinal-ordinal inconsistency thresholds in discrete and continuous pairwise comparisons

Abstract

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 \( I_{CD} \) 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.