On the transitivity and consistency approximated thresholds of some consistency indices for pairwise comparison matrices

Abstract

Several measures have been proposed in the literature to verify the rationality and consistency of judgements expressed by means of pairwise comparisons. However, some of these indices lack a meaningful interpretation because of the absence of thresholds associated with them. In this paper, we first introduce the consistency approximated thresholds for the Salo–Hamalainen index by deriving an inconsistency tolerance level, which is analogous to that proposed for the Consistency Ratio by Saaty, for the Geometric Consistency Index by Aguaron–Moreno Jimenez, and for the Consistency Measure by Koczkodaj. Then, via a simulation, we propose a transitivity threshold, for n × n matrices, which may give meaningful information about the degree of misclassification and the reliability of preferences while avoiding the need to revise the judgements. On varying n, we derive the transitivity thresholds for Salo–Hamalainen and for the other consistency indices proposed in literature.