A chi-square-based inconsistency index for pairwise comparison matrices

Abstract

In this paper, we introduce a new method for evaluating the inconsistency level of a pairwise comparison matrix. The classical Chi-square index suggests an interesting formal similarity for a consistent pairwise comparison matrix and, as a consequence, a method for measuring the relative deviation of the elicited preferences from a set of consistent preferences defined on the basis of the similarity mentioned above. Contrary to some previously introduced Chi-square-based approaches, no optimisation problems are involved. We verify that the new index satisfies some recently introduced characterising properties of inconsistency indices. Then, by means of numerical simulations, we compare our index with some other well-known inconsistency indices and we focus, in particular, on the comparison with Saaty’s consistency index. We discuss some numerical results showing that the new index is closely related with Saaty’s one but it is more stable with respect to the number of alternatives.