A novel perspective on the inconsistency indices of reciprocal relations and pairwise comparison matrices

The paper offers a new perspective on the construction of inconsistency indices for reciprocal pairwise comparison matrices representing reciprocal relations on a set of n≥3 alternatives (or criteria). We introduce a general framework for defining inconsistency indices in which a parametric generating function plays a central role, by scaling the natural measures of local inconsistency and thereby obtaining the actual local contributions to the inconsistency index. The general framework is applied in an analogous way to the two classical approaches to local inconsistency measurement, in which the natural local inconsistency measures are either based on index pairs or index triplets. In this construction, our framework encompasses four different classical inconsistency indices, associated with particular parameter choices for the generating function. We examine in detail the relevant features of the general framework proposed, from the study of the parametric generating function to the properties satisfied by the two parametric families of inconsistency indices obtained. The role of the generating function parameter is discussed and it is proven to have an emphasis effect with respect to the original values of the natural measures of local inconsistency. The present framework combines the algebraic structure of reciprocal pairwise comparison matrices with a parametric scaling mechanism which interpolates and generalizes the classical inconsistency indices in the literature. The parametric nature of the construction is directly interpretable in relation to how local inconsistencies are measured and how they actually contribute, via the generating function, to the inconsistency indices of reciprocal pairwise comparison matrices.