$$\mathcal {G}$$ G -distance and $$\mathcal {G}$$ G -decomposition for improving $$\mathcal {G}$$ G -consistency of a Pairwise Comparison Matrix

Abstract

Pairwise comparisons have been a long standing technique for comparing alternatives/criteria and their role has been pivotal in the development of modern decision making methods. Since consistency ensures rational decisions, in literature several approaches are proposed for the revision of the Pairwise Comparison Matrix in order to improve its consistency. In order to obtain general results, suitable for several kinds of Pairwise Comparison Matrices proposed in literature, we focus on matrices defined over a general unifying framework, that is an Abelian linearly ordered group. In this context, firstly, we provide $$\mathcal {G}$$ -distance between Pairwise Comparison Matrices and $$\mathcal {G}$$ -decomposition of a Pairwise Comparison Matrix in its $$\mathcal {G}$$ -consistent and totally $$\mathcal {G}$$ -inconsistent components. Then, we show how a $$\mathcal {G}$$ -inconsistent Pairwise Comparison Matrix can be revised according to the associated $$\mathcal {G}$$ -consistent component; the revision process takes into account $$\mathcal {G}$$ -distance from the former in order to better represent decision maker’s preferences.