Distance-based characterization of inconsistency in pairwise comparisons

Abstract

This paper deals with the evaluation of preference consistency in decision making, assuming that decision makers express their preferences by means of pairwise comparisons in the set of alternatives. Preferences can be expressed using one of the various known representations, such as fuzzy preference relations or multiplicative pairwise comparison matrices. A geometrical characterization of inconsistency evaluation is proposed by considering a pairwise comparison matrix as a point in the vector space of square matrices of order n and by using different metrics to measure deviation of this matrix from full consistency. An inconsistency index is defined as the minimum distance of a pairwise comparison matrix from a consistent one, according to a fixed metric. Consequently, to each choice of a particular metric corresponds an inconsistency index. Geometrical properties of the subset of consistent matrices are investigated. © 2012 Springer-Verlag.