Generalized consistency and intensity vectors for comparison matrices

A crucial problem in a decision-making process is the determination of a scale of relative importance for a set X = {x1, x2,..., xn} of alternatives either with respect to a criterion C or an expert E. A widely used tool in Multicriteria Decision Making is the pairwise comparison matrix A = (aij), where aij is a positive number expressing how much the alternative xi is preferred to the alternative xj. Under a suitable hypothesis of no indifference and transitivity over the matrix A = (aij), the actual qualitative ranking on the set X is achievable. Then a vector w may represent the actual ranking at two different levels: as an ordinal evaluation vector, or as an intensity vector encoding information about the intensities of the preferences. In this article we focus on the properties of a pairwise comparison matrix A = (aij) linked to the existence of intensity vectors.