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 ={x_{1}, x_{2}, \ldots, x_{n} } 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=(a_{ij}) where a_{ij} is a positive number expressing how much the alternative x_{i} is preferred to the alternative x_{j}. Under suitable hypothesis of no indifference and transitivity over the matrix A = (a_{ij}), the actual qualitative ranking on the set X is achievable. Then a vector w may represent the actual ranking at two different levels: as ordinal evaluation vector, or as intensity vector encoding information about the intensities of the preferences. In this paper we focus on the properties of a pairwise comparison matrix A=(a_{ij}) linked to the existence of intensity vectors.