Inefficient weights from pairwise comparison matrices with arbitrarily small inconsistency

Having a pairwise comparison matrix in a multi-attribute decision problem, two basic problems arise: how to compute the weight vector, and, how to associate an inconsistency index to the matrix. Two key concepts of the Analytic Hierarchy Process, the eigenvector method and inconsistency index are discussed. (In)efficiency is a well-known property in multiple objective optimization. We introduce a restriction of it in the paper. Given a pairwise comparison matrix, a weight vector is called internally inefficient if there exists another weight vector such that every pairwise ratio of the latter’s components is between the corresponding element of the pairwise comparison matrix and the pairwise ratio of the former’s components, and there exists at least one position, where the latter’s approximation is strictly better than the former’s. A class of pairwise comparison matrices with arbitrarily small inconsistency is provided such that the eigenvector method yields an internally inefficient weight vector. The paper is closed by another pairwise comparison matrix with an internally inefficient eigenvector and open questions.