Consistency adjustments for pairwise comparison matrices

Abstract

SUMMARYThis paper is concerned with the development of a 'best' rank one transitive approximation to a general paired comparison matrix in a least-squares sense. A direct attack on the non-linear problem is frequently replaced by a sub-optimal linear problem and, here, the best procedure of this kind is obtained. The Newton-Kantorovich method for the solution of the non-linear problem is also studied, including the role of the best linear approximation as a starting point for this method. Numerical examples are included.

A method for approximating symmetrically reciprocal matrices by transitive matrices

“…We call this the transitive approximation problem and denote it by (TA). Note that in [6] one considered the nonlinear problem…”

“…, y n such that b ij ≈ y j /y i for each i, j . This problem was discussed in [6]. This approximation problem is not a straightforward task as the set of all transitive matrices is a complicated nonlinear manifold (in the space of real n × n matrices).…”

“…It is then natural to ask for numbers giving the absolute weight of each alternative that "explain" these relative comparisons best possible. This problem was discussed in [3,6] and in both papers a least squares approach was taken. We refer to [6] for a further problem background and references to related papers.…”

“…This problem was discussed in [3,6] and in both papers a least squares approach was taken. We refer to [6] for a further problem background and references to related papers. The goal of this paper is to suggest another method for finding these absolute weights and we use a different distance measure than in the least squares approach.…”

On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices

“…Newton's method of successive approximation can be applied [16], but a good initial point is required.…”