Singular value decomposition in AHP

Abstract

The analytic hierarchy process (AHP) has been accepted as a leading multiattribute decision-aiding model both by practitioners and academics. The foundation of the AHP is the Saaty’s eigenvector method (EM) and associated inconsistency index that are based on the largest eigenvalue and associated eigenvector of an ( n× n) positive reciprocal matrix. The elements of the matrix are the decision maker’s (DM) numerical estimates of the preference of n alternatives with respect to a criterion when they are compared pairwise using the 1–9 AHP fundamental comparison scale. The components of the normalized eigenvector provide approximations of the unknown weights of the criteria (alternatives), and the deviation of the largest eigenvector from n yields a measure of how inconsistent the DM is with respect to the pairwise comparisons. Singular value decomposition (SVD) is an important tool of matrix algebra that has been applied to a number of areas, e.g., principal component analysis, canonical correlation in statistics, the determination of the Moore–Penrose generalized inverse, and low rank approximation of matrices. In this paper, using the SVD and the theory of low rank approximation of a (pairwise comparison) matrix, we offer a new approach for determining the associated weights. We prove that the rank one left and right singular vectors, that is the vectors associated with the largest singular value, yield theoretically justified weights. We suggest that an inconsistency measure for these weights is the Frobenius norm of the difference between the original pairwise comparison matrix and one formed by the SVD determined weights. How this measure can be applied in practice as a means of measuring the confidence the DM should place in the SVD weights is still an open question. We illustrate the SVD approach and compare it to the EM for some numerical examples.