Estimation of priorities in the AHP through taxicab decomposition

A pairwise comparison matrix used in the Analytic Hierarchy Process (AHP) can be seen as a contingency table that allows to verify if the judgements expressed are affected by errors. In literature, several techniques have been proposed in order to obtain a robust estimation of priority vectors one of them is to solve a singular value decomposition (SVD) of the matrix of preferences and to consider the eigenvector associated to higher eigenvalue as an estimate of the priority vector. For our purposes we consider a particular kind of reduced rank matrix approximation method derived from generalized SVD. Generalized SVD of a data set can be derived in a stepwise manner and is based on the Euclidean matrix norm. The aim of this paper is to use a particular SVD based on the taxicab norm, named taxicab singular value decomposition (TSVD). Compared to the other methods applied in the AHP context, TSVD has the advantage that the estimations of priority vectors are not influenced by errors. This paper is organized as follows. In section 1 we introduce the pairwise comparison matrices in the AHP context; in sections 2 and 3 we present an exposition of the SVD and TSDV methods for estimating priorities; in section 4 we compare the weights obtained with the above methods by analyzing a data set in the presence of an outlier; in section 5 we finish with some concluding remarks.