JOURNAL OF MATHEMATICAL PSYCHOLOGY 28, 205-214 (1984) Inconsistency and Rank Preservation THOMAS L. SAATY AND Lurs G. VARGAS University of Pitrsburgh Conditions for rank preservation in a positive reciprocal matrix that is inconsistent are provided. Three methods of deriving ratio estimates are examined: the eigenvalue, the logarithmic least squares, and the least squares methods. It is shown that only the principal eigenvector directly deals with the question of inconsistency and captures the rank order inherent in the inconsistent data. cs 1984 Academic Press, Inc. 1. INTRODUCTION In the last few years several authors have advocated particular best ways for approximating a positive reciprocal matrix A = (a,), uji = l/a, by a vector x = (x1 )...) x,) such that the matrix of ratios (xi/xj) is a best approximation to A in some sense. This surge of interest has arisen out of the many applications that have been made of the Analytic Hierarchy Process for decisions in complex situations (Saaty, 1980, 1982). There are infinitely many possible ways to generate approx- imations for A. Three most widely used ones which have been strongly advocated are: the method of least squares (LSM) (Cogger and Yu, 1983; Jensen, 1983; McMeekin, 1979) which finds x by minimizing the Euclidean metric n x (Uij - xi/xj)*; i,j= I the method of logarithmic least squares (LLSM) (de Grann, 1980; Fichtner, 1983; Williams & Crawford, 1980) which minimizes i (log aij - log xi/xi)‘; i,j= 1 and the eigenvector method (EM) (Saaty, 1977, 1980, 1982) which solves the problem Ax = A,,, x, where A,,,,, is the principal eigenvalue of A. Because A is Address all reprint requests to Thomas L. Saaty, Department of Mathematics, University of Pittsburgh, Room 354, Mervil Hall, Pittsburgh, Pennsylvania 15260. 205 0022-2496184 $3.00 Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.