The logarithmic least squares and the generalized pseudoinverse in estimating ratios

This paper concerns the pairwise-comparison method used in Analytic Hierarchy Process. The logarithmic least square method is one of the methods used to rank a finite number of stimuli based on their pairwise-comparison. In the case of one decision-maker the problem can be solved using the geometric mean method. It is then assumed that the solution is geometrically normalized. In the case of multiple decision makers a set of linear equations is obtained and if we have a different number of judgments for each pair of the compared objects the geometric normalization assumption can not be used directly. The aim of this paper is to show that applying the generalized pseudoinverse we obtain the solution that is geometrically normalized and consistent with the case of one decision maker. To define the pseudoinverse the spectral decomposition is used. The structure of the general solution is presented and the existence of the general solution is discussed.