Logarithmic Least Squares criterion revisited for general matrices of pairwise comparisons

Abstract

In many applications of expert judgements, the collected matrices of pairwise comparisons, though inconsistent, are forced to satisfy the reciprocity property. To remove inconsistency and establish weights (priorities) for compared objects, the “geometric mean method” is frequently used. The method derives from the Logarithmic Least Squares (LLS) criterion, but it is seldom known that its main formula simplifies to a row-based geometric mean exactly due to the reciprocity, whereas the general solution is different. General matrices of comparisons do not have to be reciprocal nor even have 1s on the main diagonal. In 1999, Koczkodaj and Orłowski [1] provided an algorithm, named RCGM (Row/Column Geometric Means), that addresses the general case. In our opinion, that proposition has not been given the deserved recognition. The goal of this paper is threefold: (i) to show a straightforward derivation of the correct and general solution to LLS — the solution is equivalent to RCGM algorithm and can be expressed by a single formula, (ii) to emphasize that direct computations of geometric means, involving multiplications, can be numerically unstable, hence, logarithm-based computations are needed; and last, but certainly not least, (iii) to show that general judgement matrices allow for more accurate reconstruction of object weights (priorities) than reciprocal ones. That last goal is demonstrated via thorough experiments with various random models imposed on pairwise comparisons and errors underlying them.