Key words: Paired Comparisons, Relative Weights, Multiple Attribute Decision Making, AHP, Rank Preservation

Abstract

Paired comparisons method is used to rank decision variants in multi-attribute decision-making problems and is based on comparisons of particular decision variants between each other. The purpose of this paper is to discuss the properties of the logarithmic least squares method (also called geometric mean method) which is commonly applied in calculation of the weight vector in paired comparisons evaluation. The geometric mean method gives a unique, geometrically normalized solution independent on the scale inversion.

INTRODUCTION

Paired comparisons method is used to rank decision variants in multi-attribute decision- making problems and bases on comparing particular decision variants. In a decision process an expert or experts are asked to associate to each pair of variants a number chosen from a given scale. The number, called judgement, express a relative preference of one variant in a pair over the second one. Basing on expert judgements a square judgement matrix is created. An evaluation of decision variants ranking leads to an approximation of a judgement matrix by a matrix of weight ratios and a normalization of the obtained solution. In order to calculate weight vector mainly two methods are applied: maximal eigenvalue and logarithmic least squares one (Saaty, 1980). In this paper, the analysis of properties of the latter method (also called geometric mean one) is performed.

PAIRED COMPARISONS

Assume that there are n variants F1, F2, …, Fn, and an expert is asked to provide his opinions concerning each pair of them, expressing intensity of importance of one factor in a pair over the second one with a use of the preference scale from the Table 14.1. Thus a judgment matrix R can be created where rij is an estimate for the relative significance of the factors (Fi, Fj), provided by the expert.