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Proceedings of GO 2005, pp. 57 – 60.Least Squares approximation ofpairwise comparison matrices ∗Sándor Bozókipresently a PhD student at Corvinus University of Budapest, pursuing research at the Laboratory and the Department ofOperations Research and Decision Systems, Computer and Automation Research Institute, Hungarian Academy of Sciences(MTA SZTAKI), bozoki@oplab.sztaki.huAbstractKeywords:One of the most important steps in solving Multi-Attribute Decision Making (MADM) problems isto derive the weights of importance of the attributes. The decision maker is requested to comparethe importance for each pair of attributes. The result expressed in numbers is written in a pairwisecomparison matrix. The aim is to determine a weight vector w = (w 1, w 2, , w n), which reflectsthe preferences of the decision maker, in the positive orthant of the n-dimensional Euclidean space.We examine a distance minimizing method, the Least Squares Method (LSM). The LSM objectivefunction is nonlinear and, usually, non-convex, thus its minima is not unique, in general. It is shownthat the LSM-minimization problem can be transformed into a multivariate polynomial system.We consider the resultant and the generalized resultant methods, which can be applied in the caseof small-size matrices, and the homotopy continuation proposed by Tien-Yien Li and Tangan Gao.Numerical experience show that the homotopy method finds all the roots of polynomial systems,hence all the minima of the LSM objective functions. At present the maximum size of matricesregarding which the LSM approximation problem can be solved by using the homotopy method is8 × 8. We show that LSM works even if some elements are missing from the pairwise comparisonmatrix. The paper ends with few numerical examples.Pairwise comparison matrix, Least Squares Method, Polynomial systems.1. IntroductionOne of the most studied methodology in Multi-Attribute Decision Making is the Analytic HierarchyProcess developed by Thomas L. Saaty [21]. Using AHP, difficult decision problemscan be broken into smaller parts by the hierarchical criterion-tree, one level of the tree canbe handled by pairwise comparison matrices. The idea of using pairwise comparison matricesis that decision makers may not tell us the explicit weights of the criteria or the cardinalpreferences of the alternatives but they can make pairwise comparisons.A pairwise comparison matrix A = [a ij ] i,j=1..n is defined as⎛⎞1 a 12 a 13 . . . a 1na 21 1 a 23 . . . a 2nA =a 31 a 32 1 . . . a 3n⎜⎝.. . . ..⎟. ⎠a n1 a n2 a n3 . . . 1∈ R n×n+ ,∗ This research was supported by the Hungarian National Research Foundation, Grant No. OTKA-T043241.