View - Universidad de Almería

Proceedings of GO 2005, pp. 53 – 56.On weights estimation in MultipleCriteria Decision Analysis ∗R. Blanquero, 1 E. Carrizosa, 1 E. Conde, 1 and F. Messine 21 Facultad de Matemáticas, Universidad de Sevilla, Spain{rblanquero,ecarrizosa,educon}@us.es2 ENSEEIHT-IRIT, 2 rue C Camichel, 31071 Toulouse Cedex, FranceFrederic.Messine@n7.frAbstractKeywords:Several Multiple-Criteria Decision Making methods require, as starting point, weights measuringthe relative importance of the criteria. A common approach to obtain such weights is to derive themfrom a pairwise comparison matrix A.There is a vast literature on proposals of mathematical-programming methods to infer weightsfrom A, such as the eigenvector method or the least (logarithmic) squares. Since distinct proceduresyield distinct results (weights) we pose the problem of describing the set of weights obtained by“sensible” methods: those which are Pareto-optimal for the nonconvex (vector-) optimization problemof simultaneous minimization of discrepancies.A characterization of the set of Pareto-optimal solutions is given. Moreover, although the abovementionedoptimization problems may be multimodal, standard Global Optimization methods cancome up with a globally optimal vector of weights in reasonable time.Pairwise comparison matrices, weights, Interval Analysis.1. IntroductionSeveral strategies have been suggested in the literature to associate with a set D = {d 1 , . . . , d n }of decisions weights x 1 , x 2 , . . . , x n reflecting decision-maker’s preferences. In the AnalyticHierarchy Process (AHP), [11, 12, 14, 15], an n × n matrix A,⎛⎞a 11 a 12 · · · a 1na 21 a 22 · · · a 2nA = ⎜⎝.. . ..⎟. ⎠a n1 a n2 · · · a nnis obtained after asking the decision-maker (DM) to quantify the ratio of his/her preferencesof one decision over another. In other words, for every pair of decisions d i , d j , the term a ij > 0is requested satisfyinga ij ≈ x ix j(1)The matrix A so obtained must be a positive reciprocal matrix, i.e.,a ji = 1a ij> 0 for all i, j = 1, 2, . . . , n.∗ This research has been partially supported by grants BFM2002-04525-C02-02 and BFM2002-11282-E MCYT, Spain