View - Universidad de Almería

56 R. Blanquero, E. Carrizosa, E. Conde, and F. Messinewith ε(s, [a L , a U ]) defined as the distance between s and the closest point in [a L , a U ],⎧⎨ 0, if s ∈ [a L , a U ]ε(s, [a L , a U ]) = a L − s, if s ≤ a L(8)⎩s − a U , if s ≥ a U{= max 0,∣ s − aU + a L2 ∣ − aU − a L }(9)2Obviously (7) coincides with (2) in what we call the degenerate case in which all intervalsA ij are degenerate, A ij = {a ij }.In this context, theoretical results are given without needing the definition of a norm. Weshow that well-known strategies of Deterministic Global Optimization, namely, Branch andBound algorithms with bounds based on Interval Analysis, provide in reasonable time globallyoptimal solutions to problem (7), and thus, as particular case, to model (2), for differentchoices of γ. The standard interval Branch and Bound algorithm has been modified here inorder to determine the accuracy of the so-obtained optimal solution and, using this, the numericalexperiments show perfectly that some problems can efficiently be solved even if thesize is rather important.References[1] Bryson, N. (1995). "A goal programming method for generating priority vectors". Journal of the OperationalResearch Society, 46:641–648.[2] Chankong, V., and Y. Haimes (1983). Multiobjective Decision Making, North–Holland, New York.[3] Choo, E.U. and Wedley, W.C. (2004). "A common framework for deriving preference values from pairwisecomparison matrices". Computers & Operations Research, 31:893–908.[4] Conde, E. (2002). "Mean utility in the assurance region model". European Journal of Operational Research140:93–103.[5] Cook, W.D., and Kress, M. (1988). "Deriving weights from pairwise comparison ratio matrices: An axiomaticapproach". European Journal of Operational Research, 37:320–355.[6] Expert Choice, at http://www.expertchoice.com[7] Flp, J. "Global Optimization methods for approximation by consistent matrices". Presented at the ISMP2003, Copenhaguen.[8] Haines, L.M. (1998). "A statistical approach to the analytic hierarchy process with interval judgements. (I).Distributions on feasible regions". Journal of the Operational Research Society 110:112–125.[9] Lootsma, F.A. (1996). "A model for the relative importance of the criteria in the Multiplicative AHP andSMART". European Journal Of Operational Research, 94:467–476.[10] Ramanathan, R.(1997). "A note on the use of goal programming for the Multiplicative AHP". Journal of MulticriteriaDecision Analysis, 6:296–307.[11] Saaty, T.L. (1977). "A scaling method for priorities in hierarchical structures". Journal of Mathematical Psychology15:234–281.[12] Saaty, T.L. (1980). Multicriteria Decision Making: The Analytic Hierarchy Process. McGraw-Hill, New York.[13] Saaty, T.L. (1990). "Eigenvector and logaritmic least squares". European Journal of Operational Research 48:156–160.[14] Saaty, T.L. (1994). Fundamentals of Decision Making. RSW Publications, Pittsburg, PA.[15] Saaty, T.L. (1994). "How to make a decision: The Analytic Hierarchy Process". Interfaces 24:19–43.[16] Salo, A. and Hamalainen, R.P. (1995). "Preference programming through approximate ratio comparisons".European Journal of Operational Research, 82:458–475.