A tropical optimization approach in the analysis of pairwise comparison matrices

AbstractWe propose a new approach to solve the problem of rating alterna-tives based on their pairwise comparison. The problem is formulatedin terms of tropical algebra, and then reduced to the approximationof pairwise comparison matrices by reciprocal matrices of unit rank.We represent the approximation problem in a common form for bothmultiplicative and additive comparison scales. To solve the problemobtained, tropical optimization techniques are applied to provide newcomplete direct solutions to the rating problems in a compact vectorform, which extend known solutions and involve less computationalefforts. The results are illustrated with numerical examples.Key-Words: tropical mathematics, idempotent semifield, matrixapproximation, reciprocal matrix, pairwise comparisons, ranking of al-ternatives.MSC(2010): 65K10, 15A80, 65K05, 41A50, 90B50 1 Introduction Tropical (idempotent) mathematics, which focuses on the theory and ap-plications of idempotent semirings [1, 2, 3, 4, 5, 6, 7], find increasing usein solving optimization problems in operations research, including problemsof project scheduling, location analysis and decision making. The problemsare formulated in the framework of tropical mathematics to minimize ormaximize functions defined on vectors over idempotent semifields (see, eg,an overview in [8]).One of the applications of tropical optimization is the analysis of pref-erences based on pairwise comparison data in decision making. A problemis examined in [9, 10, 11], which aims at rating alternatives on the basis oftheir pairwise comparison matrix. The solution involves the calculation of

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