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We apply methods and techniques of tropical optimization to develop a newtheoretical and computational framework for the implementation of the AnalyticHierarchy Process in multi-criteria problems of rating alternatives frompairwise comparison data. The framework involves the Chebyshev approximation ofpairwise comparison matrices by consistent matrices in the logarithmic scale.We reduce the log-Chebyshev approximation to multidimensional tropicaloptimization problems, and offer complete direct solutions to the problems inthe framework of tropical mathematics. The results obtained provide aclosed-form solution to the rating problem of interest as either a unique scorevector (up to a positive factor) or as a set of different score vectors. Tohandle the problem when the solution is not unique, we develop tropicaloptimization techniques to find those vectors from the solution set that arethe most and least differentiating between the alternatives with the highestand lowest scores, and thus can be well representative of the entire solution.

Tropical optimization techniques in multi-criteria decision making with Analytical Hierarchy Process