We apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the Analytic Hierarchy Process in multi-criteria problems of rating alternatives from pairwise comparison data. The framework involves the Chebyshev approximation of pairwise comparison matrices by consistent matrices in the logarithmic scale. We reduce the log-Chebyshev approximation to multidimensional tropical optimization problems, and offer complete direct solutions to the problems in the framework of tropical mathematics. The results obtained provide a closed-form solution to the rating problem of interest as either a unique score vector (up to a positive factor) or as a set of different score vectors. To handle the problem when the solution is not unique, we develop tropical optimization techniques to find those vectors from the solution set that are the most and least differentiating between the alternatives with the highest and lowest scores, and thus can be well representative of the entire solution.