  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. In this framework, we first consider the minimax
Chebyshev approximation of pairwise comparison matrices by consistent matrices
in the logarithmic scale. Recasting this approximation problem as a problem of
tropical pseudo-quadratic programming we then write out a closed-form solution
to it. This solution might be either a unique score vector (up to a positive
factor) or a set of different score vectors. To handle the problem when the
solution is not unique, we develop tropical optimization techniques of
maximizing and minimizing the Hilbert seminorm 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 are well
representative of the entire solution set.
