  We describe a new approach based on tropical optimization techniques to solve
the problem of rating alternatives from pairwise comparison data. The problem
is formulated to approximate, in the log-Chebyshev sense, pairwise comparison
matrices by reciprocal matrices of unit rank, and then represented in general
terms of tropical mathematics as a tropical optimization problem. The
optimization problem takes a common, unified form for both multiplicative and
additive comparison scales. We apply recent results in tropical optimization to
offer new complete solutions to the rating problems under various assumptions
about the pairwise comparison matrices. The solutions are given in a compact
vector form, which extends known solutions and involves modest computational
efforts. The results obtained are illustrated with numerical examples.
Specifically, we show by example that the partial solution known before may
miss better results provided by the new complete solution. An example to
demonstrate a tropical analogue of the analytical hierarchy process decision
scheme is also given.
