Methods of tropical optimization in rating alternatives based on pairwise comparisons

TL;DR

This paper introduces tropical optimization methods for rating alternatives using pairwise comparison matrices, providing a closed-form solution and analyzing the set of possible score vectors for better decision-making.

Contribution

It applies tropical optimization to pairwise comparison problems, deriving a direct solution and characterizing the set of score vectors when the solution is not unique.

Findings

Closed-form solution for rating problems

Characterization of non-unique solution sets

Identification of most and least differentiating score vectors

Abstract

We apply methods of tropical optimization to handle problems of rating alternatives on the basis of the log-Chebyshev approximation of pairwise comparison matrices. We derive a direct solution in a closed form, and investigate the obtained solution when it is not unique. Provided the approximation problem yields a set of score vectors, rather than a unique (up to a constant factor) one, we find those vectors in the set, which least and most differentiate between the alternatives with the highest and lowest scores, and thus can be representative of the entire solution.