Rating alternatives from pairwise comparisons by solving tropical optimization problems

TL;DR

This paper introduces a novel tropical optimization approach for rating alternatives based on pairwise comparisons, providing efficient solutions for both constrained and unconstrained problems with practical numerical examples.

Contribution

It formulates pairwise comparison rating problems within tropical mathematics, extending existing methods with more efficient, direct solutions applicable to both multiplicative and additive scales.

Findings

New complete solutions for tropical optimization problems

Reduced computational effort compared to previous methods

Numerical examples demonstrating practical application

Abstract

We consider problems of rating alternatives based on their pairwise comparison under various assumptions, including constraints on the final scores of alternatives. The problems are formulated in the framework of tropical mathematics to approximate pairwise comparison matrices by reciprocal matrices of unit rank, and written in a common form for both multiplicative and additive comparison scales. To solve the unconstrained and constrained approximation problems, we apply recent results in tropical optimization, which provide new complete direct solutions given in a compact vector form. These solutions extend known results and involve less computational effort. As an illustration, numerical examples of rating alternatives are presented.