Using tropical optimization techniques to evaluate alternatives via pairwise comparisons

TL;DR

This paper introduces a novel tropical optimization approach for rating alternatives based on pairwise comparison data, providing complete solutions that improve upon previous partial methods and extending the analytical hierarchy process.

Contribution

The paper develops a unified tropical optimization framework for rating from pairwise comparisons, offering complete solutions and demonstrating advantages over existing partial solutions.

Findings

New complete solutions for pairwise comparison rating problems.

Tropical optimization methods extend existing approaches and improve results.

Numerical examples illustrate the effectiveness of the proposed solutions.

Abstract

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.