Tropical implementation of the Analytical Hierarchy Process decision method

TL;DR

This paper introduces a novel tropical optimization framework for the Analytic Hierarchy Process, providing a closed-form solution for pairwise comparison matrices and methods to select representative score vectors.

Contribution

It develops a new tropical optimization approach for implementing AHP, including closed-form solutions and techniques for handling non-unique solutions.

Findings

Closed-form solution for pairwise comparison approximation

Methods for selecting most and least differentiating score vectors

Framework applicable to multi-criteria decision problems

Abstract

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.