The Analytic Hierarchy Process, Max Algebra and Multi-objective Optimisation

TL;DR

This paper extends the max-algebraic approach to the Analytic Hierarchy Process for multi-criteria decision making, establishing conditions for optimal solutions and their existence using spectral properties.

Contribution

It generalizes the max-algebraic framework to multi-objective AHP and links solution optimality to matrix commutativity and spectral radius concepts.

Findings

Existence of globally optimal solutions related to matrix commutativity

Min-max solutions characterized by generalized spectral radius

Pareto optimal solutions are guaranteed to exist

Abstract

The Analytic Hierarchy Process (AHP) is widely used for decision making involving multiple criteria. Elsner and van den Driessche introduced a max-algebraic approach to the single criterion AHP. We extend this to the multi-criteria AHP, by considering multi-objective generalisations of the single objective optimisation problem solved in these earlier papers. We relate the existence of globally optimal solutions to the commutativity properties of the associated matrices; we relate min-max optimal solutions to the generalised spectral radius; and we prove that Pareto optimal solutions are guaranteed to exist.