Basic solutions of systems with two max-linear inequalities

We give an explicit description of the basic solutions of max-linear systems with two inequalities.Comment: 16 page

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

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.Comment: 1 figur

On visualisation scaling, subeigenvectors and Kleene stars in max algebra

The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is strict visualisation scaling, which means finding, for a given nonnegative matrix A, a diagonal matrix X such that all elements of X^{-1}AX are less than or equal to the maximum cycle geometric mean of A, with strict inequality for the entries which do not lie on critical cycles. In this paper such scalings are described by means of the max-algebraic subeigenvectors and Kleene stars of nonnegative matrices as well as by some concepts of convex geometry.Comment: 22 page