Discrete representation of non-dominated sets in multi-objective linear programming - Shao, Lizhen; Ehrgott, Matthias (European journal of operational research, 2016)
•Discrete representation of non-dominated sets relates to NP-hard location problems.•Known methods do not have the coverage property or bound the uniformity level.•Under some assumptions the RNBI method has bounded uniformity level and coverage error.•Empirical tests on applied problems and random examples validate the RNBI method. In this paper we address the problem of representing the continuous but non-convex set of non-dominated points of a multi-objective linear programme by a finite subset of such points. We prove that a related decision problem is NP-complete. Moreover, we illustrate the drawbacks of the known global shooting, normal boundary intersection and normal constraint methods concerning the coverage error and uniformity level of the representation by examples. We propose a method which combines the global shooting and normal boundary intersection methods. By doing so, we overcome their limitations, but preserve their advantages. We prove that our method computes a set of evenly distributed non-dominated points for which the coverage error and the uniformity level can be guaranteed. We apply this method to an optimisation problem in radiation therapy and present illustrative results for some clinical cases. Finally, we present numerical results on randomly generated examples.