The planar multiple obnoxious facilities location problem: A Voronoi based heuristic

Abstract

Consider a situation where a given number of facilities must be located in a convex polygon with the objective of maximizing the minimum distance between facilities and a given set of communities subject to the facilities being farther than a certain distance from one another. This continuous multiple obnoxious facility location problem is very difficult to solve by commercial nonlinear optimizers. We propose a mathematical formulation of two variants of the problem and a heuristic approach based on Voronoi diagrams and a binary linear program. We compare our heuristic with popular state of the art solvers which utilize the multi-start approach: interior point, sparse nonlinear optimizer (SNOPT), GA, IPOPT, and NOMAD running in Matlab. Each problem instance is solved using 100 randomly generated starting solutions, and the best solution is selected. We found that our results are much better and were obtained in a fraction of the time required by other methods. The multiple obnoxious location problem is a perfect example where all-purpose nonlinear non-convex solvers perform poorly and are significantly outperformed by custom designed heuristics.