Optimized location of light sources to cover a rectangular region

Abstract

AbstractThe problem we considered was proposed by an industrial partner. The aim is to locate light sources around a rectangular field such that the areas these illuminate cover the whole field. We assume these illuminated areas to be rectangular as well, parallel to the field. Covering an area with multiple lights is allowed. There are several types of light sources, priced differently with different sizes of their illuminated area. We aim to minimize the cost of the cover. We propose a constraint generation approach for solving this covering problem. We formulate a MIP model to locate the light sources such that a finite number of predetermined points have to be covered. The result does not necessarily solve the original problem, i.e. it does not cover the whole field. Therefore, a constraint generation model is built to calculate a non-covered point such that the first model has to improve its previous solution to cover this new point as well. If no uncovered point is found, the result is an optimal covering, thus we stop. We have also designed some set of additional constraints to exclude symmetrical solutions, to speed up the algorithm. We analyze the efficiency of the additional constraints and report some computational results in realistic settings.