RAIRO-Oper. Res. 54 (2020) 873–882 RAIRO Operations Research https://doi.org/10.1051/ro/2019041 www.rairo-ro.org NEW PROPOSALS FOR MODELLING AND SOLVING THE PROBLEM OF COVERING SOLIDS USING SPHERES OF DIFFERENT RADII Pedro Henrique Gonz´ alez Silva 1,4,∗ , Ana Fl´ avia U. S. Macambira 2 , Renan Vicente Pinto 3 , Luidi Simonetti 4 , Nelson Maculan 4 and Philippe Michelon 5 Abstract. Given a solid T , represented by a compact set in R 3 , the aim of this work is to find a covering of T by a finite set of spheres of different radii. Some level of intersection between the spheres is necessary to cover the solid. Moreover, the volume occupied by the spheres on the outside of T is limited. This problem has an application in the planning of a radio-surgery treatment known by Gamma Knife and can be formulated as a non-convex optimization problem with quadratic constraints and linear objective function. In this work, two new linear mathematical formulations with binary variables and a hybrid method are proposed. The hybrid method combines heuristic, data mining and an exact method. Computational results show that the proposed methods outperform the ones presented in the literature. Mathematics Subject Classification. 90C10. Received September 27, 2017. Accepted April 7, 2019. 1. Introduction Covering problems are often related to the determination of the position of certain objects in order to cover the area or volume of a greater object. To achieve this objective, a level of intersection between the objects is allowed. This class of problems is naturally formulated as a minimization problem. A classical application of covering a two-dimensional region with circles arises in the field of telecommunications. In this context, the equipment covers a circular geographic area and the objective is to attend or to cover a city by installing the smallest amount of equipments. Another class of problems that is closely related to the covering problems is known as packing problems. In fact, packing and covering are a pair of primal x dual problems. Packing problems consists in positioning the maximum number of objects inside a bigger object, often called container. The objects cannot overlap. A classical application of the packing problem is stacking the maximum number of oranges in a box. Keywords. Problem of covering solids, mathematical programming, hybrid method. 1 Departamento de Inform´ atica, Centro Federal de Educa¸c˜ ao Tecnol´ogica Celso Suckow da Fonseca, Brazil. 2 Departamento de Estat´ ıstica, Universidade Federal da Para´ ıba, Brazil. 3 Departamento de Matem´ atica, Universidade Federal Rural do Rio de Janeiro, Brazil. 4 Programa de Engenharia de Sistemas e Computa¸ c˜ao, Universidade Federal do Rio de Janeiro, Brazil. 5 Laboratoire de Math´ ematiques d’Avignon, Universit´ e d’Avignon et des Pays de Vaucluse, France. * Corresponding author: pehgonzalez@gmail.com; pegonzalez@eic.cefet-rj.br Article published by EDP Sciences c  EDP Sciences, ROADEF, SMAI 2020