New proposals for modelling and solving the problem of covering solids using spheres of different radii

Autor:	Philippe Michelon, Nelson Maculan, Luidi Simonetti, Ana Flávia Uzeda dos Santos Macambira, Pedro Henrique González Silva, Renan Vicente Pinto
Rok vydání:	2020
Předmět:	Optimization problem Compact space Quadratic equation Cover (topology) Intersection (set theory) Heuristic (computer science) Binary number Applied mathematics Management Science and Operations Research Finite set Computer Science Applications Theoretical Computer Science Mathematics
Zdroj:	RAIRO - Operations Research. 54:873-882
ISSN:	1290-3868 0399-0559
DOI:	10.1051/ro/2019041
Popis:	Given a solid T, represented by a compact set in ℝ3, 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.
Databáze:	OpenAIRE
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