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Vietnam Journal of Mathematics 40:2&3(2012) 345-354

 

A Model to the Ellipsoidal Filling Problem

Roberto Quirino do Nascimento1, Ana Flavia U. S. Macambira1,

Lucidio dos Anjos Formiga Cabral1 and Nelson Maculan2

1Universidade Federal da Paraiba, Departamento de Informatica,

58.051-900 - Joao Pessoa, Paraiba, Brazil

2 Universidade Federal do Rio de Janeiro,

COPPE-Engenharia de Sistemas e Computação,

Caixa Postal 68511, 21941-972 Rio de Janeiro, RJ, Brasil

Dedicated to Professor Phan Quoc Khanh on the occasion of his 65th birthday

Received November 6, 2011

Revised April 16, 2012

Abstract. The ellipsoidal filling problem consists in covering an ellipsoid with spheres whose radii belong to a discrete set. The discrete nature of the radii of the spheres is one of the difficulties inherent to this problem when one tries to solve it and another dificulty is ensuring that every point of the ellipsoid is covered by at least one sphere. Despite these difficulties, a good reason to study this problem is its application in configuring Gamma ray machines, used in brain tumors treatments. This problem is a semi-infinite nonlinear discrete one and we present a weak version that uses the idea of the Facility Location Problem to determine a likely location for the center of the spheres in such a way that the ellipsoid must be filled by them.

2000 Mathematics Subject Classification. 90C11.

Keywords. Discrete optimization, global optimization, Gamma Knife.

 

 

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