New and improved results for packing identical unitary radius circles within triangles, rectangles and strips

Authors:
Ernesto G. Birgin;Jan M. Gentil
Affiliations:
Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil;Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil
Venue:
Computers and Operations Research
Year:
2010

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Abstract

The focus of study in this paper is the class of packing problems. More specifically, it deals with the placement of a set of N circular items of unitary radius inside an object with the aim of minimizing its dimensions. Differently shaped containers are considered, namely circles, squares, rectangles, strips and triangles. By means of the resolution of non-linear equations systems through the Newton-Raphson method, the herein presented algorithm succeeds in improving the accuracy of previous results attained by continuous optimization approaches up to numerical machine precision. The computer implementation and the data sets are available at http://www.ime.usp.br/~egbirgin/packing/.