Cylinder packing by simulated annealing

Abstract

This paper is motivated by the problem of loading identical items of circular base (tubes, rolls, ...) into a rectangular base (the pallet). For practical reasons, all the loaded items are considered to have the same height. The resolution of this problem consists in determining the positioning pattern of the circular bases of the items on the rectangular pallet, while maximizing the number of items. This pattern will be repeated for each layer stacked on the pallet. Two algorithms based on the metaheuristic Simulated Annealing have been developed and implemented. The tuning of these algorithms parameters implied running intensive tests in order to improve its efficiency. The algorithms developed were easily extended to the case of non-identical circles.Keywords: cylinder packing, combinatorial optimization, simulated annealing. ResumoEste artigo aborda o problema de posicionamento de objetos de base circular (tubos, rolos, ...) sobre uma base retangular de maiores dimensões. Por razões práticas, considera-se que todos os objetos a carregar apresentam a mesma altura. A resolução do problema consiste na determinação do padrão de posicionamento das bases circulares dos referidos objetos sobre a base de forma retangular, tendo como objetivo a maximização do número de objetos estritamente posicionados no interior dessa base. Este padrão de posicionamento será repetido em cada uma das camadas a carregar sobre a base retangular. Apresentam-se dois algoritmos para a resolução do problema. Estes algoritmos baseiam-se numa metaheurística, Simulated Annealling, cuja afinação de parâmetros requereu a execução de testes intensivos com o objetivo de atingir um elevado grau de eficiência no seu desempenho. As características dos algoritmos implementados permitiram que a sua extensão à consideração de círculos com raios diferentes fosse facilmente conseguida.Palavras-chave: empacotamento de cilindros, otimização combinatória, simulated annealing.

Optimizing the packing of cylinders into a rectangular container: A nonlinear approach

“…Both subsets seems to be the harder ones, as mentioned in [7]. The reason for which no problems from subsets 3.x, 4.x and 5.x were improved is because, probably, since these are easier problems, the number of cylinders packed in [7] are the optimal ones. Figure 4 shows the packings found for the 6 improved problems of this second set.…”

“…The second set of problems was taken from [7,8] and is described in Table 2. As mentioned in [7,8], the test instances were generated such as to be representative of three possible relations between the rectangular box and the circle area: in problems 1.x, 2.x, 3.x, 4.x and 5.x, the circle area is approximately 1%, 2.5%, 5%, 10% and 15% of the rectangular box, respectively.…”

“…As mentioned in [7,8], the test instances were generated such as to be representative of three possible relations between the rectangular box and the circle area: in problems 1.x, 2.x, 3.x, 4.x and 5.x, the circle area is approximately 1%, 2.5%, 5%, 10% and 15% of the rectangular box, respectively. In terms of the box shape, problems x.1, x.2, x.3 are long-rectangular, problems x.4, x.5, x.6 are short-rectangular and problems x.7, x.8, x.9 are squares.…”

“…This requires further research. Table 2: Performance of GENPACK for the problems presented in [7]. Table 3: Performance of GENPACK for packing circles into circles.…”

“…The paper [10] is concerned with the relationship between the way circles are packed into a rectangular box and the efficiency of the resultant palletisation. In [7] algorithms based on Simulated Annealing are presented. Furthermore, the same authors [8] propose a new upper bound for this problem.…”