An algorithm for polygon placement using a bottom-left strategy

Abstract This paper describes a fast and efficient implementation of a bottom-left (BL) placement algorithm for polygon packing. The algorithm allows pieces to be nested within the partial layout produced by previously placed pieces, and produces an optimal BL layout in the sense that the positions considered are guaranteed to contain the bottom-left position of the infinite set of possibilities. Full details of the way in which these positions are calculated are given. Computational experiments comparing the results of different orderings on a variety of datasets from the literature are reported, and these illustrate that problems having in excess of 100 pieces of several piece types can be solved within one minute on a modern desktop PC. The procedure can easily be incorporated into algorithms that apply more sophisticated piece selection procedures.

[1]  S. Jakobs,et al.  European Journal Ofoperational Research on Genetic Algorithms for the Packing of Polygons , 2022 .

[2]  José Fernando Oliveira,et al.  TOPOS – A new constructive algorithm for nesting problems , 2000, OR Spectr..

[3]  A. M. Gomes,et al.  A New Constructive Algorithm for Nesting Problems , 2001 .

[4]  Donna J. Brown,et al.  An Improved BL Lower Bound , 1980, Inf. Process. Lett..

[5]  Kathryn A. Dowsland,et al.  The irregular cutting-stock problem - a new procedure for deriving the no-fit polygon , 2001, Comput. Oper. Res..

[6]  Joaquim A. Jorge,et al.  Marker-making using automatic placement of irregular shapes for the garment industry , 1990, Comput. Graph..

[7]  Henry J. Lamousin,et al.  Nesting of two-dimensional irregular parts using a shape reasoning heuristic , 1997, Comput. Aided Des..

[8]  Giorgio Ausiello,et al.  Algorithm Design for Computer System Design , 1984, International Centre for Mechanical Sciences.

[9]  Ronald L. Rivest,et al.  Orthogonal Packings in Two Dimensions , 1980, SIAM J. Comput..

[10]  J Błażewicz,et al.  An Improved Version of Tabu Search for Irregular Cutting Problems , 1993 .

[11]  K. A. Dowsland,et al.  Jostling for position: local improvement for irregular cutting patterns , 1998, J. Oper. Res. Soc..

[12]  David S. Johnson,et al.  Approximation Algorithms for Bin-Packing — An Updated Survey , 1984 .

[13]  李幼升,et al.  Ph , 1989 .

[14]  Richard Carl Art.  An approach to the two dimensional irregular cutting stock problem. , 1966 .

[15]  Hongfei Teng,et al.  An improved BL-algorithm for genetic algorithm of the orthogonal packing of rectangles , 1999, Eur. J. Oper. Res..

[16]  V. Milenkovic,et al.  Compaction and separation algorithms for non-convex polygons and their applications☆ , 1995 .

[17]  Pijush K. Ghosh,et al.  An algebra of polygons through the notion of negative shapes , 1991, CVGIP Image Underst..

[18]  Lawrence Davis,et al.  Applying Adaptive Algorithms to Epistatic Domains , 1985, IJCAI.

[19]  José Fernando Oliveira,et al.  Algorithms for Nesting Problems , 1993 .