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A fully general, exact algorithm for nesting irregular shapes

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  • Donald Jones

Abstract

This paper introduces a fully general, exact algorithm for nesting irregular shapes. Both the shapes and material resource can be arbitrary nonconvex polygons. Moreover, the shapes can have holes and the material can have defective areas. Finally, the shapes can be arranged using both translations and arbitrary rotations (as opposed to a finite set of rotation angles, such as 0 $$^\circ $$ ∘ and 180 $$^\circ $$ ∘ ). The insight that has made all this possible is a novel way to relax the constraint that the shapes not overlap. The key idea is to inscribe a few circles in each irregular shape and then relax the non-overlap constraints for the shapes by replacing them with non-overlap constraints for the inscribed circles. These relaxed problems have the form of quadratic programming problems (QPs) and can be solved to optimality to provide valid lower bounds. Valid upper bounds can be found via local search with strict non-overlap constraints. If the shapes overlap in the solution to the relaxed problem, new circles are inscribed in the shapes to prevent this overlapping configuration from recurring, and the QP is then resolved to obtain improved lower bounds. Convergence to any fixed tolerance is guaranteed in a finite number of iterations. A specialized branch-and-bound algorithm, together with some heuristics, are introduced to find the initial inscribed circles that approximate the shapes. The new approach, called “QP-Nest,” is applied to three problems as proof of concept. The most complicated of these is a problem due to Milenkovic that has four nonconvex polygons with 94, 72, 84, and 74 vertices, respectively. QP-Nest is able prove global optimality when nesting the first two or the first three of these shapes. When all four shapes are considered, QP-Nest finds the best known solution, but cannot prove optimality. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Donald Jones, 2014. "A fully general, exact algorithm for nesting irregular shapes," Journal of Global Optimization, Springer, vol. 59(2), pages 367-404, July.
  • Handle: RePEc:spr:jglopt:v:59:y:2014:i:2:p:367-404
    DOI: 10.1007/s10898-013-0129-z
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    References listed on IDEAS

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    1. Costa, M. Teresa & Gomes, A. Miguel & Oliveira, José F., 2009. "Heuristic approaches to large-scale periodic packing of irregular shapes on a rectangular sheet," European Journal of Operational Research, Elsevier, vol. 192(1), pages 29-40, January.
    2. Burke, E.K. & Hellier, R.S.R. & Kendall, G. & Whitwell, G., 2007. "Complete and robust no-fit polygon generation for the irregular stock cutting problem," European Journal of Operational Research, Elsevier, vol. 179(1), pages 27-49, May.
    3. Dowsland, Kathryn A. & Dowsland, William B., 1995. "Solution approaches to irregular nesting problems," European Journal of Operational Research, Elsevier, vol. 84(3), pages 506-521, August.
    4. Li, Zhenyu & Milenkovic, Victor, 1995. "Compaction and separation algorithms for non-convex polygons and their applications," European Journal of Operational Research, Elsevier, vol. 84(3), pages 539-561, August.
    5. J A Bennell & J F Oliveira, 2009. "A tutorial in irregular shape packing problems," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 60(1), pages 93-105, May.
    6. Edmund Burke & Robert Hellier & Graham Kendall & Glenn Whitwell, 2006. "A New Bottom-Left-Fill Heuristic Algorithm for the Two-Dimensional Irregular Packing Problem," Operations Research, INFORMS, vol. 54(3), pages 587-601, June.
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    Cited by:

    1. Luiz H. Cherri & Adriana C. Cherri & Edilaine M. Soler, 2018. "Mixed integer quadratically-constrained programming model to solve the irregular strip packing problem with continuous rotations," Journal of Global Optimization, Springer, vol. 72(1), pages 89-107, September.
    2. Akang Wang & Christopher L. Hanselman & Chrysanthos E. Gounaris, 2018. "A customized branch-and-bound approach for irregular shape nesting," Journal of Global Optimization, Springer, vol. 71(4), pages 935-955, August.
    3. Romanova, Tatiana & Litvinchev, Igor & Pankratov, Alexander, 2020. "Packing ellipsoids in an optimized cylinder," European Journal of Operational Research, Elsevier, vol. 285(2), pages 429-443.
    4. Igor Kierkosz & Maciej Łuczak, 2019. "A one-pass heuristic for nesting problems," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 29(1), pages 37-60.
    5. Sato, André Kubagawa & Martins, Thiago Castro & Gomes, Antonio Miguel & Tsuzuki, Marcos Sales Guerra, 2019. "Raster penetration map applied to the irregular packing problem," European Journal of Operational Research, Elsevier, vol. 279(2), pages 657-671.
    6. Burak Beyhan & Cüneyt Güler & Hidayet Tağa, 2020. "An algorithm for maximum inscribed circle based on Voronoi diagrams and geometrical properties," Journal of Geographical Systems, Springer, vol. 22(3), pages 391-418, July.
    7. Leao, Aline A.S. & Toledo, Franklina M.B. & Oliveira, José Fernando & Carravilla, Maria Antónia & Alvarez-Valdés, Ramón, 2020. "Irregular packing problems: A review of mathematical models," European Journal of Operational Research, Elsevier, vol. 282(3), pages 803-822.
    8. Yizhe Yang & Bingshan Liu & Haochen Li & Xin Li & Gong Wang & Shan Li, 2023. "A nesting optimization method based on digital contour similarity matching for additive manufacturing," Journal of Intelligent Manufacturing, Springer, vol. 34(6), pages 2825-2847, August.
    9. Akang Wang & Chrysanthos E. Gounaris, 2021. "On tackling reverse convex constraints for non-overlapping of unequal circles," Journal of Global Optimization, Springer, vol. 80(2), pages 357-385, June.

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