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Approximate Packing: Integer Programming Models, Valid Inequalities and Nesting

In: Optimized Packings with Applications

Author

Listed:
  • Igor Litvinchev

    (Russian Academy of Sciences)

  • Luis Infante

    (Nuevo Leon Sate University)

  • Lucero Ozuna

    (Nuevo Leon Sate University)

Abstract

Using a regular grid to approximate a container, packing objects is reduced to assigning objects to the nodes of the grid subject to non-overlapping constraints. The packing problem is then stated as a large scale linear 0-1 optimization problem. Different formulations for non-overlapping constraints are presented and compared. Valid inequalities are proposed to strengthening formulations. This approach is applied for packing circular and L-shaped objects. Circular object is considered in a general sense as a set of points that are all the same distance (not necessary Euclidean) from a given point. Different shapes, such as ellipses, rhombuses, rectangles, octagons, etc., are treated similarly by simply changing the definition of the norm used to define the distance. Nesting objects inside one another is also considered. Numerical results are presented to demonstrate the efficiency of the proposed approach.

Suggested Citation

  • Igor Litvinchev & Luis Infante & Lucero Ozuna, 2015. "Approximate Packing: Integer Programming Models, Valid Inequalities and Nesting," Springer Optimization and Its Applications, in: Giorgio Fasano & János D. Pintér (ed.), Optimized Packings with Applications, edition 1, chapter 0, pages 187-205, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-18899-7_9
    DOI: 10.1007/978-3-319-18899-7_9
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    Citations

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    Cited by:

    1. 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.
    2. A. Pankratov & T. Romanova & I. Litvinchev, 2019. "Packing ellipses in an optimized convex polygon," Journal of Global Optimization, Springer, vol. 75(2), pages 495-522, October.

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