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Packing ellipses in an optimized convex polygon

Author

Listed:
  • A. Pankratov

    (Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine)

  • T. Romanova

    (Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine)

  • I. Litvinchev

    (Nuevo Leon State University (UANL))

Abstract

Packing ellipses with arbitrary orientation into a convex polygonal container which has a given shape is considered. The objective is to find a minimum scaling (homothetic) coefficient for the polygon still containing a given collection of ellipses. New phi-functions and quasi phi-functions to describe non-overlapping and containment constraints are introduced. The packing problem is then stated as a continuous nonlinear programming problem. A solution approach is proposed combining a new starting point algorithm and a new modification of the LOFRT procedure (J Glob Optim 65(2):283–307, 2016) to search for locally optimal solutions. Computational results are provided to demonstrate the efficiency of our approach. The computational results are presented for new problem instances, as well as for instances presented in the recent paper ( http://www.optimization-online.org/DB_FILE/2016/03/5348.pdf , 2016).

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:75:y:2019:i:2:d:10.1007_s10898-019-00777-y
    DOI: 10.1007/s10898-019-00777-y
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    References listed on IDEAS

    as
    1. Josef Kallrath & Steffen Rebennack, 2014. "Cutting ellipses from area-minimizing rectangles," Journal of Global Optimization, Springer, vol. 59(2), pages 405-437, July.
    2. Yuriy Stoyan & Alexander Pankratov & Tatiana Romanova, 2016. "Cutting and packing problems for irregular objects with continuous rotations: mathematical modelling and non-linear optimization," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 67(5), pages 786-800, May.
    3. E. G. Birgin & R. D. Lobato & J. M. Martínez, 2016. "Packing ellipsoids by nonlinear optimization," Journal of Global Optimization, Springer, vol. 65(4), pages 709-743, August.
    4. Josef Kallrath, 2017. "Packing ellipsoids into volume-minimizing rectangular boxes," Journal of Global Optimization, Springer, vol. 67(1), pages 151-185, January.
    5. E. G. Birgin & R. D. Lobato & J. M. Martínez, 2017. "A nonlinear programming model with implicit variables for packing ellipsoids," Journal of Global Optimization, Springer, vol. 68(3), pages 467-499, July.
    6. 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.
    7. Yuriy Stoyan & Tatiana Romanova & Alexander Pankratov & Andrey Chugay, 2015. "Optimized Object Packings Using Quasi-Phi-Functions," Springer Optimization and Its Applications, in: Giorgio Fasano & János D. Pintér (ed.), Optimized Packings with Applications, edition 1, chapter 0, pages 265-293, Springer.
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    Cited by:

    1. Andreas Fischer & Igor Litvinchev & Tetyana Romanova & Petro Stetsyuk & Georgiy Yaskov, 2023. "Quasi-Packing Different Spheres with Ratio Conditions in a Spherical Container," Mathematics, MDPI, vol. 11(9), pages 1-19, April.
    2. 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.
    3. Josef Kallrath & Tatiana Romanova & Alexander Pankratov & Igor Litvinchev & Luis Infante, 2023. "Packing convex polygons in minimum-perimeter convex hulls," Journal of Global Optimization, Springer, vol. 85(1), pages 39-59, January.

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