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Packing of concave polyhedra with continuous rotations using nonlinear optimisation

Author

Listed:
  • Romanova, T.
  • Bennell, J.
  • Stoyan, Y.
  • Pankratov, A.

Abstract

We study the problem of packing a given collection of arbitrary, in general concave, polyhedra into a cuboid of minimal volume. Continuous rotations and translations of polyhedra are allowed. In addition, minimal allowable distances between polyhedra are taken into account. We derive an exact mathematical model using adjusted radical free quasi phi-functions for concave polyhedra to describe non-overlapping and distance constraints. The model is a nonlinear programming formulation. We develop an efficient solution algorithm, which employs a fast starting point algorithm and a new compaction procedure. The procedure reduces our problem to a sequence of nonlinear programming subproblems of considerably smaller dimension and a smaller number of nonlinear inequalities. The benefit of this approach is borne out by the computational results, which include a comparison with previously published instances and new instances.

Suggested Citation

  • Romanova, T. & Bennell, J. & Stoyan, Y. & Pankratov, A., 2018. "Packing of concave polyhedra with continuous rotations using nonlinear optimisation," European Journal of Operational Research, Elsevier, vol. 268(1), pages 37-53.
    • Handle: RePEc:eee:ejores:v:268:y:2018:i:1:p:37-53
    DOI: 10.1016/j.ejor.2018.01.025
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    References listed on IDEAS

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    1. Wascher, Gerhard & Hau[ss]ner, Heike & Schumann, Holger, 2007. "An improved typology of cutting and packing problems," European Journal of Operational Research, Elsevier, vol. 183(3), pages 1109-1130, December.
    2. Galrão Ramos, A. & Oliveira, José F. & Gonçalves, José F. & Lopes, Manuel P., 2016. "A container loading algorithm with static mechanical equilibrium stability constraints," Transportation Research Part B: Methodological, Elsevier, vol. 91(C), pages 565-581.
    3. Giorgio Fasano, 2013. "A global optimization point of view to handle non-standard object packing problems," Journal of Global Optimization, Springer, vol. 55(2), pages 279-299, February.
    4. S. Torquato & Y. Jiao, 2009. "Dense packings of the Platonic and Archimedean solids," Nature, Nature, vol. 460(7257), pages 876-879, August.
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    Cited by:

    1. Romanova, Tatiana & Stoyan, Yurij & Pankratov, Alexander & Litvinchev, Igor & Plankovskyy, Sergiy & Tsegelnyk, Yevgen & Shypul, Olga, 2021. "Sparsest balanced packing of irregular 3D objects in a cylindrical container," European Journal of Operational Research, Elsevier, vol. 291(1), pages 84-100.
    2. Alexander Pankratov & Tatiana Romanova & Igor Litvinchev, 2020. "Packing Oblique 3D Objects," Mathematics, MDPI, vol. 8(7), pages 1-17, July.
    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. 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.
    5. 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.

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