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Optimized Object Packings Using Quasi-Phi-Functions

In: Optimized Packings with Applications

Author

Listed:
  • Yuriy Stoyan

    (National Academy of Sciences of Ukraine)

  • Tatiana Romanova

    (National Academy of Sciences of Ukraine)

  • Alexander Pankratov

    (National Academy of Sciences of Ukraine)

  • Andrey Chugay

    (National Academy of Sciences of Ukraine)

Abstract

In this chapter we further develop the main tool of our studies,phi-functions. We define new functions, called quasi-phi-functions, that we use for analytic description of relations of geometric objects placed in a container taking into account their continuous rotations, translations, and distance constraints. The new functions are substantially simpler than phi-functions for some types of objects. They also are simple enough for some types of objects for which phi-functions could not be constructed. In particular, we derive quasi-phi-functions for certain 2D&3D-objects. We formulate a basic optimal packing problem and introduce its exact mathematical model in the form of a nonlinear continuous programming problem, using our quasi-phi-functions. We propose a general solution strategy, involving: a construction of feasible starting points, a generation of nonlinear subproblems of a smaller dimension and decreased number of inequalities; a search for local extrema of our problem using subproblems. To show the advantages of our quasi-phi-functions we apply them to two packing problems, which have a wide spectrum of industrial applications: packing of a given collection of ellipses into a rectangular container of minimal area taking into account distance constraints; packing of a given collection of 3D-objects, including cuboids, spheres, spherocylinders and spherocones, into a cuboid container of minimal height. Our efficient optimization algorithms allow us to get local optimal object packings and reduce considerably computational cost. We applied our algorithms to several inspiring instances: our new benchmark instances and known test cases.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:spochp:978-3-319-18899-7_13
    DOI: 10.1007/978-3-319-18899-7_13
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    Citations

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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. 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. Lastra-Díaz, Juan J. & Ortuño, M. Teresa, 2024. "Mixed-integer programming models for irregular strip packing based on vertical slices and feasibility cuts," European Journal of Operational Research, Elsevier, vol. 313(1), pages 69-91.
    4. 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.
    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.
    6. 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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