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Quasi-Packing Different Spheres with Ratio Conditions in a Spherical Container

Author

Listed:
  • Andreas Fischer

    (Institute of Numerical Mathematics, Technische Universität Dresden, 01062 Dresden, Germany)

  • Igor Litvinchev

    (Graduate Program in Systems Engineering, Nuevo Leon State University (UANL), Av. Universidad s/n, Col. Ciudad Universitaria, San Nicolas de los Garza CP 66455, Nuevo Leon, Mexico)

  • Tetyana Romanova

    (A. Pidhornyi Institute of Mechanical Engineering Problems of the National Academy of Sciences of Ukraine, Pozhars’koho St., 2/10, 61046 Kharkiv, Ukraine
    Department of Systems Engineering, Kharkiv National University of Radioelectronics, 14 Nauky Ave., 61166 Kharkiv, Ukraine)

  • Petro Stetsyuk

    (V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, 03187 Kyiv, Ukraine)

  • Georgiy Yaskov

    (A. Pidhornyi Institute of Mechanical Engineering Problems of the National Academy of Sciences of Ukraine, Pozhars’koho St., 2/10, 61046 Kharkiv, Ukraine)

Abstract

This paper considers the optimized packing of different spheres into a given spherical container under non-standard placement conditions. A sphere is considered placed in the container if at least a certain part of the sphere is in the container. Spheres are allowed to overlap with each other according to predefined parameters. Ratio conditions are introduced to establish correspondence between the number of packed spheres of different radii. The packing aims to maximize the total number of packed spheres subject to ratio, partial overlapping and quasi-containment conditions. A nonlinear mixed-integer optimization model is proposed for this ratio quasi-packing problem. A heuristic algorithm is developed that reduces the original problem to a sequence of continuous open dimension problems for quasi-packing scaled spheres. Computational results for finding global solutions for small instances and good feasible solutions for large instances are provided.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:2033-:d:1132385
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    References listed on IDEAS

    as
    1. Igor Litvinchev & Edith Lucero Ozuna Espinosa, 2014. "Integer Programming Formulations for Approximate Packing Circles in a Rectangular Container," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-6, March.
    2. Galiev, Shamil I. & Lisafina, Maria S., 2013. "Linear models for the approximate solution of the problem of packing equal circles into a given domain," European Journal of Operational Research, Elsevier, vol. 230(3), pages 505-514.
    3. Castillo, Ignacio & Kampas, Frank J. & Pintér, János D., 2008. "Solving circle packing problems by global optimization: Numerical results and industrial applications," European Journal of Operational Research, Elsevier, vol. 191(3), pages 786-802, December.
    4. Hifi, Mhand & Yousef, Labib, 2019. "A local search-based method for sphere packing problems," European Journal of Operational Research, Elsevier, vol. 274(2), pages 482-500.
    5. Mhand Hifi & Rym M'Hallah, 2009. "A Literature Review on Circle and Sphere Packing Problems: Models and Methodologies," Advances in Operations Research, Hindawi, vol. 2009, pages 1-22, July.
    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.
    7. 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.
    8. Yuriy Stoyan & Alexandr Pankratov & Tatiana Romanova & Giorgio Fasano & János D. Pintér & Yurij E. Stoian & Andrey Chugay, 2019. "Optimized Packings in Space Engineering Applications: Part I," Springer Optimization and Its Applications, in: Giorgio Fasano & János D. Pintér (ed.), Modeling and Optimization in Space Engineering, pages 395-437, Springer.
    9. A. Sutou & Y. Dai, 2002. "Global Optimization Approach to Unequal Global Optimization Approach to Unequal Sphere Packing Problems in 3D," Journal of Optimization Theory and Applications, Springer, vol. 114(3), pages 671-694, September.
    10. Luiz J.P. Araújo & Ender Özcan & Jason A.D. Atkin & Martin Baumers, 2019. "Analysis of irregular three-dimensional packing problems in additive manufacturing: a new taxonomy and dataset," International Journal of Production Research, Taylor & Francis Journals, vol. 57(18), pages 5920-5934, September.
    11. Ashish Raj & Yu-hsien Chen, 2011. "The Wiring Economy Principle: Connectivity Determines Anatomy in the Human Brain," PLOS ONE, Public Library of Science, vol. 6(9), pages 1-11, September.
    12. Jie Wang, 1999. "Packing of Unequal Spheres and Automated Radiosurgical Treatment Planning," Journal of Combinatorial Optimization, Springer, vol. 3(4), pages 453-463, December.
    13. Yuriy Stoyan & Alexandr Pankratov & Tatiana Romanova, 2017. "Placement Problems for Irregular Objects: Mathematical Modeling, Optimization and Applications," Springer Optimization and Its Applications, in: Sergiy Butenko & Panos M. Pardalos & Volodymyr Shylo (ed.), Optimization Methods and Applications, pages 521-559, Springer.
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