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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
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