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Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 6601-6630. doi: 10.3934/mbe.2020344
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Optimized packing multidimensional hyperspheres: a unified approach

  • 1.
    Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine, 2/10 Pozharskogo st., Kharkiv 61046, Ukraine
  • 2.
    Kharkiv National University of Radioelectronics, 14 Nauky ave., Kharkiv 61166, Ukraine
  • 3.
    Kharkiv Aviation Institute, National Aerospace University, 17 Chkalov st., Kharkiv 61070, Ukraine
  • 4.
    Computing Center, Russian Academy of Sciences, Vavilov 40, Moscow, Russia
  • 5.
    Nuevo Leon State University, Monterrey, Nuevo Leon, CP 66455, Mexico
  • 6.
    Technological Institute of Sonora (ITSON), Obregón-City, Sonora, Mexico
  • Received: 20 June 2020 Accepted: 07 September 2020 Published: 28 September 2020

  • In this paper an optimized multidimensional hyperspheres packing problem (HPP) is considered for a bounded container. Additional constraints, such as prohibited zones in the container or minimal allowable distances between spheres can also be taken into account. Containers bounded by hyper- (spheres, cylinders, planes) are considered. Placement constraints (non-intersection, containment and distant conditions) are formulated using the phi-function technique. A mathematical model of HPP is constructed and analyzed. In terms of the general typology for cutting & packing problems, two classes of HPP are considered: open dimension problem (ODP) and knapsack problem (KP). Various solution strategies for HPP are considered depending on: a) objective function type, b) problem dimension, c) metric characteristics of hyperspheres (congruence, radii distribution and values), d) container's shape; e) prohibited zones in the container and/or minimal allowable distances. A solution approach is proposed based on multistart strategies, nonlinear programming techniques, greedy and branch-and-bound algorithms, statistical optimization and homothetic transformations, as well as decomposition techniques. A general methodology to solve HPP is suggested. Computational results for benchmark and new instances are presented.

    Citation: Yuriy Stoyan, Georgiy Yaskov, Tatiana Romanova, Igor Litvinchev, Sergey Yakovlev, José Manuel Velarde Cantú. Optimized packing multidimensional hyperspheres: a unified approach[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6601-6630. doi: 10.3934/mbe.2020344

    Related Papers:

  • In this paper an optimized multidimensional hyperspheres packing problem (HPP) is considered for a bounded container. Additional constraints, such as prohibited zones in the container or minimal allowable distances between spheres can also be taken into account. Containers bounded by hyper- (spheres, cylinders, planes) are considered. Placement constraints (non-intersection, containment and distant conditions) are formulated using the phi-function technique. A mathematical model of HPP is constructed and analyzed. In terms of the general typology for cutting & packing problems, two classes of HPP are considered: open dimension problem (ODP) and knapsack problem (KP). Various solution strategies for HPP are considered depending on: a) objective function type, b) problem dimension, c) metric characteristics of hyperspheres (congruence, radii distribution and values), d) container's shape; e) prohibited zones in the container and/or minimal allowable distances. A solution approach is proposed based on multistart strategies, nonlinear programming techniques, greedy and branch-and-bound algorithms, statistical optimization and homothetic transformations, as well as decomposition techniques. A general methodology to solve HPP is suggested. Computational results for benchmark and new instances are presented.


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