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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

Special Issues: Computational systems for sustainable development in Computing and Engineering

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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