Packing Multidimensional Spheres in an Optimized Hyperbolic Container

Yuriy Stoyan, Georgiy Yaskov, Tetyana Romanova, Igor Litvinchev (), Yurii E. Stoian, José Manuel Velarde Cantú () and Mauricio López Acosta
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Yuriy Stoyan: Anatolii Pidhornyi Institute of Power Machines and Systems of the National Academy of Sciences of Ukraine, 61046 Kharkiv, Ukraine
Georgiy Yaskov: Anatolii Pidhornyi Institute of Power Machines and Systems of the National Academy of Sciences of Ukraine, 61046 Kharkiv, Ukraine
Tetyana Romanova: Anatolii Pidhornyi Institute of Power Machines and Systems of the National Academy of Sciences of Ukraine, 61046 Kharkiv, Ukraine
Igor Litvinchev: Faculty of Mechanical and Electrical Engineering, Autonomous University of Nuevo Leon, Monterrey 66455, Mexico
Yurii E. Stoian: Anatolii Pidhornyi Institute of Power Machines and Systems of the National Academy of Sciences of Ukraine, 61046 Kharkiv, Ukraine
José Manuel Velarde Cantú: Department of Industrial Engineering, Technological Institute of Sonora (ITSON), Navojoa 85800, Mexico
Mauricio López Acosta: Department of Industrial Engineering, Technological Institute of Sonora (ITSON), Navojoa 85800, Mexico

Mathematics, 2025, vol. 13, issue 23, 1-23

Abstract: The problem of packing multidimensional spheres in a container defined by a hyperbolic surface is introduced. The objective is to minimize the height of the hyperbolic container under non-overlapping and containment conditions for the spheres, considering minimal allowable distances between them. To the best of our knowledge, no mathematical models addressing optimized packing spheres in hyperbolic containers have been proposed before. Our approach is based on a space dimensionality reduction transformation. This transformation relies on projecting a multidimensional hyperboloid into a lower-dimensional space sequentially up to two-dimensional case. Employing the phi-function technique, packing spheres in the hyperbolic container is formulated as a nonlinear programming problem. The latter is solved using a model-based heuristic combined with a decomposition approach. Numerical results are presented for a wide range of parameters, i.e., space dimension, number of spheres, and metric characteristics of the hyperbolic container. The results demonstrate efficiency of the proposed modeling and solution approach highlighting new opportunities for packing problems within non-traditional geometries.

Keywords: optimized packing; multidimensional spheres; hyperbolic container; phi-function technique; nonlinear optimization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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