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Mathematics
Volume 13
Issue 23
10.3390/math13233747
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Packing Multidimensional Spheres in an Optimized Hyperbolic Container

by
Yuriy Stoyan
1,
Georgiy Yaskov
1,2,
Tetyana Romanova
1,2,3,4,
Igor Litvinchev
5,*,
Yurii E. Stoian
1,
José Manuel Velarde Cantú
6,* and
Mauricio López Acosta
6
1
Anatolii Pidhornyi Institute of Power Machines and Systems of the National Academy of Sciences of Ukraine, 61046 Kharkiv, Ukraine
2
Department of Applied Mathematics, Faculty of Computer Science, Kharkiv National University of Radio Electronics, 61166 Kharkiv, Ukraine
3
Leeds University Business School, University of Leeds, Leeds LS2 9JT, UK
4
School of Energy, Information and Transport Infrastructure, O. M. Beketov National University of Urban Economy in Kharkiv, 61002 Kharkiv, Ukraine
5
Faculty of Mechanical and Electrical Engineering, Autonomous University of Nuevo Leon, Monterrey 66455, Mexico
6
Department of Industrial Engineering, Technological Institute of Sonora (ITSON), Navojoa 85800, Mexico
*
Authors to whom correspondence should be addressed.
Mathematics 2025, 13(23), 3747; https://doi.org/10.3390/math13233747
Submission received: 20 October 2025 / Revised: 13 November 2025 / Accepted: 17 November 2025 / Published: 21 November 2025
(This article belongs to the Special Issue Advanced Applications of Mathematical Modeling and Optimization in Logistics and Manufacturing)
Versions Notes

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 optimized packing; multidimensional spheres; hyperbolic container; phi-function technique; nonlinear optimization

Share and Cite

MDPI and ACS Style

Stoyan, Y.; Yaskov, G.; Romanova, T.; Litvinchev, I.; Stoian, Y.E.; Velarde Cantú, J.M.; Acosta, M.L. Packing Multidimensional Spheres in an Optimized Hyperbolic Container. Mathematics 2025, 13, 3747. https://doi.org/10.3390/math13233747

AMA Style

Stoyan Y, Yaskov G, Romanova T, Litvinchev I, Stoian YE, Velarde Cantú JM, Acosta ML. Packing Multidimensional Spheres in an Optimized Hyperbolic Container. Mathematics. 2025; 13(23):3747. https://doi.org/10.3390/math13233747

Chicago/Turabian Style

Stoyan, Yuriy, Georgiy Yaskov, Tetyana Romanova, Igor Litvinchev, Yurii E. Stoian, José Manuel Velarde Cantú, and Mauricio López Acosta. 2025. "Packing Multidimensional Spheres in an Optimized Hyperbolic Container" Mathematics 13, no. 23: 3747. https://doi.org/10.3390/math13233747

APA Style

Stoyan, Y., Yaskov, G., Romanova, T., Litvinchev, I., Stoian, Y. E., Velarde Cantú, J. M., & Acosta, M. L. (2025). Packing Multidimensional Spheres in an Optimized Hyperbolic Container. Mathematics, 13(23), 3747. https://doi.org/10.3390/math13233747

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