Quasi-packing different spheres with ratio conditions in a spherical container

Research output: Contribution to journalResearch articleContributedpeer-review


  • Andreas Fischer - , Chair of Numerical Optimization (Author)
  • Igor Litvinchev - , Autonomous University of Nuevo León (Author)
  • Tetyana Romanova - , A. Pidhornyi Institute of Mechanical Engineering Problems (Author)
  • Petro Stetsyuk - , National Academy of Sciences of Ukraine, V. M. Glushkov Institute of Cybernetics, Kyiv (Author)
  • Georgiy Yaskov - , National Academy of Sciences of Ukraine, A. Pidhornyi Institute of Mechanical Engineering Problems, Kharkiv (Author)


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.


Original languageEnglish
Article number2033
Number of pages19
Issue number9
Publication statusPublished - May 2023

External IDs

Scopus 85159210770
WOS 000986944000001
Mendeley d8a41461-8caa-3975-8dec-51158209b7d6


Research priority areas of TU Dresden

Subject groups, research areas, subject areas according to Destatis


  • Optimization, Partial overlapping, Quasi-containment, Ratio condition, Sphere packing, Spherical container, spherical container, optimization, partial overlapping, quasi-containment, ratio condition, sphere packing

Library keywords