Cybernetics and Systems Analysis

, Volume 54, Issue 2, pp 221–231 | Cite as

Combinatorial Configurations in Balance Layout Optimization Problems

  • I. V. Grebennik
  • A. A. Kovalenko
  • T. E. Romanova
  • I. A. Urniaieva
  • S. B. Shekhovtsov


The balance layout optimization problem for a given set of 3D objects in a container divided by horizontal racks into subcontainers is considered. For analytical description of non-overlapping and containment constraints, the phi-function technique is used. Combinatorial configurations describing the combinatorial structure of the problem are defined. Based on the introduced configurations, a mathematical model is constructed that takes into account not only the placement constraints and mechanical properties of the system but also the combinatorial features of the problem associated with generation of partitions of the set of objects placed inside the subcontainers. A solution strategy is proposed. The results of numerical experiments are provided.


balance layout combinatorial configurations 3D objects phi-function method mathematical model optimization 


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  1. 1.
    B. Chazelle, H. Edelsbrunner, and L. J. Guibas, “The complexity of cutting complexes,” Discrete & Computational Geometry, Vol. 4, No. 2, 139–181 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    N. Chernov, Y. Stoyan, T. Romanova, and A. Pankratov, “Phi-functions for 2D objects formed by line segments and circular arcs,” Advances in Operations Research, DOI: (2012).
  3. 3.
    N. Chernov, Y. Stoyan, and T. Romanova, “Mathematical model and efficient algorithms for object packing problem,” Computational Geometry: Theory and Applications, Vol. 43, No. 5, 535–553 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    G. Fasano and J. D. Pintér (eds.), Modeling and Optimization in Space Engineering, Springer, Optimization and Its Applications, Vol. 73 (2013).Google Scholar
  5. 5.
    C. Che, Y. Wang, and H. Teng, “Test problems for quasi-satellite packing: Cylinders packing with behavior constraints and all the optimal solutions known,” URL:
  6. 6.
    G. Fasano and J. D. Pintér (eds.), Space Engineering. Modeling and Optimization with Case Studies, Springer Optimization and its Applications, Vol. 114 (2016).Google Scholar
  7. 7.
    Z. Sun and H. Teng, “Optimal layout design of a satellite module,” Engineering Optimization, Vol. 35, No. 5, 513–530 (2003).CrossRefGoogle Scholar
  8. 8.
    K. Lei, “Constrained layout optimization based on adaptive particle swarm optimizer,” in: Zhihua Cai, Zhenhua Li, Zhuo Kang, and Yong Liu (eds.), Proc. ISICA 2009, Advances in Computation and Intelligence, LNCS, Vol. 5821, 434–442 (2009).Google Scholar
  9. 9.
    Yu. Stoyan and T. Romanova, “Mathematical models of placement optimization: Two- and three-dimensional problems and applications,” in: G. Fasano and J. D. Pintér (eds.), Modeling and Optimization in Space Engineering, Springer Optimization and its Applications, Vol. 73, Chap. 15, 363–388 (2013).Google Scholar
  10. 10.
    I. V. Sergienko and V. P. Shilo, Discrete Optimization Problems: Challenges, Methods, Solutions, and Analysis [in Russian], Naukova Dumka, Kyiv (2003).Google Scholar
  11. 11.
    N. V. Semenova and L. M. Kolechkina, Vector Discrete Optimization Problems on Combinatorial Sets: Methods of the Analysis and Solution [in Ukrainian], Naukova Dumka, (2009).Google Scholar
  12. 12.
    A. A. Kovalenko, T. E. Romanova, and P. I. Stetsyuk, “Balance layout problem for 3D-objects: Mathematical model and solution methods,” Cybern. Syst. Analysis, Vol. 51, No. 4, 556–565 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yu. Stoyan, T. Romanova, A. Pankratov, A. Kovalenko, and P. Stetsyuk, “Modeling and optimization of balance layout problems,” in: G. Fasano and J. D. Pintér (eds.), Space Engineering. Modeling and Optimization with Case Studies, Springer Optimization and its Applications Vol. 114, 177–208 (2016).Google Scholar
  14. 14.
    E. Reingold, J. Nievergelt, and N. Deo, Combinatorial Algorithms: Theory and Practice, Pearson Education (1977).Google Scholar
  15. 15.
    Yu. G. Stoyan and I. V. Grebennik, “Description and generation of combinatorial sets having special characteristics,” Intern. J. of Biomedical Soft Computing and Human Sciences, Special Volume “Bilevel Programming, Optimization Methods, and Applications to Economics,” Vol. 18, No. 1, 83–88 (2013).Google Scholar
  16. 16.
    Yu. Stoyan, A. Pankratov, and T. Romanova, “Quasi-phi-functions and optimal packing of ellipses,” J. of Global Optimization, Vol. 65, No. 2, 283–307 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    I. Grebennik and O. Lytvynenko, “Random generation of combinatorial sets with special properties,” An Intern. Quarterly J. on Economics of Technology and Modelling Processes (ECONTECHMOD), Vol. 5, No. 4, 43–48 (2016).Google Scholar
  18. 18.
    A. Wachter and L. T. Biegler, “On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,” Mathematical Programming, Vol. 106, No. 1, 25–57 (2006).MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • I. V. Grebennik
    • 1
  • A. A. Kovalenko
    • 1
  • T. E. Romanova
    • 2
  • I. A. Urniaieva
    • 1
  • S. B. Shekhovtsov
    • 3
  1. 1.Kharkiv National University of Radio ElectronicsKharkivUkraine
  2. 2.A. Podgorny Institute for Mechanical Engineering ProblemsKharkivUkraine
  3. 3.Kharkiv National University of Home AffairsKharkivUkraine

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