Cybernetics and Systems Analysis

, Volume 55, Issue 6, pp 943–948 | Cite as

Fragmentary Structures in a Two-Dimensional Strip Packing Problem

  • I. V. KozinEmail author
  • S. E. Batovskyi


The general problem of two-dimensional packing in a semi-bounded strip is considered. It is shown that the problem can be considered as an optimization problem on a fragmentary structure and is reduced to the problem of combinatorial optimization on a set of permutations. A universal approach to representing two-dimensional figures and an algorithm for packing them in a strip are considered. An approach to modifying the original problem to attain an optimal solution is proposed.


discrete optimization fragmentary structure two-dimensional strip packing evolutionary algorithm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness [Russian translation], Mir, Moscow (1982).Google Scholar
  2. 2.
    F. Glover, E. Taillard, and D. de Werra, “A user’s guide to tabu search,” Annals of Operation Research, Vol. 41, Iss. 1, 1–28 (1993).CrossRefGoogle Scholar
  3. 3.
    J. H. Holland, Adaptation in Natural and Artificial Systems, MIT Press, Boston, MA (1992).CrossRefGoogle Scholar
  4. 4.
    V. M. Kurejchik, “Genetic Algorithms. Condition. Problems. Prospects,” News of RAS, Theory and Control Systems, No. 1, 144–160 (1999).Google Scholar
  5. 5.
    J. F. Goncalves, “A hybrid genetic algorithm-heuristic for a two-dimensional orthogonal packing problem,” European Journal of Operational Research, Vol. 183, No. 3, 1212–1229 (2007).MathSciNetCrossRefGoogle Scholar
  6. 6.
    S. D. Shtovba, “Ant algorithms: Theory and application,” Programmirovanie, No. 4, 1–16 (2005).CrossRefGoogle Scholar
  7. 7.
    M. Dorigo, Optimization, Learning, and Natural Algorithms, PhD Thesis, Dipartimento di Elettronica, Politechnico di Milano, Italy (1992).Google Scholar
  8. 8.
    A. Lodi, S. Martello, and M. Monaci, “Two-dimensional packing problems: A survey,” European Journal of Operation Research, Vol. 141, No. 2, 241–252 (2002).MathSciNetCrossRefGoogle Scholar
  9. 9.
    S. Jain and H. C. Gea, “Two dimensional packing problems using genetic algorithms,” Engineering with Computers, Vol. 14, No. 3, 206–213 (1998).CrossRefGoogle Scholar
  10. 10.
    V. A. Vanidovsky and O. B. Lebedev, “Two-dimensional strip packing based on simulation of adaptive behavior of an ant colony,” News of SFU, Technical Sciences, No. 7 (156), 34–42 (2014).Google Scholar
  11. 11.
    V. M. Kartak, “The problem of packing rectangles: An exact algorithm based on the matrix representation,” Bulletin of USATU, Vol. 9, No. 4 (22), 104–110 (2007).Google Scholar
  12. 12.
    A. S. Rudnev, “Probabilistic tabu search algorithm for the packing circles and rectangles into the strip,” Discrete Analysis and Operations Research, Vol. 16, No. 4, 61–86 (2009).Google Scholar
  13. 13.
    Yu. G. Stoyan and M.V. Zlotnik, “The placement of circles and nonconvex polygons with rotations in a rectangle of minimal length,” Reports of the National Academy of Sciences of Ukraine, No. 2, 37–42 (2007).Google Scholar
  14. 14.
    I. V. Kozin, N. K. Maksyshko, and V. A. Perepelitsa, “Fragmentary Structures in Discrete Optimization Problems,” Cybernetics and Systems Analysis, Vol. 53, No. 6, 931–936 (2017).MathSciNetCrossRefGoogle Scholar
  15. 15.
    I. V. Kozin, “Evolutionary-fragmentary model of the pentomino packing problem,” Discrete Analysis and Operations Research, Vol. 21, No. 6, 35–50 (2014).Google Scholar
  16. 16.
    T. E. Romanova, E. A. Stupak, and M. V. Zlotnik, “A mathematical model and a method to solve the packing optimization problem for arbitrary 2D objects in rectangular domains,” Reports of the National Academy of Sciences of Ukraine, No. 1, 48–53 (2009).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Zaporizhzhya National UniversityZaporozhyeUkraine

Personalised recommendations