Journal of Global Optimization

, Volume 65, Issue 2, pp 283–307 | Cite as

Quasi-phi-functions and optimal packing of ellipses

  • Y. Stoyan
  • A. Pankratov
  • T. RomanovaEmail author


We further develop our phi-function technique for solving Cutting and Packing problems. Here we introduce quasi-phi-functions for an analytical description of non-overlapping and containment constraints for 2D- and 3D-objects which can be continuously rotated and translated. These new functions can work well for various types of objects, such as ellipses, for which ordinary phi-functions are too complicated or have not been constructed yet. We also define normalized quasi-phi-functions and pseudonormalized quasi-phi-functions for modeling distance constraints. To show the advantages of our new quasi-phi-functions we apply them to the problem of placing a given collection of ellipses into a rectangular container of minimal area. We use radical free quasi-phi-functions to reduce it to a nonlinear programming problem and develop an efficient solution algorithm. We present computational results that compare favourably with those published elsewhere recently.


Quasi-phi-functions Object continuous rotations Non-overlapping Distance constraints Ellipse packing Mathematical model Nonlinear optimization 



T. Romanova, Yu. Stoyan and A. Pankratov acknowledge the support of the Science and Technology Center in Ukraine and the National Academy of Sciences of Ukraine, Grant 5710.


  1. 1.
    Wascher, G., Hauner, H., Schumann, H.: An improved typology of cutting and packing problems. Eur. J. Oper. Res. 183(3,16), 1109–1130 (2007)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bennell, J.A., Oliveira, J.F.: The geometry of nesting problems: a tutorial. Eur. J. Oper. Res. 184, 397–415 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Sugihara, K., Sawai, M., Sano, H., Kim, D.-S., Kim, D.: Disk packing for the estimation of the size of a wire bundle. Jpn J. Ind. Appl. Math. 21(3), 259–278 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Burke, E.K., Hellier, R., Kendall, G., Whitwell, G.: Irregular packing using the line and arc no-fit polygon. Oper. Res. 58(4), 948–970 (2010)CrossRefzbMATHGoogle Scholar
  5. 5.
    Milenkovic, V.J., Sacks, E.: Two approximate Minkowski sum algorithms. Int. J. Comput. Geom. Appl. 20(4), 485–509 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lodi, A., Martello, S., Vigo, D.: Two-dimensional packing problems: a survey. Eur. J. Oper. Res. 141, 241–252 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bennell, J.A., Scheithauer, G., Stoyan, Yu., Romanova, T.: Tools of mathematical modelling of arbitrary object packing problems. J. Ann. Oper. Res. Publ. Springer Neth. 179(1), 343–368 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chernov, N., Stoyan, Y., Romanova, T.: Mathematical model and efficient algorithms for object packing problem. Comput. Geom. Theory Appl. 43(5), 535–553 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    . Chernov, N., Stoyan, Y., Romanova, T., Pankratov, A.: Phi-functions for 2D-objects formed by line segments and circular arcs. Adv. Oper. Res. 2012, 26. Article ID 346358 (2012). doi: 10.1155/2012/346358
  10. 10.
    Bennell, J., Scheithauer, G., Stoyan, Y., Romanova, T., Pankratov, A.: Optimal clustering of a pair of irregular objects. J. Glob. Optim. (2014). doi: 10.1007/s10898-014-0192-0
  11. 11.
    StoyanYu, Chugay, A.: Mathematical modeling of the interaction of non-oriented convex polytopes. Cybern. Syst. Anal. 48(6), 837–845 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Stoyan, Y., Chugay, A.: Construction of radical free phi-functions for spheres and non-oriented polytopes. Rep. NAS Ukraine 12, 35–40 (2011). (In Russian)Google Scholar
  13. 13.
    Kallrath, J.: Cutting circles and polygons from area-minimizing rectangles. J. Glob. Optim. 43, 299–328 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kallrath, J., Rebennack, S. (2013) Cutting ellipses from area-minimizing rectangles. J. Glob. Optim. (2013). doi: 10.1007/s10898-013-0125-3
  15. 15.
    Chazelle, B., Edelsbrunner, H., Guibas, L.J.: The complexity of cutting complexes. Discrete Comput. Geom. 4(2), 139–181 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wachter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Stoyan, Y., Yaskov, G.: Packing congruent hyperspheres into a hypersphere. J. Glob. Optim. 52(4), 855–868 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematical Modeling and Optimal DesignInstitute for Mechanical Engineering Problems of the National Academy of Sciences of UkraineKharkovUkraine

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