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Advances in Operations Research
Volume 2012 (2012), Article ID 346358, 26 pages
doi:10.1155/2012/346358
Research Article

Phi-Functions for 2D Objects Formed by Line Segments and Circular Arcs

N. Chernov,1 Yu. Stoyan,2 T. Romanova,2 and A. Pankratov2

1Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA
2Department of Mathematical Modeling, Institute for Mechanical Engineering Problems of The National Academy of Sciences of Ukraine, Kharkov, Ukraine

Received 21 October 2011; Revised 30 January 2012; Accepted 31 January 2012

Academic Editor: Ching-Jong Liao

Copyright © 2012 N. Chernov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. G. Birgin, J. M. Martínez, F. H. Nishihara, and D. P. Ronconi, “Orthogonal packing of rectangular items within arbitrary convex regions by nonlinear optimization,” Computers & Operations Research, vol. 33, no. 12, pp. 3535–3548, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. A. M. Gomes and J. F. Oliveira, “Solving irregular strip packing problems by hybridising simulated annealing and linear programming,” European Journal of Operational Research, vol. 171, pp. 811–829, 2006.
  3. V. J. Milenkovic and K. Daniels, “Translational polygon containment and minimal enclosure using mathematical programming,” International Transactions in Operational Research, vol. 6, no. 5, pp. 525–554, 1999. View at Publisher · View at Google Scholar
  4. J. A. Bennell and J. F. Oliveira, “The geometry of nesting problems: a tutorial,” European Journal of Operational Research, vol. 184, no. 2, pp. 397–415, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. G. Wäscher, H. Haußner, and H. Schumann, “An improved typology of cutting and packing problems,” European Journal of Operational Research, vol. 183, pp. 1109–1130, 2007.
  6. E. Burke, R. Hellier, G. Kendall, and G. Whitwell, “A new bottom-left-fill heuristic algorithm for the two-dimensional irregular packing problem,” Operations Research, vol. 54, no. 3, pp. 587–601, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. V. Milenkovic and E. Sacks, “Two approximate Minkowski sum algorithms,” International Journal of Computational Geometry & Applications, vol. 20, no. 4, pp. 485–509, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. V. J. Milenkovic, “Rotational polygon overlap minimization and compaction,” Computational Geometry, vol. 10, no. 4, pp. 305–318, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. V. J. Milenkovic, “Rotational polygon containment and minimum enclosure using only robust 2D constructions,” Computational Geometry, vol. 13, no. 1, pp. 3–19, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. E. Burke, R. Hellier, G. Kendall, and G. Whitwell, “Irregular packing using the line and arc no-fit polygon,” Operations Research, vol. 58, pp. 948–970, 2010.
  11. J. Bennell, G. Scheithauer, Y. Stoyan, and T. Romanova, “Tools of mathematical modeling of arbitrary object packing problems,” Annals of Operations Research, vol. 179, pp. 343–368, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Y. Stoyan, G. Scheithauer, N. Gil, and T. Romanova, “Φ-functions for complex 2D-objects,” 4OR. Quarterly Journal of the Belgian, French and Italian Operations Research Societies, vol. 2, no. 1, pp. 69–84, 2004. View at Zentralblatt MATH
  13. Yu. Stoyan, J. Terno, G. Scheithauer, N. Gil, and T. Romanova, “Phi-functions for primary 2D-objects,” Studia Informatica Universalis, vol. 2, pp. 1–32.
  14. N. Chernov, Yu. Stoyan, and T. Romanova, “Mathematical model and efficient algorithms for object packing problem,” Computational Geometry, vol. 43, no. 5, pp. 535–553, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Y. G. Stoyan and A. Chugay, “Packing cylinders and rectangular parallelepipeds with distances between them,” European Journal of Operational Research, vol. 197, pp. 446–455, 2008.
  16. T. Romanova, G. Scheithauer, and A. Krivulya, “Covering a polygonal region by rectangles,” Computational Optimization and Applications, vol. 48, no. 3, pp. 675–695, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. http://www.math.uab.edu/~chernov/CP/.
  18. A. Wächter 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, pp. 25–57, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. H. Minkovski, “Dichteste gitterförmige Lagerung kongruenter Körper,” Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, pp. 311–355, 1904.