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Optimal Packing of General Ellipses in a Circle

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Modeling and Optimization: Theory and Applications (MOPTA 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 213))

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Abstract

Our objective is to find the optimal non-overlapping packing of a collection of general (non-identical) ellipses, with respect to a container circle that has minimal radius. Following the review of selected topical literature, we introduce a model development approach based on using embedded Lagrange multipliers. Our optimization model has been implemented using the computing system Mathematica. We present illustrative numerical results using the LGO nonlinear (global and local) optimization software package linked to Mathematica. Our study shows that the Lagrangian modeling approach combined with nonlinear optimization tools can effectively handle challenging ellipse packing problems with hundreds of decision variables and non-convex constraints.

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References

  1. Szabó, P.G., Csendes, T., Casado, L.G., García, I.: Equal circles packing in a square I – problem setting and bounds for optimal solutions. In: Giannessi, F., Pardalos, P.M., Rapcsák, T. (eds.) Optimization Theory: Recent Developments from Mátraháza. Kluwer, Dordrecht (2001)

    Google Scholar 

  2. Szabó, P.G., Markót, M.C., Csendes, T.: Global optimization in geometry – circle packing into the square. In: Audet, P., Hansen, P., Savard, G. (eds.) Essays and Surveys in Global Optimization. Kluwer, Dordrecht (2005)

    Google Scholar 

  3. Szabó, P.G., Markót, M.C., Csendes, T., Specht, E., Casado, L.G., García, I.: New Approaches to Circle Packing in a Square with Program Codes. Springer, New York (2007)

    MATH  Google Scholar 

  4. Markót, M.C.: Optimal packing of 28 equal circles in a unit square – the first reliable solution. Numerical Algorithms. 37, 253–261 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Riskin, M.D., Bessette, K.C., Castillo, I.: A logarithmic barrier approach to solving the dashboard planning problem. INFOR 41, 245–257 (2003)

    Google Scholar 

  6. Castillo, I., Sim, T.: A spring-embedding approach for the facility layout problem. J. Oper. Res. Soc. 55, 73–81 (2004)

    Article  MATH  Google Scholar 

  7. Pintér, J.D., Kampas, F.J.: Nonlinear optimization in Mathematica with MathOptimizer Professional. Math. Educ. Res. 10, 1–18 (2005)

    Google Scholar 

  8. Pintér, J.D., Kampas, F.J.: Mathoptimizer professional: key features and illustrative applications. In: Liberti, L., Maculan, N. (eds.) Global Optimization: From Theory to Implementation, pp. 263–280. Springer, New York (2006)

    Google Scholar 

  9. Kampas, F.J., Pintér, J.D.: Configuration analysis and design by using optimization tools in Mathematica. The Math J. 10, 128–154 (2006)

    Google Scholar 

  10. Addis, B., Locatelli, M., Schoen, F.: Efficiently packing unequal disks in a circle. Oper. Res. Lett. 36, 37–42 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Castillo, I., Kampas, F.J., Pintér, J.D.: Solving circle packing problems by global optimization: numerical results and industrial applications. Eur. J. Oper. Res. 191, 786–802 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grosso, A., Jamali, A.R.M.J.U., Locatelli, M., Schoen, F.: Solving the problem of packing equal and unequal circles in a circular container. J. Glob. Optim. 47, 63–81 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hifi, M., M’Hallah, R.: A literature review on circle and sphere packing problems: models and methodologies. Adv. Oper. Res. 2009, 22, 150624 (2009). doi:https://doi.org/10.1155/2009/150624

  14. Fasano, G.: Solving Non-standard Packing Problems by Global Optimization and Heuristics. Springer, Cham (2014)

    Book  MATH  Google Scholar 

  15. Fasano, G., Pintér, J.D. (eds.): Optimized Packings with Applications. Springer, New York (2015)

    MATH  Google Scholar 

  16. Gensane, T., Honvault, P.: Optimal packings of two ellipses in a square. Forum Geom. 14, 371–380 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Birgin, E.G., Bustamante, L.H., Flores Callisaya, H., Martínez, J.M.: Packing circles within ellipses. Int. Trans. Oper. Res. 20, 365–389 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Litvinchev, I., Infante, L., Ozuna, L.: Packing circular-like objects in a rectangular container. J. Comput. Syst. Sci. Int. 54, 259–267 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Galiev, S.I., Lisafina, M.S.: Numerical optimization methods for packing equal orthogonally oriented ellipses in a rectangular domain. Comput. Math. Math. Phys. 53, 1748–1762 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kallrath, J., Rebennack, S.: Cutting ellipses from area-minimizing rectangles. J. Glob. Optim. 59, 405–437 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Uhler, C., Wright, S.J.: Packing ellipsoids with overlap. SIAM Rev. 55, 671–706 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kampas, F.J., Castillo, I. Pintér, J.D.: Optimized ellipse packings in regular polygons using embedded Lagrange multipliers (2017). Submitted for publication

    Google Scholar 

  23. Pintér, J.D.: Globally optimized spherical point arrangements: model variants and illustrative results. Ann. Oper. Res. 104, 213–230 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Stortelder, W.J.H., de Swart, J.J.B., Pintér, J.D.: Finding elliptic Fekete point sets: two numerical solution approaches. J. Comput. Appl. Math. 130, 205–216 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pintér, J.D., Kampas, F.J.: Benchmarking nonlinear optimization software in technical computing environments. I. Global optimization in Mathematica with MathOptimizer Professional. TOP. 21, 133–162 (2013)

    Article  MATH  Google Scholar 

  26. Pintér, J.D.: Global Optimization in Action. Kluwer Academic Publishers, Dordrecht (1996). Now distributed by Springer Science + Business Media, New York

    Book  MATH  Google Scholar 

  27. Pintér, J.D.: LGO − a program system for continuous and Lipschitz global optimization. In: Bomze, I., Csendes, T., Horst, R., Pardalos, P.M. (eds.) Developments in Global Optimization, pp. 183–197. Kluwer Academic Publishers, Dordrecht (1997)

    Chapter  Google Scholar 

  28. Pintér, J.D.: Global optimization: software, test problems, and applications. In: Pardalos, P.M., Romeijn, H.E. (eds.) Handbook of Global Optimization, vol. 2, pp. 515–569. Kluwer Academic Publishers, Dordrecht (2002)

    Chapter  Google Scholar 

  29. Pintér, J.D.: Nonlinear optimization in modeling environments: software implementations for compilers, spreadsheets, modeling languages, and integrated computing systems. In: Jeyakumar, V., Rubinov, A.M. (eds.) Continuous Optimization: Current Trends and Applications, pp. 147–173. Springer, New York (2005)

    Chapter  Google Scholar 

  30. Pintér, J.D.: Nonlinear optimization with GAMS/LGO. J. Glob. Optim. 38, 79–101 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Pintér, J.D.: Software development for global optimization. In: Pardalos, P.M., Coleman, T.F. (eds.) Global Optimization: Methods and Applications, Fields Institute Communications, vol. 55, pp. 183–204. American Mathematical Society, Providence, RI (2009)

    Google Scholar 

  32. Pintér, J.D., Linder, D., Chin, P.: Global optimization toolbox for maple: an introduction with illustrative applications. Optim. Methods Softw. 21, 565–582 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Çaĝlayan, M.O., Pintér, J.D.: Development and calibration of a currency trading strategy using global optimization. J. Glob. Optim. 56, 353–371 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pintér, J.D., Horváth, Z.: Integrated experimental design and nonlinear optimization to handle computationally expensive models under resource constraints. J. Glob. Optim. 57, 191–215 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pintér, J.D.: How difficult is nonlinear optimization? A practical solver tuning approach, with illustrative results. Ann. Oper. Res. 1–23 (2017). https://doi.org/10.1007/s10479-017-2518-z. Preprint available at www.optimization-online.org/DB_FILE/2014/06/4409.pdf

    Google Scholar 

  36. Pintér, J.D.: LGO – a Model Development and Solver System for Global-Local Nonlinear Optimization, User’s Guide, Current edn. Pintér Consulting Services, Inc., Halifax (2016)

    Google Scholar 

  37. Wolfram Research: Mathematica (Release 11). Wolfram Research, Inc., Champaign, IL (2016)

    Google Scholar 

  38. Pintér, J.D., Kampas, F.J.: Getting Started with Mathoptimizer professional. Pintér Consulting Services, Inc., Halifax (2015)

    Google Scholar 

  39. GCC: GCC, the GNU Compiler Collection. (2016). https://gcc.gnu.org/

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Author information

Authors and Affiliations

  1. Physicist at Large Consulting LLC, Bryn Mawr, PA, USA

    Frank J. Kampas

  2. Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA

    János D. Pintér

  3. Pintér Consulting Services Inc., Halifax, NS, Canada

    János D. Pintér

  4. Lazaridis School of Business and Economics, Wilfrid Laurier University, Waterloo, ON, Canada

    Ignacio Castillo

Authors
  1. Frank J. Kampas
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  2. János D. Pintér
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  3. Ignacio Castillo
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Corresponding author

Correspondence to Ignacio Castillo .

Editor information

Editors and Affiliations

  1. Industrial and Systems Engineering Department, Lehigh University, Bethlehem, Pennsylvania, USA

    Martin Takáč

  2. Industrial and Systems Engineering Department, Lehigh University, Bethlehem, Pennsylvania, USA

    Tamás Terlaky

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Kampas, F.J., Pintér, J.D., Castillo, I. (2017). Optimal Packing of General Ellipses in a Circle. In: Takáč, M., Terlaky, T. (eds) Modeling and Optimization: Theory and Applications. MOPTA 2016. Springer Proceedings in Mathematics & Statistics, vol 213. Springer, Cham. https://doi.org/10.1007/978-3-319-66616-7_2

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