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Packing circles into perimeter-minimizing convex hulls

  • Josef Kallrath
  • Markus M. Frey
Article
  • 24 Downloads

Abstract

We present and solve a new computational geometry optimization problem in which a set of circles with given radii is to be arranged in unspecified area such that the length of the boundary, i.e., the perimeter, of the convex hull enclosing the non-overlapping circles is minimized. The convex hull boundary is established by line segments and circular arcs. To tackle the problem, we derive a non-convex mixed-integer non-linear programming formulation for this circle arrangement or packing problem. Moreover, we present some theoretical insights presenting a relaxed objective function for circles with equal radius leading to the same circle arrangement as for the original objective function. If we minimize only the sum of lengths of the line segments, for selected cases of up to 10 circles we obtain gaps smaller than \(10^{-4}\) using BARON or LINDO embedded in GAMS, while for up to 75 circles we are able to approximate the optimal solution with a gap of at most \(14\%\).

Keywords

Global optimization Non-convex nonlinear programming Circular packing problem Convex hull Perimeter minimization Non-overlap constraints Computational geometry Isoperimetric inequality 

Notes

Acknowledgements

We thank Julius Näumann (Student, TU Darmstadt, Darmstadt, Germany), Prof. Dr. Julia Kallrath and Jan-Erik Justkowiak (Student, Hochschule Darmstadt, Darmstadt, Germany) and Dr. Fritz Näumann (Consultant, Weisenheim am Berg, Germany) for their careful reading of and feedback on the manuscript.

Supplementary material

References

  1. 1.
    Ahn, H.K., Cheong, O.: Aligning two convex figures to minimize area or perimeter. Algorithmica 62(1), 464–479 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bennell, J., Scheithauer, G., Stoyan, Y., Romanova, T., Pankratov, A.: Optimal clustering of a pair of irregular objects. J. Global Optim. 61(3), 497–524 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bezdek, K.: Lectures on Sphere Arrangements—the Discrete Geometric Side, vol. 32. Springer, New York (2013)CrossRefGoogle Scholar
  4. 4.
    Böröczky, J.R.: Finite Packing and Covering. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  5. 5.
    Castillo, I., Kampas, F., Pintér, J.: Solving circle packing problems by global optimization: numerical results and industrial application. Eur. J. Oper. Res. 191(3), 786–802 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Conway, J., Sloane, N.: Sphere Packings, Lattices and Groups. A Series of Comprehensive Studies in Mathematics. Springer, Berlin (1988)CrossRefGoogle Scholar
  7. 7.
    Dyckhoff, H.: A typology of cutting and packing problems. Eur. J. Oper. Res. 44(4), 145–159 (1990)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hifi, M., M’Hallah, R.: Adaptive and restarting techniques-based algorithms for circular packing problems. Comput. Optim. Appl. 39, 17–35 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hifi, M., Paschos, V., Zissimopoulos, V.: A simulated annealing approach for the circular cutting problem. Eur. J. Oper. Res. 159, 430–448 (2004)CrossRefGoogle Scholar
  10. 10.
    Kallrath, J.: Combined strategic design and operative planning in the process industry. Comput. Chem. Eng. 33, 1983–1993 (2009)Google Scholar
  11. 11.
    Kallrath, J.: Cutting circles and polygons from area-minimizing rectangles. J. Global Optim. 43, 299–328 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kallrath, J.: Polylithic modeling and solution approaches using algebraic modeling systems. Optim. Lett. 5, 453–466 (2011).  https://doi.org/10.1007/s11590-011-0320-4 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kallrath, J.: Packing ellipsoids into volume-minimizing rectangular boxes. J. Global Optim. 67(1), 151–185 (2017).  https://doi.org/10.1007/s10898-015-0348-6 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Libeskind, S.: Euclidean and transformational geometry: a deductive inquiry. Jones & Bartlett Learning, LLC (2008). https://books.google.de/books?id=JiTse0Nm_-IC
  15. 15.
    Nash, E., Pir, A., Sottile, F., Ying, L.: Convex hull of two circles in \(r^3\). Tech. rep. Algebraic Geometry (2017)Google Scholar
  16. 16.
    Rappaport, D.: A convex hull algorithm for discs, an application. Comput. Geom. Theory Appl. 1(3), 171–187 (1992)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Stetsyuk, P., Romanova, T., Scheithauer, G.: On the global minimum in a balanced circular packing problem. Optim. Lett. 10, 1347–1360 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Stoyan, Y., Yaskov, G.: Packing congruent hyperspheres into a hypersphere. J. Global Optim. 52(4), 855–868 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Stoyan, Y., Yaskov, G.: Packing unequal circles into a strip of minimal length with a jump algorithm. Optim. Lett. 8(3), 949–970 (2014).  https://doi.org/10.1007/s11590-013-0646-1 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Thue, A.: Über die dichteste Zusammenstellung von kongruenten Kreisen in einer Ebene. J. Dybwad (1910). https://books.google.de/books?id=IUJyQwAACAAJ

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

Authors and Affiliations

  1. 1.BASF SE, Advanced Business Analytics, G-FSS/OAO-B009LudwigshafenGermany
  2. 2.Department of AstronomyUniversity of FloridaGainesvilleUSA
  3. 3.Technische Universität München, TUM-School of ManagementMunichGermany

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