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Packing circles into perimeter-minimizing convex hulls

Author

Listed:
  • Josef Kallrath

    (BASF SE, Advanced Business Analytics, G-FSS/OAO-B009
    University of Florida)

  • Markus M. Frey

    (BASF SE, Advanced Business Analytics, G-FSS/OAO-B009
    Technische Universität München, TUM-School of Management)

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 $$10^{-4}$$ 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 $$14\%$$ 14 % .

Suggested Citation

  • Josef Kallrath & Markus M. Frey, 2019. "Packing circles into perimeter-minimizing convex hulls," Journal of Global Optimization, Springer, vol. 73(4), pages 723-759, April.
  • Handle: RePEc:spr:jglopt:v:73:y:2019:i:4:d:10.1007_s10898-018-0724-0
    DOI: 10.1007/s10898-018-0724-0
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    References listed on IDEAS

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    1. Hifi, Mhand & Paschos, Vangelis Th. & Zissimopoulos, Vassilis, 2004. "A simulated annealing approach for the circular cutting problem," European Journal of Operational Research, Elsevier, vol. 159(2), pages 430-448, December.
    2. Yuriy Stoyan & Georgiy Yaskov, 2012. "Packing congruent hyperspheres into a hypersphere," Journal of Global Optimization, Springer, vol. 52(4), pages 855-868, April.
    3. Josef Kallrath, 2017. "Packing ellipsoids into volume-minimizing rectangular boxes," Journal of Global Optimization, Springer, vol. 67(1), pages 151-185, January.
    4. Dyckhoff, Harald, 1990. "A typology of cutting and packing problems," European Journal of Operational Research, Elsevier, vol. 44(2), pages 145-159, January.
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    Cited by:

    1. Josef Kallrath & Tatiana Romanova & Alexander Pankratov & Igor Litvinchev & Luis Infante, 2023. "Packing convex polygons in minimum-perimeter convex hulls," Journal of Global Optimization, Springer, vol. 85(1), pages 39-59, January.

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