Journal of Global Optimization

, Volume 59, Issue 2–3, pp 405–437 | Cite as

Cutting ellipses from area-minimizing rectangles

Article

Abstract

A set of ellipses, with given semi-major and semi-minor axes, is to be cut from a rectangular design plate, while minimizing the area of the design rectangle. The design plate is subject to lower and upper bounds of its widths and lengths; the ellipses are free of any orientation restrictions. We present new mathematical programming formulations for this ellipse cutting problem. The key idea in the developed non-convex nonlinear programming models is to use separating hyperlines to ensure the ellipses do not overlap with each other. For small number of ellipses we compute feasible points which are globally optimal subject to the finite arithmetic of the global solvers at hand. However, for more than 14 ellipses none of the local or global NLP solvers available in GAMS can even compute a feasible point. Therefore, we develop polylithic approaches, in which the ellipses are added sequentially in a strip-packing fashion to the rectangle restricted in width, but unrestricted in length. The rectangle’s area is minimized in each step in a greedy fashion. The sequence in which we add the ellipses is random; this adds some GRASP flavor to our approach. The polylithic algorithms allow us to compute good, near optimal solutions for up to 100 ellipses.

Keywords

Global optimization Non-convex nonlinear programming  Mixed integer programming Cutting stock problem Packing problem Shape constraints Non-overlap constraints  Design problem Polylithic solution approach Computational geometry 

References

  1. 1.
    Adjiman, C.S., Androulakis, I.P., Floudas, C.A.: Global optimization of mixed integer nonlinear problems. AIChE J. 46, 1769–1797 (2000)CrossRefGoogle Scholar
  2. 2.
    Adjiman, C.S., Androulakis, I.P., Maranas, C.D., Floudas, C.A.: A global optimization method aBB for process design. Comput. Chem. Eng. 20(Suppl. 1), S419–424 (1996)CrossRefGoogle Scholar
  3. 3.
    Androulakis, I.P., Maranas, C.D., Floudas, C.A.: aBB: a global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7, 337–363 (1995)CrossRefGoogle Scholar
  4. 4.
    Dyckhoff, H.: A typology of cutting and packing problems. Eur. J. Oper. Res. 44, 145–159 (1990)CrossRefGoogle Scholar
  5. 5.
    Feo, T., Resende, M.: Greedy randomized adaptive search procedures. J. Glob. Optim. 6, 109–133 (1995)CrossRefGoogle Scholar
  6. 6.
    Floudas, CA.: Private communication (2012)Google Scholar
  7. 7.
    Floudas, C.A., Akrotirianakis, I.G., Caratzoulas, S., Meyer, C.A., Kallrath, J.: Global optimization in the 21st century: advances and challenges for problems with nonlinear dynamics. Comput. Chem. Eng. 29, 1185–1202 (2005)CrossRefGoogle Scholar
  8. 8.
    Gensane, T., Honvault, P.: Optimal Packings of Two Ellipses in a Square. Technical report, L.M.P.A. (2012)Google Scholar
  9. 9.
    Hettich, R., Kortanek, K.O.: Semi-infinite programming. SIAM Rev. 35, 380–429 (1993)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. Glob. Optim. 43, 299–328 (2009)CrossRefGoogle Scholar
  12. 12.
    Kallrath, J.: Polylithic modeling and solution approaches using algebraic modeling systems. Optim. Lett. 5, 453–466 (2011). doi:10.1007/s11590-011-0320-4 CrossRefGoogle Scholar
  13. 13.
    Liberti, L.: Reformulation and convex relaxation techniques for global optimization. Ph.D. Thesis, Imperial College London, London, UK (2004)Google Scholar
  14. 14.
    Liberti, L., Pantelides, C.: An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms. J. Glob. Optim. 36, 161–189 (2006)CrossRefGoogle Scholar
  15. 15.
    Lindo Systems: Lindo API: User’s Manual. Lindo Systems, Chicago (2004)Google Scholar
  16. 16.
    Lopez, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491–518 (2007)CrossRefGoogle Scholar
  17. 17.
    Maranas, C.D., Floudas, C.A.: Finding all solutions of nonlinearly constrained systems of equations. J. Glob. Optim. 7, 143–182 (1995)CrossRefGoogle Scholar
  18. 18.
    Markót, M.C.: Interval methods for verifying structural optimality of circle packing configurations in the unit square. J. Comput. Appl. Math. 199, 353–357 (2007)CrossRefGoogle Scholar
  19. 19.
    Markót, M.C., Csendes, T.: A new verified optimization technique for the “packing circles in a unit square” problems. SIAM J. Optim. 16, 193–219 (2005)CrossRefGoogle Scholar
  20. 20.
    Miller, P.: Globally optimal packing of nonconvex two-dimensional shapes by approximation with ellipses. Princeton University, Princeton, NJ, Senior Thesis (2012)Google Scholar
  21. 21.
    Misener, R., Floudas, C.: GloMIQO: global mixed-integer quadratic optimizer. J. Glob. Optim. pp. 1–48 (2012). doi:10.1007/s10898-012-9874-7
  22. 22.
    Rebennack, S., Kallrath, J.: Continuous piecewise linear \(\delta \)-approximations for MINLP problems. I. Minimal breakpoint systems for univariate functions. Tech. rep., Colorado School of Mines Division of Economics and Business Working Paper Series 2012–12 (2012)Google Scholar
  23. 23.
    Rebennack, S., Kallrath, J.: Continuous piecewise linear \(\delta \)-approximations for MINLP problems. II. Bivariate and multivariate functions. Tech. rep., Colorado School of Mines Division of Economics and Business Working Paper Series 2012–13 (2012)Google Scholar
  24. 24.
    Rebennack, S., Kallrath, J., Pardalos, P.M.: Column enumeration based decomposition techniques for a class of non-convex MINLP problems. J. Glob. Optim. 43, 277–297 (2009)CrossRefGoogle Scholar
  25. 25.
    Tawarmalani, M., Sahinidis, N.V.: Convexification and global optimization in continuous and mixed-integer nonlinear programming: theory, algorithms, software, and applications. Nonconvex optimization and its applications series. Kluwer, Dordrecht, The Netherlands (2002)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.BASF SE, Scientific Computing, GVM/S-B009Ludwigshafen am RheinGermany
  2. 2.Division of Economics and BusinessColorado School of MinesGoldenUSA

Personalised recommendations