Abstract
In this paper we propose a Monotonic Basin Hopping approach and its population-based variant Population Basin Hopping to solve the problem of packing equal and unequal circles within a circular container with minimum radius. Extensive computational experiments have been performed both to analyze the problem at hand, and to choose in an appropriate way the parameter values for the proposed methods. Different improvements with respect to the best results reported in the literature have been detected.
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Addis B., Locatelli M., Schoen F.: Efficiently packing unequal disks in a circle. Oper. Res. Lett. 36(1), 37–42 (2008)
Addis B., Locatelli M., Schoen F.: Disk packing in a square: a new global optimization approach. INFORMS J. Comput. 20, 516–524 (2008)
Birgin E.G., Sobral F.N.C.: Minimizing the object dimensions in circle and sphere packing problems. Comput. Oper. Res. 35, 2357–2375 (2008)
Boll D.V., Donovan J., Graham R.L., Lubachevsky B.D.: Improving dense packings of equal disks in a square. Electron. J. Comb. 7, R46 (2000)
Casado, L.G., García, I., Szabó, P.G., Csendes, T.: Equal circles packing in square II—new results for up to 100 circles using the TAMSASS-PECS Algorithm. In: “Optimizarion Theory; Recent Developments from Mátraháza. Applied Optimization Book Series, Vol. 59, pp 207–224, Kluwer Academic Publishers (2001).
Castillo I., Kampas F.J., Pinter J.D.: Solving circle packing problems by global optimization: numerical results and industrial applications. Eur. J. Oper. Res. 191(3), 786–802 (2008)
de Groot, C., Peikert, R., Würtz, D., Monagan, M.: Packing circles in a square: a review and new results, System Modelling and Optimization, Proceedings of 15th IFIP Conference, Zürich, 45–54 (1991)
Graham R.L., Lubachevsky B.D.: Repeated patterns of dense packings of equal disks in a square. Electron. J. Comb. 3, 1–16 (1996)
Gill P.E., Murray W., Saunders M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM J. Optim. 12, 979–1006 (2002)
Grosso A., Locatelli M., Schoen F.: A population based approach for hard global optimization problems based on dissimilarity measures. Math. Program. 110(2), 373–404 (2007)
Hifi M., M’Hallah R.: Adaptive and restarting techniques based algorithms for circular packing problems. Comput. Optim. Appl. 39, 17–35 (2008)
Huang H., Huang W., Zhang Q., Xu D.: An improved algorithm for the packing of unequal circles within a larger containing circle. Eur. J. Oper. Res. 141, 440–453 (2002)
Huang, W., Li, Y., Jurkowiak, B., Li, C.M., Xu, R.C.: A two-level search strategy for packing unequal circles into a circle container. In: Proceedings of the International Conference on Principles and Practice of Constraint Programming, pp. 868–872. Springer, Berlin (2003)
Kallrath, J.: Cutting circles and polygons from area-minimizing rectangles, J. Glob. Optim. (2008, in press). doi:10.1007/s10898-007-9274-6
Leary R.H.: Global optimization on funneling landscapes. J. Glob. Optim. 18, 367–383 (2000)
Lee J., Scheraga H.A., Rackovsky S.: New optimization method for conformational energy calculations on polypeptides: conformational space annealing. J. Comput. Chem. 18(9), 1222–1232 (1997)
Locatelli M., Raber U.: Packing equal circles in a square: a deterministic global optimization approach. Discrete Appl. Math. 122, 139–166 (2002)
Lourenco H.R., Martin O., Stützle T.: Iterated local search. In: Glover, F., Kochenberger, G. (eds) Handbook of Metaheuristics, pp. 321–353. Kluwer, Norwell (2002)
Markot 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)
Mladenovic N., Plastria F., Uroevi D.: Reformulation descent applied to circle packing problems. Comput. Oper. Res. 32, 2419–2434 (2005)
Nurmela K.J., Oestergard P.R.J.: More optimal packings of equal circles in a square. Discrete Comput. Geom. 22, 439–457 (1999)
Nurmela K.J., Oestergard P.R.J.: Packing up to 50 equal circles in a square. Discrete Comput. Geom. 18, 111–120 (1997)
Packomania web site maintained by Specht, www.packomania.com
Pintér J.D., Kampas F.J.: Nonlinear optimization in mathematica with mathoptimizer professional. Math. Educ. Res. 10, 1–18 (2005)
Stoyan Y., Yaskow G.: Mathematical model and solution method of optimization problem of placement of rectangles and circles taking into account special constraints. Int. Trans. Oper. Res. 5, 45–57 (1998)
Stoyan Y.G., Yaskov G.N.: A mathematical model and a solution method for the problem of placing various-sized circles into a strip. Eur. J. Oper. Res. 156, 590–600 (2004)
Szabó P.G., Markót M.C., Csendes T.: Global optimization in geometry-circle packing into the square. In: Audet, C., Hansen, P., Savard, G. (eds) Essays and Surveys in Global Optimization, pp. 233–266. Kluwer, Dordrecht (2005)
Szabó P.G., Markót M.C., Csendes T., Specht E., Casado L.G., Garcia I.: New Approaches to Circle Packing in a Square, Optimization and Its Applications. Springer, Berlin (2007)
Wales D.J., Doye J.P.K.: Global optimization by basin-hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms. J. Phys. Chem. A 101, 5111–5116 (1997)
Wang H., Huang W., Zhang Q., Xu D.: An improved algorithm for the packing of unequal circles within a larger containing circle. Eur. J. Oper. Res. 141, 440–453 (2002)
Zhang D., Deng A.: An effective hybrid algorithm for the problem of packing circles into a larger containing circle. Comput. Oper. Res. 32, 1941–1951 (2005)
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Grosso, A., Jamali, A.R.M.J.U., Locatelli, M. et al. Solving the problem of packing equal and unequal circles in a circular container. J Glob Optim 47, 63–81 (2010). https://doi.org/10.1007/s10898-009-9458-3
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DOI: https://doi.org/10.1007/s10898-009-9458-3