Abstract
In this paper we consider the problem of packing a fixed number of identical circles inside the unit circle container, where the packing is complicated by the presence of fixed size circular prohibited areas. Here the objective is to maximise the radius of the identical circles. We present a heuristic for the problem based upon formulation space search. Computational results are given for six test problems involving the packing of up to 100 circles. One test problem has a single prohibited area made up from the union of circles of different sizes. Four test problems are annular containers, which have a single inner circular prohibited area. One test problem has circular prohibited areas that are disconnected.
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Amirgaliyeva, Z., Mladenović, N., Todosijević, R., Urosević, D.: Solving the maximum min-sum dispersion by alternating formulations of two different problems. Eur. J. Oper. Res. 260(2), 444–459 (2017)
Birgin, E.G., Gentil, J.M.: New and improved results for packing identical unitary radius circles within triangles, rectangles and strips. Comput. Oper. Res. 37(7), 1318–1327 (2010)
Birgin, E.G., Sobral, F.N.C.: Minimizing the object dimensions in circle and sphere packing problems. Comput. Oper. Res. 35(7), 2357–2375 (2008)
Brimberg, J., Drezner, Z., Mladenović, N., Salhi, S.: A new local search for continuous location problems. Eur. J. Oper. Res. 232(2), 256–265 (2014)
Butenko, S., Yezerska, O., Balasundaram, B.: Variable objective search. J. Heuristics 19(4), 697–709 (2013)
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)
Duarte, A., Pantrigo, J.J., Pardo, E.G., Sánchez-Oro, J.: Parallel variable neighbourhood search strategies for the cutwidth minimization problem. IMA J. Manag. Math. 27(1), 55–73 (2016)
Erromdhani, R., Jarboui, B., Eddaly, M., Rebai, A., Mladenović, N.: Variable neighborhood formulation search approach for the multi-item capacitated lot-sizing problem with time windows and setup times. Yugosl. J. Oper. Res. 27(3), 310–322 (2017)
Gill, P.E., Murray, W., Saunders, M.A.: User’s guide for SNOPT version 7: software for large-scale nonlinear programming. http://web.stanford.edu/group/SOL/guides/sndoc7.pdf. Last accessed 14 June 2018 (2008)
Hansen, P., Mladenović, N., Brimberg, J., Perez, J.A.M.: Variable neighborhood search. In: Gendreau, M., Potvin, J.-Y. (eds.) Handbook of Metaheuristics. International Series in Operations Research and Management Science, vol. 146, pp. 61–86. Springer, Berlin (2010)
Hansen, P., Mladenović, N., Todosijević, T., Hanafi, S.: Variable neighborhood search: basics and variants. EURO J. Comput. Optim. 5(3), 423–454 (2017)
Hertz, A., Plumettaz, M., Zufferey, N.: Variable space search for graph coloring. Discrete Appl. Math. 156(13), 2551–2560 (2008)
Hertz, A., Plumettaz, M., Zufferey, N.: Corrigendum to variable space search for graph coloring [Discrete Appl. Math. 156 (2008) 2551–2560]. Discrete Appl. Math. 157(7), 1335–1336 (2009)
Hifi, M., M’Hallah, R.: A literature review on circle and sphere packing problems: models and methodologies. Adv. Oper. Res. https://doi.org/10.1155/2009/150624. (2009)
Holmström., K., Göran, A.O., Edvall, M.M.: User’s guide for TOMLAB/SNOPT. http://tomopt.com/docs/TOMLAB_SNOPT.pdf. Last accessed 14 June 2018 (2008)
Huang, W.Q., Ye, T.: Greedy vacancy search algorithm for packing equal circles in a square. Oper. Res. Lett. 38(5), 378–382 (2010)
Kochetov, Y., Kononova, P., Paschenko, M.: Formulation space search approach for the teacher/class timetabling problem. Yugosl. J. Oper. Res. 18(1), 1–11 (2008)
Li, Y., Akeb, H.: Basic heuristics for packing a great number of equal circles. Working Paper Available From the Second Author at ISC Paris Business School, 22, Bd du Fort de Vaux, 75017 Paris, France (2005)
López, C.O.: Formulation space search for two-dimensional packing problems. Ph.D. thesis, Brunel University (2013)
López, C.O., Beasley, J.E.: A heuristic for the circle packing problem with a variety of containers. Eur. J. Oper. Res. 214(3), 512–525 (2011)
López, C.O., Beasley, J.E.: Packing unequal circles using formulation space search. Comput. Oper. Res. 40(5), 1276–1288 (2013)
López, C.O., Beasley, J.E.: A note on solving MINLP’s using formulation space search. Optim. Lett. 8(3), 1167–1182 (2014)
López, C.O., Beasley, J.E.: A formulation space search heuristic for packing unequal circles in a fixed size circular container. Eur. J. Oper. Res. 251(1), 64–73 (2016)
López, C.O., Beasley, J.E.: Packing unequal rectangles and squares in a fixed size circular container using formulation space search. Comput. Oper. Res. 94, 106–117 (2018)
Mladenović, N., Plastria, F., Urošević, D.: Reformulation descent applied to circle packing problems. Comput. Oper. Res. 32(9), 2419–2434 (2005)
Mladenović, N., Plastria, F., Urošević, D.: Formulation space search for circle packing problems. In: “Engineering Stochastic Local Search Algorithms. Designing, Implementing and Analyzing Effective Heuristics”, Proceedings of the International Workshop, SLS 2007, Brussels, Belgium, Sep 6–8, 2007. Lecture Notes in Computer Science volume 4638, pp. 212–216 (2007)
Pardo, E.G., Mladenović, N., Pantrigo, J.J., Duarte, A.: Variable formulation search for the cutwidth minimization problem. Appl. Soft Comput. 13(5), 2242–2252 (2013)
Pedroso, J.P., Cunha, S., Tavares, J.N.: Recursive circle packing problems. Int. Trans. Oper. Res. 23(1–2), 355–368 (2016)
Specht, E.: http://www.packomania.com. Last accessed 14 June 2018 (2010)
Stoyan, Y., Pankratov, A., Romanova, T.: Quasi-phi-functions and optimal packing of ellipses. J. Glob. Optim. 65(2), 283–307 (2016)
Stoyan, Y., Yaskov, G.: Packing equal circles into a circle with circular prohibited areas. Int. J. Comput. Math. 89(10), 1355–1369 (2012)
Stoyan, Y., Yaskov, G.: Packing congruent hyperspheres into a hypersphere. J. Glob. Optim. 52(4), 855–868 (2012)
Stoyan, Y.G., Zlotnik, M.V., Chugay, A.M.: Solving an optimization packing problem of circles and non-convex polygons with rotations into a multiply connected region. J. Oper. Res. Soc. 63(3), 379–391 (2012)
Szabó, P.G., Markót, M.C., Csendes, T., Specht, E., Casado, L.G., Garcia, I.: New Approaches to Circle Packing in a Square: With Program Codes, vol. 6. Springer Optimization and its Applications, Berlin (2007)
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(2), 440–453 (2002)
Wieman, H.: SSD cable packing. http://www-rnc.lbl.gov/~wieman/SSDpackingcombined.pdf. Last accessed 14 June 2018 (2008)
Zhuang, X.Y., Yan, L., Chen, L.: Packing equal circles in a damaged square. In: International Joint Conference on Neural Networks (IJCNN), pp. 1–6. IEEE, New York (2015)
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The first author has grant support from the programme UNAM-DGAPA-PAPIIT-IA106916.
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López, C.O., Beasley, J.E. Packing a fixed number of identical circles in a circular container with circular prohibited areas. Optim Lett 13, 1449–1468 (2019). https://doi.org/10.1007/s11590-018-1351-x
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DOI: https://doi.org/10.1007/s11590-018-1351-x