Kubach T., Bortfeldt A., Gehring H.: Parallel greedy algorithms for packing unequal circles into a strip or a rectangle. Kallrath J.: Cutting circles and polygons from area-minimizing rectangles. Huang W.Q., Li Y., Akeb H., Li C.M.: Greedy algorithms for packing unequal circles into a rectangular container. Hifi M., Paschos V.Th., Zissimopoulos V.: A simulated annealing approach for the circular cutting problem. Hifi, M., M’Hallah, R.: A literature review on circle and sphere packing problems: models and methodologies. Hifi M., M’Hallah R.: Approximate algorithms for constrained circular cutting problems. He Y., Wu Y., de Souza R.: A global search framework for practical three-dimensional packing with variable carton orientations. In: Rawlins, G (ed.) Foundations of Genetic Algorithms, Morgan Kaufmann, California (1991) Goldberg D.E., Deb K.: A comparative analysis of selection schemes used in genetic algorithms. Goldberg D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. George J.A., George J.M., Lamar B.W.: Packing different-sized circles into a rectangular container. Gen M., Cheng R.: Genetic Algorithms and Engineering Optimization. 160, 19–33 (2005)Ĭastillo I., Kampas F.J., Pinter J.D.: Solving circle packing problems by global optimization: Numerical results and industrial applications. 61(2), 373–381 (2011)īirgin E.G., Martinez J.M., Ronconi D.P.: Optimizing the packing of cylinders into a rectangular container: a nonlinear approach. 15, 685–704 (2008)Īkeb H., Hifi M., Negre S.: An augmented beam search-based algorithm for the circular open dimension problem. The most significant contributions of this work are: firstly, we proposed three types of packing positions for selection and used human intelligence to convert an arbitrary circle sequence into a feasible compact layout secondly, diverse position selection criteria have been tested, and it is found that the criterion commonly used in the literature is not the best thirdly, the traditional genetic algorithm is adapted with lower crossover rate but higher mutation rate particularly, and a minor-adjustment operator with the purpose of exploring the neighborhood of the current best solutions is introduced.Īkeb H., Hifi M.: Algorithms for the circular two-dimensional open dimension. This article aims to challenge the existing methods for the benchmark instances. Even via heuristic methods, the solution time for industrial-scale instances sometimes is too long to be acceptable. However, the non-convex constraints in this problem make it intractable using exact analytical approaches. Packing non-identical circles inside a rectangle witnesses a wide range of industrial applications.

Pack circles in rectangle