Abstract
A problem of packing a limited number of unequal circles in a fixed size rectangular container is considered. The aim is to maximize the (weighted) number of circles placed into the container or minimize the waste. Frequently, the problem is formulated as a nonconvex continuous optimization problem which is solved by heuristic techniques combined with local search procedures. A new formulation is proposed using a regular grid approximating the container and considering the nodes of the grid as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0-1 optimization problem. The binary variables represent the assignment of centers to the nodes of the grid. The resulting binary problem is then solved by commercial software. Two families of valid inequalities are proposed to strengthen the formulation. Numerical results are presented to demonstrate the efficiency of the proposed approach.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Akeb, H., Hifi, M.: Solving the circular open dimension problem using separate beams and look-ahead strategies. Computers & Operations Research 40, 1243–1255 (2013)
Baltacioglu, E., Moore, J.T., Hill, R.R.: The distributor´s three-dimensional pallet-packing problem: a human intelligence-based heuristic approach. International Journal of Operational Research 1, 249–266 (2006)
Beasley, J.E.: An exact two-dimensional non-guillotine cutting tree search procedure. Operations Research 33, 49–64 (1985)
Birgin, E.G., MartÃnez, J.M., Nishihara, F.H., Ronconi, D.P.: Orthogonal packing of rectangular items within arbitrary convex regions by nonlinear optimization. Computers & Operations Research 33, 3535–3548 (2006)
Birgin, E.G., Gentil, J.M.: New and improved results for packing identical unitary radius circles within triangles, rectangles and strips. Computers & Operations Research 37, 1318–1327 (2010)
Castillo, I., Kampas, F.J., Pinter, J.D.: Solving circle packing problems by global optimization: Numerical results and industrial applications. European Journal of Operational Research 191, 786–802 (2008)
Costa, A.: Valid constraints for the point packing in a square problem. Discrete Applied Mathematics 161, 2901–2909 (2013)
Fasano, G.: Solving Non-standard Packing Problems by Global Optimization and Heuristics. Springer, Berlin (2014)
Frazer, H.J., George, J.A.: Integrated container loading software for pulp and paper industry. European Journal of Operational Research 77, 466–474 (1994)
Galiev, S.I., Lisafina, M.S.: Linear models for the approximate solution of the problem of packing equal circles into a given domain. European Journal of Operational Research 230, 505–514 (2013)
George, J.A., George, J.M., Lamar, B.W.: Packing different–sized circles into a rectangular container. European Journal of Operational Research 84, 693–712 (1995)
George, J.A.: Multiple container packing: a case study of pipe packing. Journal of the Operational Research Society 47, 1098–1109 (1996)
Hifi, M., M’Hallah, R.: A literature review on circle and sphere packing problems: Models and methodologies. Advances in Operations Research Article ID 150624, 22 pages (2009), doi:10.1155/2009/150624
Litvinchev, I., Ozuna, E.L.: Packing circles in a rectangular container. In: Proc. Intl. Congr. on Logistics and Supply Chain, Queretaro, Mexico, pp. 24–25 (October 2013)
Litvinchev, I., Ozuna, E.L.: Integer programming formulations for approximate packing circles in a rectangular container. Mathematical Problems in Engineering (2014), doi:10.1155/2014/317697
Litvinchev, I., Rangel, S., Saucedo, J.: A Lagrangian bound for many-to-many assignment problem. Journal of Combinatorial Optimization 19, 241–257 (2010)
Lopez, C.O., Beasley, J.E.: A heuristic for the circle packing problem with a variety of containers. European Journal of Operational Research 214, 512–525 (2011)
Lopez, C.O., Beasley, J.E.: Packing unequal circles using formulation space search. Computers & Operations Research 40, 1276–1288 (2013)
Stoyan, Y.G., Yaskov, G.N.: Packing congruent spheres into a multi-connected polyhedral domain. International Transactions in Operational Research 20, 79–99 (2013)
Wolsey, L.A.: Integer Programming. Wiley, New York (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Litvinchev, I., Infante, L., Ozuna Espinosa, E.L. (2014). Approximate Circle Packing in a Rectangular Container: Integer Programming Formulations and Valid Inequalities. In: González-RamÃrez, R.G., Schulte, F., Voß, S., Ceroni DÃaz, J.A. (eds) Computational Logistics. ICCL 2014. Lecture Notes in Computer Science, vol 8760. Springer, Cham. https://doi.org/10.1007/978-3-319-11421-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-11421-7_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11420-0
Online ISBN: 978-3-319-11421-7
eBook Packages: Computer ScienceComputer Science (R0)