Abstract
Most integer programming problems can be formulated in several ways. Some formulations are better suited for solution by exact methods, because they have either (i) a strong LP relaxation, (ii) few symmetries in the solution space, or both. However, solving one formulation, we can often branch and/or add cutting planes which are implicitly based on variables of other formulations, working in fact on intersection of several polytopes. Traditional examples of this approach can be found in, e.g., (capacitated) routing and network planning where decomposed models operate with paths or trees, and thus need to be solved by column generation, but original models operate on separate edges. We consider such a ‘capacity-extended formulation’, the so-called arc-flow model, of the 1D cutting stock problem. Its variables are known to induce effective branching constraints leading to small and stable branch&bounsd trees. In this work we explore Chvátal-Gomory cuts on its variables. The results are positive only for small instances. Moreover, we compare the results to the cuts constructed on the variables of the direct model. The latter are more involved but also more effective.
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
Gilmore, P.C., Gomory, R.E.: A linear programming approach to the cutting-stock problem. Operations Research 9 (1961) 849–859
Rietz, J.: Investigations of MIRUP for vector packing problems. PhD thesis, Institute of Numerical Mathematics and Optimization, Freiberg University (2003)
Valério de Carvalho, J.M.: Exact solution of cutting-stock problems using column generation and branch-and-bound. International Transactions in Operations Research 5 (1998) 35–44
Valério de Carvalho, J.M.: Exact solution of bin-packing problems using column generation and branch-and-bound. Annals of Operation Research 86 (1999) 629–659
Alves, C., Valério de Carvalho, J.M.: A stabilized branch-and-price-and-cut algorithm for the multiple length cutting stock problem. Computers and Operations Research 35(4) (2008) 1315–1328
Degraeve, Z., Peeters, M.: Optimal integer solutions to industrial cutting-stock problems: Part 2, benchmark results. INFORMS Journal on Computing 15(1) (2003) 58–81
Belov, G., Scheithauer, G.: A branch-and-cut-and-price algorithm for one-dimensional stock cutting and two-dimensional two-stage cutting. European Journal of Operational Research 171(1) (2006) 85–106
Vance, P.: Branch-and-price algorithms for the one-dimensional cutting stock problem. Comput. Opt. Appl. 9(3) (1998) 212–228
Vanderbeck, F.: On Dantzig-Wolfe decomposition in integer programming and ways to perform branching in a branch-and-price algorithm. Operations Research 48(1) (2000) 111–128
Belov, G., Letchford, A., Uchoa, E.: A node-flow model for 1D stock cutting: Robust branch-cut-and-price. Technical report, Universidade Federal Fluminense (2005)
Scheithauer, G., Terno, J., Müller, A., Belov, G.: Solving one-dimensional cutting stock problems exactly with a cutting plane algorithm. Journal of the Operational Research Society 52 (2001) 1390–1401
Belov, G., Scheithauer, G.: A cutting plane algorithm for the one-dimensional cutting stock problem with multiple stock lengths. European Journal of Operational Research 141(2) (2002) 274–294
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. John Wiley and Sons, New York (1988)
Letchford, A.N., Lodi, A.: Strengthening Chvátal-Gomory cuts and Gomory fractional cuts. Operations Research Letters 30(2) (2002) 74–82
Forrest, J., de la Nuez, D., Lougee-Heimer, R.: CLP user guide. Technical report, IBM Research (2004)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Betriebswirtschaftlicher Verlag Dr. Th. Gabler | GWV Fachverlage GmbH, Wiesbaden
About this chapter
Cite this chapter
Belov, G., Scheithauer, G., Alves, C., Valério de Carvalho, J.M. (2008). Gomory Cuts from a Position-Indexed Formulation of 1D Stock Cutting. In: Bortfeldt, A., Homberger, J., Kopfer, H., Pankratz, G., Strangmeier, R. (eds) Intelligent Decision Support. Gabler. https://doi.org/10.1007/978-3-8349-9777-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-8349-9777-7_1
Publisher Name: Gabler
Print ISBN: 978-3-8349-0930-5
Online ISBN: 978-3-8349-9777-7
eBook Packages: Business and EconomicsBusiness and Management (R0)