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Exact Algorithm for Minimising the Number of Setups in the One-Dimensional Cutting Stock Problem

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  • François Vanderbeck

    (Mathématiques Appliquées Bordeaux (MAB), Université Bordeaux 1, 351 Cours de la Libération, F-33405 Talence Cedex, France)

Abstract

The cutting stock problem is that of finding a cutting of stock material to meet demands for small pieces of prescribed dimensions while minimising the amount of waste. Because changing over from one cutting pattern to another involves significant setups, an auxiliary problem is to minimise the number of different patterns that are used. The pattern minimisation problem is significantly more complex, but it is of great practical importance. In this paper, we propose an integer programming formulation for the problem that involves an exponential number of binary variables and associated columns, each of which corresponds to selecting a fixed number of copies of a specific cutting pattern. The integer program is solved using a column generation approach where the subproblem is a nonlinear integer program that can be decomposed into multiple bounded integer knapsack problems. At each node of the branch-and-bound tree, the linear programming relaxation of our formulation is made tighter by adding super-additive inequalities. Branching rules are presented that yield a balanced tree. Incumbent solutions are obtained using a rounding heuristic. The resulting branch-and-price-and-cut procedure is used to produce optimal or approximately optimal solutions for a set of real-life problems.

Suggested Citation

  • François Vanderbeck, 2000. "Exact Algorithm for Minimising the Number of Setups in the One-Dimensional Cutting Stock Problem," Operations Research, INFORMS, vol. 48(6), pages 915-926, December.
    • Handle: RePEc:inm:oropre:v:48:y:2000:i:6:p:915-926
    DOI: 10.1287/opre.48.6.915.12391
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    References listed on IDEAS

    as
    1. François Vanderbeck, 2000. "On Dantzig-Wolfe Decomposition in Integer Programming and ways to Perform Branching in a Branch-and-Price Algorithm," Operations Research, INFORMS, vol. 48(1), pages 111-128, February.
    2. P. C. Gilmore & R. E. Gomory, 1961. "A Linear Programming Approach to the Cutting-Stock Problem," Operations Research, INFORMS, vol. 9(6), pages 849-859, December.
    3. A. A. Farley, 1990. "A Note on Bounding a Class of Linear Programming Problems, Including Cutting Stock Problems," Operations Research, INFORMS, vol. 38(5), pages 922-923, October.
    4. Vanderbeck, F. & Wolsey, L. A., 1996. "An exact algorithm for IP column generation," LIDAM Reprints CORE 1242, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Goulimis, Constantine, 1990. "Optimal solutions for the cutting stock problem," European Journal of Operational Research, Elsevier, vol. 44(2), pages 197-208, January.
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