Abstract
We present an algorithm for the binary cutting stock problem that employs both column generation and branch-and-bound to obtain optimal integer solutions. We formulate a branching rule that can be incorporated into the subproblem to allow column generation at any node in the branch-and-bound tree. Implementation details and computational experience are discussed.
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References
M. Desrochers, J. Desrosiers, and M. Solomon. “A New Optimization Algorithm for the Vehicle Routing Problem with Time Windows,”Operations Research 40, 542–354, 1992.
M. Desrochers and F. Soumis. “A Column Generation Approach to the Urban Transit Crew Scheduling Problem,”Transportation Science 23, 1–13, 1989.
J. Desrosiers, F. Soumis, and M. Desrochers. “Routing with Time Windows by Column Generation”Networks 14, 545–565, 1984.
J. Desrosiers, F. Soumis, M. Desrochers, and M., Sauve. “Methods for Routing with Time Windows,”European Journal of Operational Research 23, 235–245, 1986.
Y. Dumas, J. Desrosiers, and F. Soumis, “Pickup and Delivery Models with Time Windows,”European Journal of Operational Research 54, 7–22, 1991.
A.A. Farley. “A Note on Bounding a Class of Linear Programming Problems, Including Cutting Stock Problems,”Operations Research 38, 922–923, 1990.
M.R. Garey and D.S. Johnson.Computers and Intractability: A Guide to the Theory of NP Completeness, W.H. Freeman, 1979.
P.C. Gilmore and R.E. Gomory, “A Linear Programming Approach to the Cutting-Stock Problem,”Operations Research 9, 849–859, 1961.
C. Goulimis, “Optimal Solutions for the Cutting Stock Problem,”European Journal of Operational Research 44, 197–208, 1990.
R.W. Haessler and P.E. Sweeney, “Cutting Stock Problems and Solution Procedures,”European Journal of Operational Research 54, 141–150, 1991.
E. Horowitz and S. Sahni, “Computing Partitions with Applications to the Knapsack Problem,”Journal of ACM 21, 277–292, 1974.
E.L. Johnson “Modeling and Strong Linear Programs for Mixed Integer Programming,”Algorithms and Model Formulations in Mathematical Programming, S.W. Wallace ed., Springer-Verlag, 1–43, 1989.
E.L. Johnson and M.W. Padberg, “A Note on the Knapsack Problem with Special Ordered Sets,”Operations Research Letters 1, 18–22, 1981.
O. Marcotte, “The Cutting Stock Problem and Integer Rounding,”Mathematical Programming 33, 82–92, 1985.
O. Marcotte. “An Instance of the Cutting Stock Problem, for which the Integer Rounding Property Does Not Hold,”Operations Research Letters 4, 239–243, 1986.
S. Martello and P. Toth.Knapsack Problems: Algorithms and Computer Implementations, Wiley, 1990.
G.L. Nemhauser and S. Park. “A Polyhedral Approach to Edge Coloring,”Operations Research Letters, 10, 315–322, 1991.
C.C. Ribeiro, M. Minoux, and M.C. Penna. “An Optimal Column-Generation-with-Ranking, Algorithm for Very, Large Scale Set Partitioning Problems in Traffic Assignment,”European Journal of Operational Research 41, 232–239, 1989.
G.M. Roodman, “Near-Optimal Solutions to One-dimensional Cutting Stock Problems,”Computers and Operations Research 13, 713–719, 1986.
M.W.P. Savelsbergh, G.C. Sigismondi, and G.L. Nemhauser. “A Functional Description of MINTO, A Mixed INTeger Optimizer,” Technical Report, Computational Optimization Center, Georgia Institute of Technology, Atlanta, 1992.
H. Stadtler, “A One-dimensional Cutting Stock Problem in the Aluminum Industry and its Solution,”European Journal of Operational Research 44, 209–223, 1990.
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This research was supported by NSF and AFOSR grant DDM-9115768
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Vance, P.H., Barnhart, C., Johnson, E.L. et al. Solving binary cutting stock problems by column generation and branch-and-bound. Comput Optim Applic 3, 111–130 (1994). https://doi.org/10.1007/BF01300970
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DOI: https://doi.org/10.1007/BF01300970