A branch-and-price-and-cut algorithm for the pattern minimization problem
RAIRO - Operations Research - Recherche Opérationnelle, Tome 42 (2008) no. 4, pp. 435-453.

In cutting stock problems, after an optimal (minimal stock usage) cutting plan has been devised, one might want to further reduce the operational costs by minimizing the number of setups. A setup operation occurs each time a different cutting pattern begins to be produced. The related optimization problem is known as the Pattern Minimization Problem, and it is particularly hard to solve exactly. In this paper, we present different techniques to strengthen a formulation proposed in the literature. Dual feasible functions are used for the first time to derive valid inequalities from different constraints of the model, and from linear combinations of constraints. A new arc flow formulation is also proposed. This formulation is used to define the branching scheme of our branch-and-price-and-cut algorithm, and it allows the generation of even stronger cuts by combining the branching constraints with other constraints of the model. The computational experiments conducted on instances from the literature show that our algorithm finds optimal integer solutions faster than other approaches. A set of computational results on random instances is also reported.

DOI : 10.1051/ro:2008027
Classification : 90C10, 90C57
Mots-clés : pattern minimization problem, column generation, cutting planes, branch-and-bound, dual feasible functions 
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     author = {Alves, Cl\'audio and Val\'erio de Carvalho, J. M.},
     title = {A branch-and-price-and-cut algorithm for the pattern minimization problem},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {435--453},
     publisher = {EDP-Sciences},
     volume = {42},
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Alves, Cláudio; Valério de Carvalho, J. M. A branch-and-price-and-cut algorithm for the pattern minimization problem. RAIRO - Operations Research - Recherche Opérationnelle, Tome 42 (2008) no. 4, pp. 435-453. doi : 10.1051/ro:2008027. http://www.numdam.org/articles/10.1051/ro:2008027/

[1] C. Alves, Cutting and packing: problems, models and exact algorithms. Ph.D. Thesis, Universidade do Minho (2005).

[2] J.M. Allwood and C.N. Goulimis. Reducing the number of patterns in one-dimensional cutting stock problems. Technical report, Electrical Engineering Department, Imperial College, London (1988).

[3] G. Belov. Problems, models and algorithms in one- and two- dimensional cutting. Ph.D. Thesis, Dresden University (2003).

[4] C.-L.S. Chen, S.M. Hart, and W.M. Tham. A simulated annealing heuristic for the one-dimensional cutting stock problem. Eur. J. Oper. Res. 93 (1996) 522-535. | Zbl

[5] S. Fekete and J. Schepers. New classes of fast lower bounds for bin packing problems. Math. Program. 91 (2001) 11-31. | MR | Zbl

[6] H. Foerster and G. Waescher. Pattern reduction in one-dimensional cutting stock problems. Int. J. Prod. Res. 38 (2000) 1657-1676. | Zbl

[7] T. Gau and G. Waescher. CUTGEN1: A problem generator for the standard one-dimensional cutting stock problem. Eur. J. Oper. Res. 84 (1995) 572-579. | Zbl

[8] P.C. Gilmore and R.E. Gomory. A linear programming approach to the cutting stock problem. Oper. Res. 9 (1961) 849-859. | MR | Zbl

[9] C. Goulimis, Optimal solutions for the cutting stock problem. Eur. J. Oper. Res. 44 (1990) 197-208. | Zbl

[10] R.W. Haessler, A heuristic programming solution to a nonlinear cutting stock problem. Manage. Sci. 17 (1971) 793-802. | Zbl

[11] R.W. Haessler. Controlling cutting pattern changes in one-dimensional trim problems. Oper. Res. 23 (1975) 483-493. | Zbl

[12] R.E. Johnston, Rounding algorithms for cutting stock problems. Asia-Pac. Oper. Res. J. 3 (1986) 166-171. | Zbl

[13] R.E. Johnston. Cutting patterns and cutter schedules. Asia-Pac. Oper. Res. J. 4 (1987) 3-14.

[14] S. Martello and P. Toth, Knapsack Problems. Wiley, New York (1990). | MR | Zbl

[15] G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization. Wiley, New York (1988). | MR | Zbl

[16] J. Teghem, M. Pirlot, and C. Antoniadis, Embedding of linear programming in a simulated annealing algorithm for solving a mixed integer production planning problem. J. Comput. Appl. Math. 64 (1995) 91-102. | MR | Zbl

[17] S. Umetani, M. Yagiura, and T. Ibaraki, One-dimensional cutting stock problem to minimize the number of different patterns. Eur. J. Oper. Res. 146 (2003) 388-402. | MR | Zbl

[18] F. Vanderbeck. Exact algorithm for minimising the number of setups in the one-dimensional cutting stock problem. Oper. Res. 48 (2000) 915-926. | MR | Zbl

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