Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T23:19:06.390Z Has data issue: false hasContentIssue false
  • English
  • Français

A branch-and-price-and-cut algorithmfor the pattern minimization problem

Published online by Cambridge University Press:  04 April 2009

Cláudio Alves  and
J.M. Valério de Carvalho
Cláudio Alves
Affiliation:
Escola de Engenharia, Universidade do Minho, 4710-057 Braga, Portugal; [email protected]
J.M. Valério de Carvalho
Affiliation:
Escola de Engenharia, Universidade do Minho, 4710-057 Braga, Portugal; [email protected]
Get access

Abstract

In cutting stock problems, after an optimal (minimal stock usage)cutting plan has been devised, one might want to further reduce theoperational costs by minimizing the number of setups. A setupoperation occurs each time a different cutting pattern begins to beproduced. The related optimization problem is known as the PatternMinimization Problem, and it is particularly hard to solve exactly.In this paper, we present different techniques to strengthen aformulation proposed in the literature. Dual feasible functions areused for the first time to derive valid inequalities from differentconstraints of the model, and from linear combinations of constraints. A new arcflow formulation is also proposed. This formulation is used todefine the branching scheme of our branch-and-price-and-cutalgorithm, and it allows the generation of even stronger cuts bycombining the branching constraints with other constraints of themodel. The computational experiments conducted on instances from theliterature show that our algorithm finds optimalinteger solutions faster than other approaches. A set of computationalresults on random instances is also reported.

Keywords

Pattern Minimization Problemcolumn generationcuttingplanesbranch-and-bounddual feasible functions.
Type
Research Article
Information
RAIRO - Operations Research , Volume 42 , Issue 4: ALIO/EURO V Conference on Combinatorial Optimization , October 2008 , pp. 435 - 453
Copyright
© EDP Sciences, ROADEF, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

C. Alves, Cutting and packing: problems, models and exact algorithms. Ph.D. Thesis, Universidade do Minho (2005).
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).
G. Belov. Problems, models and algorithms in one- and two- dimensional cutting. Ph.D. Thesis, Dresden University (2003).
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.
S. Fekete and J. Schepers. New classes of fast lower bounds for bin packing problems. Math. Program. 91 (2001) 11–31.
H. Foerster and G. Waescher. Pattern reduction in one-dimensional cutting stock problems. Int. J. Prod. Res. 38 (2000) 1657–1676.
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.
P.C. Gilmore and R.E. Gomory. A linear programming approach to the cutting stock problem. Oper. Res. 9 (1961) 849–859.
Goulimis, C., Optimal solutions for the cutting stock problem. Eur. J. Oper. Res. 44 (1990) 197–208. CrossRef
Haessler, R.W., A heuristic programming solution to a nonlinear cutting stock problem. Manage. Sci. 17 (1971) 793–802. CrossRef
R.W. Haessler. Controlling cutting pattern changes in one-dimensional trim problems. Oper. Res. 23 (1975) 483–493.
Johnston, R.E., Rounding algorithms for cutting stock problems. Asia-Pac. Oper. Res. J. 3 (1986) 166–171.
R.E. Johnston. Cutting patterns and cutter schedules. Asia-Pac. Oper. Res. J. 4 (1987) 3–14.
S. Martello and P. Toth, Knapsack Problems. Wiley, New York (1990).
G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization. Wiley, New York (1988).
Teghem, J., Pirlot, M., and Antoniadis, C., 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. CrossRef
Umetani, S., Yagiura, M., and Ibaraki, T., One-dimensional cutting stock problem to minimize the number of different patterns. Eur. J. Oper. Res. 146 (2003) 388–402. CrossRef
F. Vanderbeck. Exact algorithm for minimising the number of setups in the one-dimensional cutting stock problem. Oper. Res. 48 (2000) 915–926.