Computational Optimization and Applications

, Volume 42, Issue 2, pp 303–326

# Approximate and exact algorithms for the double-constrained two-dimensional guillotine cutting stock problem

• M. Hifi
• R. M’Hallah
Article

## Abstract

In this paper, we propose approximate and exact algorithms for the double constrained two-dimensional guillotine cutting stock problem (DCTDC). The approximate algorithm is a two-stage procedure. The first stage attempts to produce a starting feasible solution to DCTDC by solving a single constrained two dimensional cutting problem, CDTC. If the solution to CTDC is not feasible to DCTDC, the second stage is used to eliminate non-feasibility. The exact algorithm is a branch-and-bound that uses efficient lower and upper bounding schemes. It starts with a lower bound reached by the approximate two-stage algorithm. At each internal node of the branching tree, a tailored upper bound is obtained by solving (relaxed) knapsack problems. To speed up the branch and bound, we implement, in addition to ordered data structures of lists, symmetry, duplicate, and non-feasibility detection strategies which fathom some unnecessary branches. We evaluate the performance of the algorithm on different problem instances which can become benchmark problems for the cutting and packing literature.

## Keywords

Combinatorial optimization Cutting problems Dynamic programming Single constrained knapsack problem

## References

1. 1.
Baker, B.S., Coffman, E.G. Jr., Rivest, R.L.: Orthogonal packing in two dimensions. SIAM J. Comput. 9, 846–855 (1980)
2. 2.
Beasley, J.E.: Algorithms for unconstrained two-dimensional guillotine cutting. J. Oper. Res. Soc. 36, 297–306 (1985)
3. 3.
Belov, G., Scheithauer, G.: A branch-and-cut-and-price algorithm for one-dimensional stock cutting and two-dimensional two-stage cutting. Eur. J. Oper. Res. 171, 85–106 (2006)
4. 4.
Blazewicz, J., Moret-Salvador, A., Walkowiak, R.: Parallel tabu search approaches for two-dimensional cutting. Parallel Process. Lett. 14, 23–32 Google Scholar
5. 5.
Bortfeldt, A.: A genetic algorithm for the two-dimensional strip packing problem with rectangular pieces. Eur. J. Oper. Res. (2005), available online Google Scholar
6. 6.
Caprara, A., Monaci, M.: On the 2-dimensional knapsack problems. Oper. Res. Lett. 32, 5–14 (2004)
7. 7.
Christofides, N., Whitlock, C.: An algorithm for two-dimensional cutting problems. Oper. Res. 25, 31–44 (1977)
8. 8.
Christofides, N., Hadjiconstantinou, E.: An exact algorithm for orthogonal 2-D cutting problems using guillotine cuts. Eur. J. Oper. Res. 83, 21–38 (1995)
9. 9.
Cui, Y.: Generating optimal T-shape cutting patterns for rectangular blanks. Proc. Inst. Mech. Eng. Part B: J. Eng. Manuf. 218/B8, 857–866 (2004)
10. 10.
Cui, Y., Wang, Z., Li, J.: Exact and heuristic algorithms for staged cutting problems. Proc. Inst. Mech. Eng. Part B: J. Eng. Manuf. 219/B2, 201–208 (2005)
11. 11.
Cung, V.-D., Hifi, M.: Handling lower bound constraints in two-dimensional cutting problems. In: ISMP 2000, The 17th Symposium on Mathematical Programming, Atlanta, 7–11 August 2000 Google Scholar
12. 12.
Cung, V.-D., Hifi, M., Le Cun, B.: Constrained two-dimensional cutting stock problems: a best-first branch-and-bound algorithm. Int. Trans. Oper. Res. 7, 185–210 (2000)
13. 13.
Cung, V.-D., Hifi, M., Le Cun, B.: Constrained two-dimensional cutting stock problems: the NMVB approach and the duplicate test revisited. Working Paper, Série Bleue No 2000.127 (CERMSEM), Maison des Sciences Economiques, Université Paris 1 (2000) Google Scholar
14. 14.
Dyckhoff, H.: A typology of cutting and packing problems. Eur. J. Oper. Res. 44, 145–159 (1990)
15. 15.
Dikili, A.C.: A new approach for the solution of the two-dimensional guillotine-cutting problem in ship production. Ocean Eng. 31, 1193–1203 (2004)
16. 16.
Fayard, D., Zissimopoulos, V.: An approximation algorithm for solving unconstrained two-dimensional knapsack problems. Eur. J. Oper. Res. 84, 618–632 (1995)
17. 17.
Fayard, D., Hifi, M., Zissimopoulos, V.: An efficient approach for large-scale two-dimensional guillotine cutting stock problems. J. Oper. Res. Soc. 49, 1270–1277 (1998)
18. 18.
Fekete, S.P., Schepers, J.: A general framework for bounds for higher-dimensional orthogonal packing problems. Math. Method. Oper. Res. 60, 311–329 (2004)
19. 19.
Gilmore, P., Gomory, R.: Multistage cutting problems of two and more dimensions. Oper. Res. 13, 94–119 (1965)
20. 20.
Gilmore, P., Gomory, R.: The theory and computation of knapsack functions. Oper. Res. 14, 1045–1074 (1966)
21. 21.
Herz, J.C.: A recursive computing procedure for two-dimensional stock cutting. IBM J. Res. Dev. 16, 462–469 (1972)
22. 22.
Hifi, M.: An improvement of Viswanathan and Bagchi’s exact algorithm for cutting stock problems. Comput. Oper. Res. 24, 727–736 (1997)
23. 23.
Hifi, M., M’Hallah, R.: Strip generation algorithms for two-staged two-dimensional cutting stock problems. Eur. J. Oper. Res. 172, 515–527 (2006)
24. 24.
Hifi, M., M’Hallah, R.: An exact algorithm for constrained two-dimensional two-staged cutting problems. Oper. Res. 53, 140–150 (2005)
25. 25.
Hifi, M., Zissimopoulos, V.: A recursive exact algorithm for weighted two-dimensional cutting. Eur. J. Oper. Res. 91, 553–564 (1996)
26. 26.
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin (2004). ISBN:3-540-40286-1
27. 27.
Lodi, A., Martello, S., Monaci, M.: Two-dimensional packing problems: A survey. Eur. J. Oper. Res. 141, 241–252 (2002)
28. 28.
Lodi, A., Monaci, M.: Integer linear programming models for 2-staged two-dimensional Knapsack problems. Math. Program. 94, 257–278 (2003)
29. 29.
Morabito, R., Arenales, M.: Staged and constrained two-dimensional guillotine cutting problems: An and/or-graph approach. Eur. J. Oper. Res. 94, 548–560 (1996)
30. 30.
Mumford-Valenzuela, C.L., Vick, J., Wang, P.Y.: Heuristics for large strip packing problems with guillotine patterns: An empirical study. In: Metaheuristics: Computer Decision-Making, pp. 501–522. Kluwer Academic, Dordrecht (2003) Google Scholar
31. 31.
Suliman, S.M.A.: A sequential heuristic procedure for the two-dimensional cutting-stock problem. Int. J. Prod. Econ. 99, 177–185 (2006)
32. 32.
Viswanathan, K.V., Bagchi, A.: Best-first search methods for constrained two-dimensional cutting stock problems. Oper. Res. 41, 768–776 (1993)
33. 33.
Wang, P.Y.: Two algorithms for constrained two-dimensional cutting stock problems. Oper. Res. 31, 573–586 (1983)
34. 34.
Wäescher, G., Haussner, H., Schumann, H.: An improved typology of cutting and packing problems. Eur. J. Oper. Res. 183, 1109–1130 (2007)