A new parallel approach to the constrained two-dimensional cutting stock problem

  • Stefan Tschöke
  • Norbert Holthöfer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 980)


In this paper we present a new parallelization of an efficient best-first branch-and-bound algorithm to solve the constrained two-dimensional single stock guillotine cutting stock problem (CSP) to optimality.

The underlying sequential branch-and-bound algorithm is based on an exact version of Wang's heuristic suggested by Viswanathan and Bagchi. In our algorithm we improve the upper bound and introduce duplicate pruning.

For an efficient parallelization we developed a special communication structure, as due to the unusual branching strategy and detection of duplicates a standard parallelization of the branch-and-bound approach cannot be applied to this problem. This structure allows a dynamic and fully distributed load-balancing using a direct neighbor strategy.

Computational results on two different parallel systems are presented. The implementation is system-independent using the portable parallel branch-and-bound library (PPBB-LIB) developed in Paderborn and can easily be ported to other systems.


two-dimensional cutting parallel branch-and-bound combinatorial optimization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. Christofides, C. Whitlock. An Algorithm for Two-Dimensional Cutting Problems. Operations Research 25 (1977), pp. 30–44.Google Scholar
  2. 2.
    H. Dyckhoff. A typology of cutting and packing problems. European Journal of Operational Research 44 (1990), pp. 145–159.CrossRefGoogle Scholar
  3. 3.
    B. Gendron and T. G. Crainic. Parallel branch-and-bound algorithms: survey and synthesis. Operations Research 42 (1994), pp. 1042–1066.MathSciNetGoogle Scholar
  4. 4.
    P. C. Gilmore and R. E. Gomory. The Theory and Computation of Knapsack Functions. Operations Research 14 (1966), pp. 1045–1075.Google Scholar
  5. 5.
    B. Kröger, O. Vornberger. A Parallel Branch-and-Bound Approach for Solving a two-dimensional Cutting Stock Problem. Osnabrücker Schriften zur Mathematik, September 1990.Google Scholar
  6. 6.
    R. Lüling and B. Monien. Load Balancing for distributed branch and bound algorithms. Proceedings of 6th International Parallel Processing Symposium, pp. 543–548, March 1992.Google Scholar
  7. 7.
    R. Lüling and B. Monien. A dynamic distributed load balancing algorithm with provable good performance. Proceedings of 5th ACM Symposium on Parallel Algorithms and Architectures, pp. 164–173, 1993.Google Scholar
  8. 8.
    D. S. Nau, Vipin Kumar and L. Kanal. General Branch-and-Bound and its Relation to A * and AO *. Artificial Intelligence, Vol. 23, 1984.Google Scholar
  9. 9.
    J. F. Oliveira and J. S. Ferreira. An improved version of Wang's algorithm for two-dimensional cutting problems. European Journal of Operational Research 44 (1990), pp. 256–266.CrossRefGoogle Scholar
  10. 10.
    Paul E. Sweeney and Elizabeth Ridenour Paternoster. Cutting and Packing Problems: A Categorized, Application-Oriented Research Bibliography. Journal of the Operational Research Society 43 (1992), pp. 691–706.Google Scholar
  11. 11.
    S. Tschöke, R. Lüling, and B. Monien. Solving the traveling salesman problem with a distributed branch-and-bound algorithm on a 1024 processor network. Proceedings of International Parallel Processing Symposium, 1995.Google Scholar
  12. 12.
    S. Tschöke and T. Polzer. Portable parallel branch-and-bound library: User manual. Technical report, University of Paderborn, 1995.Google Scholar
  13. 13.
    K. V. Viswanathan and A. Bagchi. Best-first Search Methods For Constrained Two-Dimensional Cutting Stock Problems. Operations Research 41 (1993), pp. 768–776.Google Scholar
  14. 14.
    P. Y. Wang. Two Algorithms for Constrained Two-Dimensional Cutting Stock Problems. Operations Research 31 (1983), pp. 573–586.Google Scholar
  15. 15.
    C.-Z. Xu, R. Lüling, B. Monien, and F. C. M. Lau. An analytical comparison of nearest neighbor algorithms for load balancing in parallel computers. Proceedings of 9th International Parallel Processing Symposium, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Stefan Tschöke
    • 1
  • Norbert Holthöfer
    • 1
  1. 1.Department of Computer ScienceUniversity of PaderbornPaderbornGermany

Personalised recommendations