Annals of Operations Research

, Volume 86, Issue 0, pp 629–659 | Cite as

Exact solution of bin‐packing problems using column generation and branch‐and‐bound

  • J.M. Valério de Carvalho


We explore an arc flow formulation with side constraints for the one‐dimensionalbin‐packing problem. The model has a set of flow conservation constraints and a set ofconstraints that force the appropriate number of items to be included in the packing. Themodel is tightened by fixing some variables at zero level, to reduce the symmetry of thesolution space, and by introducing valid inequalities. The model is solved exactly using abranch‐and‐price procedure that combines deferred variable generation and branch‐and‐bound.At each iteration, the subproblem generates a set of columns, which altogether correspondto an attractive valid packing for a single bin. We describe this subproblem, and theway it is modified in the branch‐and‐bound phase, after the branching constraints are addedto the model. We report the computational times obtained in the solution of the bin‐packingproblems from the OR‐Library test data sets. The linear relaxation of this model provides astrong lower bound for the bin‐packing problem and leads to tractable branch‐and‐boundtrees for the instances under consideration.


Exact Solution Test Data Computational Time Variable Generation Column Generation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    R. Ahuja, T. Magnanti and J. Orlin, Network Flows: Theory, Algorithms and Applications, Prentice-Hall, Englewood Cliffs, NJ, 1993.Google Scholar
  2. [2]
    J.E. Beasley, OR-library: Distributing test problems by electronic mail, Journal Operational Research Society 41(1990)1069-1072.Google Scholar
  3. [3]
    J.M. Valério de Carvalho, A model for the one-dimensional cutting-stock problem, Working Paper, 1996.Google Scholar
  4. [4]
    J. Desrosiers, Y. Dumas, M. M. Salomon and F. Soumis, Time constrained routing and scheduling, in: Handbooks in Operations Research & Management Science 8: Network Routing, Elsevier Science, 1995.Google Scholar
  5. [5]
    J. Desrosiers, F. Soumis and M. Desrochers, Routing with time windows by column generation, Networks 14(1984)545-565.Google Scholar
  6. [6]
    E.G. Coffman. Jr., M.R. Garey and D.S. Johnson, Approximation algorithms for bin-packing — an updated survey, in: Algorithm Design for Computer System Design, Springer, Berlin, 1984.Google Scholar
  7. [7]
    E.V. Denardo, Dynamic Programming Models and Applications, Prentice-Hall, NJ, 1982.Google Scholar
  8. [8]
    E. Falkenauer, A hybrid grouping genetic algorithm for bin packing, International Journal of Computers and Operations Research (1995), to appear.Google Scholar
  9. [9]
    M. Fieldhouse, The duality gap in trim problems, SICUP Bulletin 5 (November 1990).Google Scholar
  10. [10]
    M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H.Freeman, San Francisco, 1979.Google Scholar
  11. [11]
    P.C. Gilmore and R.E. Gomory, A linear programming approach to the cutting stock problem — part ii, Operations Research 11(1963)863-888.Google Scholar
  12. [12]
    O. Marcotte, The cutting stock problem and integer rounding, Mathematical Programming 33(1985)82-92.Google Scholar
  13. [13]
    O. Marcotte, An instance of the cutting stock problem for which the rounding property does not hold, Operations Research Letters 4(1986)239-243.Google Scholar
  14. [14]
    R.M. Marsten, The design of the XMP linear programming library, ACM Transactions on Mathematical Software 7(1981)481-497.Google Scholar
  15. [15]
    S. Martello and P. Toth, Knapsack Problems: Algorithms and Computer Implementations, Wiley, New York, 1990.Google Scholar
  16. [16]
    G. Nemhauser and S. Park, A polyhedral approach to edge coloring, Operations Research Letters 10(1991)315-322.Google Scholar
  17. [17]
    G. Nemhauser and L. Wolsey, Integer and Combinatorial Optimization, Wiley, New York, 1988.Google Scholar
  18. [18]
    C.H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, NJ, 1982.Google Scholar
  19. [19]
    Salford, Salford Software, 1995.Google Scholar
  20. [20]
    J.F. Shapiro, Dynamic programming algorithms for the integer programming problem I: The integer programming problem viewed as a knapsack type problem, Operations Research 16(1968)103-121.Google Scholar
  21. [21]
    D. Simchi-Levi, New worst-case results for the bin-packing problem, Naval Research Logistics 41(1994)579-585.Google Scholar
  22. [22]
    P. Vance, C. Barnhart, E. L. Johnson and G. L. Nemhauser, Solving binary cutting stock problems by column generation and branch-and-bound, Computational Optimization and Applications 3(1994)111-130.Google Scholar
  23. [23]
    F. Vanderbeck, On integer programming decomposition and ways to enforce integrality in the master, Research Papers in Management Studies, 1994-1995, No. 29 (revised May 1996), University of Cambridge, 1996.Google Scholar
  24. [24]
    F. Vanderbeck, Computational study of a column generation algorithm for binpacking and cutting stock problems, Research Papers in Management Studies, 1996, No. 14, University of Cambridge, 1996.Google Scholar

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© Kluwer Academic Publishers 1999

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  • J.M. Valério de Carvalho

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