Skip to main content
SpringerLink
Log in

Integer multicommodity flow problems

  • Conference paper
  • First Online:
Integer Programming and Combinatorial Optimization (IPCO 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1084))

Abstract

We present a column generation model and solution approach for large integer multicommodity flow problems. We solve the model using branch-and-bound, with bounds provided by linear programs at each node of the branch-and-bound tree. Since the model contains one variable for each origin-destination path, for every commodity, the linear programming relaxation is solved using column generation, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality. Methods for speeding up the solution of the linear program are presented. Also, we devise new branching rules that allow columns to be generated efficiently at each node of the branch-and-bound tree. Computational results are presented for a set of test problems arising from a transportation application.

This research has been supported by the following grants and contracts: NSF DDM-9058074, NSF DMI-9502502

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R.K. Ahuja, T.L. Magnanti, and J.B. Orlin (1993). Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs, NJ.

    Google Scholar 

  2. L.H. Appelgren (1969). A column generation algorithm for a ship scheduling problem. Transportation Science 3, 53–68.

    Google Scholar 

  3. C. Barnhart, C.A. Hane, E.L. Johnson, and G. Sigismondi (1991). An Alternative Formulation and Solution Strategy for Multi-Commodity Network Flow Problems. Report COC-9102, Georgia Institute of Technology, Atlanta, Georgia.

    Google Scholar 

  4. C. Barnhart, E.L. Johnson, G.L. Nemhauser, M.W.P. Savelsbergh, and P.H. Vance (1995). Branch-and-Price: Column Generation for Solving Huge Integer Programs. Report COC-9502, Georgia Institute of Technology, Atlanta, Georgia.

    Google Scholar 

  5. CPLEX Optimization, Inc. (1990). Using the CPLEXTM Linear Optimizer.

    Google Scholar 

  6. G.B. Dantzig and R.M. Van Slyke (1967). Generalized Upper Bounding Techniques. Journal Computer System Sci. 1, 213–226.

    Google Scholar 

  7. G.B. Dantzig and P. Wolfe (1960) Decomposition Principle for Linear Programs. Operations Research 8, 108–111.

    Google Scholar 

  8. J. Desrosiers, Y. Dumas, M.M. Solomon, and F. Soumis (1994). Time constrained routing and scheduling. M.E. Ball, T.L. Magnanti, C. Monma, And G.L. Nemhauser (eds.). Handbooks in Operations Research and Management Science, Volume on Networks, to appear.

    Google Scholar 

  9. IBM Corporation (1990). Optimization Subroutine Library, Guide and Reference.

    Google Scholar 

  10. E.L. Johnson (1989). Modeling and Strong Linear Programs for Mixed Integer Programming. S.W. Wallace (ed.) Algorithms and Model Formulations in Mathematical Programming. NATO ASI Series 51, 1–41.

    Google Scholar 

  11. G.L. Nemhauser and L.A. Wolsey (1988). Integer and Combinatorial Optimization. Wiley, Chichester.

    Google Scholar 

  12. G.L. Nemhauser, M.W.P. Savelsbergh, and G.C. Sigismondi (1994). MINTO, a Mixed INTeger Optimizer. Operations Research Letters 15, 47–58.

    MathSciNet  Google Scholar 

  13. M. Parker and J. Ryan (1994). A column generation algorithm for bandwidth packing. Telecommunications Systems, to appear.

    Google Scholar 

  14. J.B. Rosen (1964) Primal Partition Programming for Block Diagonal Matrices. Numerische Mathematik 6, 250–260.

    Article  Google Scholar 

  15. D.M. Ryan and B.A. Foster (1981). An integer programming approach to scheduling. A. Wren (ed.) Computer Scheduling of Public Transport Urban Passenger Vehicle and Crew Scheduling, North-Holland, Amsterdam, 269–280.

    Google Scholar 

  16. P.H. Vance, C. Barnhart, E.L. Johnson, G.L. Nemhauser, D. Mahidara, A. Krishna, and R. Rebello (1994). Exceptions in Crew Planning. ORSA/TIMS Detroit, Michigan.

    Google Scholar 

  17. F. Vanderbeck and L.A. Wolsey (1994). An Exact Algorithm for IP Column Generation. CORE Discussion Papaer, Universite Catholique de Louvain, Belgium.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Massachusetts Institute of Technology, Center for Transportation Studies, 02139, Cambridge, MA, USA

    Cynthia Barnhart

  2. CAPS Logistics, 30334, Atlanta, Georgia, USA

    Christopher A. Hane

  3. Industrial and Systems Engineering, Auburn University, 36849, Auburn, AL, USA

    Pamela H. Vance

Authors
  1. Cynthia Barnhart
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christopher A. Hane
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pamela H. Vance
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

William H. Cunningham S. Thomas McCormick Maurice Queyranne

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Barnhart, C., Hane, C.A., Vance, P.H. (1996). Integer multicommodity flow problems. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 1996. Lecture Notes in Computer Science, vol 1084. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61310-2_5

Download citation

  • DOI: https://doi.org/10.1007/3-540-61310-2_5

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61310-7

  • Online ISBN: 978-3-540-68453-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation