Skip to main content
SpringerLink
Log in
Book cover

Encyclopedia of Operations Research and Management Science pp 529–531Cite as

Multicommodity network flows

  • Reference work entry
  • First Online:

  • 51 Accesses

INTRODUCTION

The multicommodity minimal cost network flow problem may be described in terms of a distribution problem over a network [V, E], where V is the node set with order n and E is the arc set with order m. The decision variable x jk denotes the flow of commodity k through arc j, and the vector of all flows of commodity k is denoted by xvk = [x 1k,..., x mk]. The unit cost of flow of commodity k through arc j is denoted by c jk and the corresponding vector of costs by cvk = [c 1k,..., c mk]. The total capacity of arc j is denoted by b j with corresponding vector bv = [b 1,..., b m]. Mathematically, the multicommodity minimal cost network flow problem may be defined as follows:

where K denotes the number of commodities, Av is a node-arc incidence matrix for [V, E], rvk is the requirements vector for commodity k, and uvk is the vector of upper bounds for decision variable xvk.

Multicommodity network flow problems are extensively studied because of their numerous applications and...

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   532.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

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

    Google Scholar 

  2. Ali, A., Helgason, R., Kennington, J., and Lall, H. (1980). “Computational Comparison Among Three Multicommodity Network Flow Algorithms,” Operations Research, 28, 995–1000.

    Google Scholar 

  3. Ali, A., Barnett, D., Farhangian, K., Kennington, J., McCarl, B., Patty, B., Shetty, B., and Wong, P. (1984). “Multicommodity Network Flow Problems: Applications and Computations,” IIE Transactions, 16, 127–134.

    Google Scholar 

  4. Assad, A.A. (1978). “Multicommodity Network Flows–A Survey,” Networks, 8, 37–91.

    Google Scholar 

  5. Barnhart, C. (1993). “Dual Ascent Methods for Large-Scale Multicommodity Flow Problems,” Naval research Logistics, 40, 305–324.

    Google Scholar 

  6. Bellmore, M., Bennington, G., and Lubore, S. (1971). “A Multivehicle Tanker Scheduling Problem,” Trans. Sci., 5, 36–47.

    Google Scholar 

  7. Castro, J. and Nabona, N. (1996). “An Implementation of Linear and Nonlinear Multicommodity Network Flows,” European Jl. Operational Research, 92, 37–53.

    Google Scholar 

  8. Chen, R. and Meyer, R. (1988). “Parallel Optimization for Traffic Assignment,” Math. Programming, 42, 327–345.

    Google Scholar 

  9. S. Clark and Surkis, J. (1968). “An Operations research Approach to Racial Desegregation of School Systems,” Socio-Econ. Plan. Sci., 1, 259–272.

    Google Scholar 

  10. Evans, J. (1978). “The Simplex Method for Integral Multicommodity Networks,” Naval Research Logistics, 25, 31–38.

    Google Scholar 

  11. Evans, J. and Jarvis, J. (1978). “Network Topology and Integral Multicommodity Flow Problems,” Net-works, 8, 107–120.

    Google Scholar 

  12. Farvolden, J.M. and Powell, W.B. (1990). “A Primal Partitioning Solution for Multicommodity Network Flow Problems,” Working Paper 90-04, Department of Industrial Engineering, University of Toronto, Canada.

    Google Scholar 

  13. Farvolden, J.M., Powell, W.B., and Lustig, I.J. (1993). “A Primal Partitioning Solution for the Arc-Chain Formulation of a Multicommodity Network Flow Problem,” Operations Research, 41, 669–693.

    Google Scholar 

  14. Gautier, A. and Granot, F. (1995). “Forest Management: A Multicommodity Flow Formulation and Sensitivity Analysis,” Management Science, 41, 1654–1668.

    Google Scholar 

  15. Gendron, B. and Crainic, T.G. (1997). “A Parallel Branch-and-Bound Algorithm for Multicommodity Location with Balancing Requirements,”? Computers and Operations Research, 24, 829–847.

    Google Scholar 

  16. Gersht, A. and Shulman, A. (1987). “A New Algorithm for the Solution of the Minimum Cost Multicommodity Flow Problem,” Proceedings of the IEEE Conference on Decision and Control, 26, 748–758.

    Google Scholar 

  17. Graves, G.W. and McBride, R.D. (1976). “The Factorization Approach to Large Scale Linear Programming,” Math. Programming, 10, 91–110.

    Google Scholar 

  18. Grigoriadis, M.D. and White, W.W. (1972). “A Partitioning Algorithm for the Multi-commodity Network Flow Problem,” Math. Programming, 3, 157–177.

    Google Scholar 

  19. Hartman, J.K. and Lasdon, L.S. (1972). “A Generalized Upper Bounding Algorithm for Multicommodity Net-work Flow Problems,” Networks, 1, 331–354.

    Google Scholar 

  20. Held, M., P. Wolfe, and Crowder, H. (1974). “Validation of Subgradient Optimization,” Math. Programming, 6, 62–88.

    Google Scholar 

  21. Kennington, J. and Shalaby, M. (1977). “An Effective Subgradient Procedure for Minimal Cost Multicommodity Flow Problems,” Management Science, 23, 994–1004.

    Google Scholar 

  22. Kennington, J.L. (1978). “A Survey of Linear Cost Multicommodity Network Flows,” Operations Research, 26, 209–236.

    Google Scholar 

  23. Kennington, J.L. and Helgason, R. (1980). Algorithms for Network Programming, John Wiley, New York.

    Google Scholar 

  24. LeBlanc, L.J. (1973). “Mathematical Programming Algorithms for Large Scale Network Equilibrium and Network Design Problems,” Unpublished Dissertation, Industrial Engineering and Management Sciences Department, Northwestern University.

    Google Scholar 

  25. Liu, C-M. (1997). “Network Dual Steepest-Edge Methods for Solving Capacitated Multicommodity Net-work Problems,” Computers and Industrial Engineering, 33, 697–700.

    Google Scholar 

  26. McBride, R. (1998). “Advances in Solving the Multi-commodity-Flow Problem,” Interfaces, 28(2), 32–41.

    Google Scholar 

  27. Naniwada, M. (1969). “Multicommodity Flows in a Communications Network,” Electronics and Communications in Japan, 52-A, 34–41.

    Google Scholar 

  28. Pinar, M.C. and Zenios, S.A. (1990). “Parallel Decomposition of Multicommodity Network Flows Using Smooth Penalty Functions,” Technical Report 90-12-06, Department of Decision Sciences, Wharton School, University of Pennsylvania, Philadelphia.

    Google Scholar 

  29. Popken, D.A. (1994). “An Algorithm for the Multiattribute, Multicommodity Flow Problem with Freight Consolidation and Inventory Costs,” Operations Research, 42, 274–286.

    Google Scholar 

  30. Potts, R.B. and Oliver, R.M. (1972). Flows in Transportation Networks, Academic Press, New York.

    Google Scholar 

  31. Schneur, R. and Orlin, J.B. (1998). “A Scaling Algorithm for Multicommodity Flow Problems,” Operations Research, 46, 231–246.

    Google Scholar 

  32. Shetty, B. and Muthukrishnan, R. (1990). “A Parallel Projection for the Multicommodity Network Model,” Jl. Operational Research, 41, 837–842.

    Google Scholar 

  33. Swoveland, C. (1971). “Decomposition Algorithms for the Multi-Commodity Distribution Problem,” Working Paper, No. 184, Western Management Science Institute, University of California, Los Angeles.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Texas A&M University, College Station, USA

    Bala Shetty

Authors
  1. Bala Shetty
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Robert H. Smith School of Business, University of Maryland, College Part, Maryland, USA

    Saul I. Gass

  2. School of Information Technology & Engineering, George Mason University, Fairfax, Virginia, USA

    Carl M. Harris

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Kluwer Academic Publishers

About this entry

Cite this entry

Shetty, B. (2001). Multicommodity network flows. In: Gass, S.I., Harris, C.M. (eds) Encyclopedia of Operations Research and Management Science. Springer, New York, NY. https://doi.org/10.1007/1-4020-0611-X_645

Download citation

  • DOI: https://doi.org/10.1007/1-4020-0611-X_645

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-7923-7827-3

  • Online ISBN: 978-1-4020-0611-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation