# Multicommodity network flows

Reference work entry

First Online: 25 October 2005

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

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## 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*v***x**^{k}= [*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*v***c**^{k}= [*c*^{1k},...,*c*^{mk}]. The total capacity of arc*j*is denoted by*b*^{j}with corresponding vector*v = [***b***b*^{1},...,*b*^{m}]. Mathematically, the multicommodity minimal cost network flow problem may be defined as follows:This is a preview of subscription content, log in to check access.

## 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

## Copyright information

© Kluwer Academic Publishers 2001

## How to cite

- Cite this entry as:
- 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

## About this entry

- First Online 25 October 2005
- DOI https://doi.org/10.1007/1-4020-0611-X
- 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