# Multicommodity network flows

Reference work entry

First Online: 25 October 2005

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

## 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.

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## 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

## 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