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The simplex algorithm for multicommodity networks
Author(s) -
Detlefsen Nina K.,
Wallace Stein W.
Publication year - 2002
Publication title -
networks
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.977
H-Index - 64
eISSN - 1097-0037
pISSN - 0028-3045
DOI - 10.1002/net.10006
Subject(s) - simplex , simplex algorithm , dimension (graph theory) , matrix (chemical analysis) , flow network , algorithm , mathematical optimization , multi commodity flow problem , simple (philosophy) , mathematics , computer science , inverse , variable (mathematics) , focus (optics) , flow (mathematics) , linear programming , combinatorics , mathematical analysis , philosophy , materials science , geometry , physics , epistemology , optics , composite material
We consider multicommodity network flow problems, where external flow is allowed to vary and where flows of individual commodities may be constrained. For this problem, we describe the simplex algorithm. The simplex algorithm is based upon the inverse of the basis matrix. We discuss an approach where we only have to invert a working matrix with dimension at most equal to the number of arcs in the network. The dimension is independent of the number of commodities. The reduced cost of a nonbasic variable is found only using the working matrix that is explicitly inverted and some simple network operations. We also discuss how to pivot in the working matrix. The focus is on the detailed interpretation of the network structures arising in terms of cycles for the individual commodities. © 2002 John Wiley & Sons, Inc.