Minimum Cost Network Flow
Problems
The transportation, assignment, transshipment,
shortest path, and maximum flow problems are all
special cases of
minimum cost network flow
problems (MCNFP)
. Any MCNFP can be
formulated as an LP. It can be also solved by a
generalization of the transportation simplex
method called the network simplex method.

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To define MCNFP, let
X
ij
= number of units of flow sent from node i to
node j through arc(i,j)
b
i
= net supply (outflow-inflow) at node i
c
ij
= cost of transporting 1 unit of flow from node i to
node j via arc(i,j)
L
ij
= lower bound of flow through arc(i,j) (if there is
no lower bound, let L
ij
=0)
U
ij
= upper bound of flow through arc(i,j) (if there is