PROC NETFLOW: Mathematical Description of NPSC :: SAS/OR(R) 9.22 User's Guide: Mathematical Programming

The NETFLOW Procedure

Mathematical Description of NPSC

If a network programming problem with side constraints has $\text{[math]}$ nodes, $\text{[math]}$ arcs, $\text{[math]}$ nonarc variables, and $\text{[math]}$ side constraints, then the formal statement of the problem solved by PROC NETFLOW is

$\text{[math]}$

where

$\text{[math]}$ is the $\text{[math]}$ arc variable objective function coefficient vector (the cost vector)
$\text{[math]}$ is the $\text{[math]}$ arc variable value vector (the flow vector)
$\text{[math]}$ is the $\text{[math]}$ nonarc variable objective function coefficient vector
$\text{[math]}$ is the $\text{[math]}$ nonarc variable value vector
$\text{[math]}$ is the $\text{[math]}$ node-arc incidence matrix of the network, where
$\text{[math]}$ if arc $\text{[math]}$ is directed from node $\text{[math]}$
$\text{[math]}$ if arc $\text{[math]}$ is directed toward node $\text{[math]}$
$\text{[math]}$ otherwise

$\text{[math]}$ is the $\text{[math]}$ node supply/demand vector, where

$\text{[math]}$ if node $\text{[math]}$ has supply capability of $\text{[math]}$ units of flow

$\text{[math]}$ if node $\text{[math]}$ has demand of $\text{[math]}$ units of flow

$\text{[math]}$ if node $\text{[math]}$ is a trans-shipment node

$\text{[math]}$ is the $\text{[math]}$ side constraint coefficient matrix for arc variables, where $\text{[math]}$ is the coefficient of arc $\text{[math]}$ in the $\text{[math]}$ th side constraint

$\text{[math]}$ is the $\text{[math]}$ side constraint coefficient matrix for nonarc variables, where $\text{[math]}$ is the coefficient of nonarc $\text{[math]}$ in the $\text{[math]}$ th side constraint

$\text{[math]}$ is the $\text{[math]}$ side constraint right-hand-side vector

$\text{[math]}$ is the $\text{[math]}$ arc lower flow bound vector

$\text{[math]}$ is the $\text{[math]}$ arc capacity vector

$\text{[math]}$ is the $\text{[math]}$ nonarc variable lower bound vector

$\text{[math]}$ is the $\text{[math]}$ nonarc variable upper bound vector

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