### Mathematical Description of NPSC

If a network programming problem with side constraints has nodes, arcs, nonarc variables, and side constraints, then the formal statement of the problem solved by PROC NETFLOW is where

• is the arc variable objective function coefficient vector (the cost vector)

• is the arc variable value vector (the flow vector)

• is the nonarc variable objective function coefficient vector

• is the nonarc variable value vector

• is the node-arc incidence matrix of the network, where

• if arc is directed from node • if arc is directed toward node • otherwise

• is the node supply/demand vector, where

• if node has supply capability of units of flow

• if node has demand of units of flow

• if node is a trans-shipment node

• is the side constraint coefficient matrix for arc variables, where is the coefficient of arc in the th side constraint

• is the side constraint coefficient matrix for nonarc variables, where is the coefficient of nonarc in the th side constraint

• is the side constraint right-hand-side vector

• is the arc lower flow bound vector

• is the arc capacity vector

• is the nonarc variable lower bound vector

• is the nonarc variable upper bound vector