The NETFLOW Procedure

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