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