Some additional information arrives that suggests the supplies and demands are not fixed values as suggested above. The logistics manager learns the following about the situation.
The situations described above represent variable external flows, that is flow that can enter or leave the network at nodes that are variable in amount. We handle such situations by adding arcs that touch only a single node in the network, either originate or terminating at a node. The list below shows the set of external flow arcs added to the network of Fig. 2.
Because the arcs originate or terminate at a node not defined (node 0), these arcs contribute to the conservation of flow constraints for only one node. Three negative values appear in the cost column representing the revenue at the cities with extra demand. In general, negative costs are equivalent to revenues.
The optimum solution for the Example is shown in Fig. 4. Reviewing the solution, we note that all products are shipped from Phoenix, while Austin produces 300 and Gainesville produces its minimum amount. All the demanders receive the maximum possible. The objective function is negative (z = -$1600), indicating a net profit for variable flows. The fixed external flows at the nodes do not contribute to the profit figure as no costs or revenues are associated with fixed flows.
Figure 4. Solution with variable node flows. z = -$1600.