### Variable External Flows

 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. Phoenix: This plant is to be discontinued. The entire inventory of 700 units must be shipped or sold for scrap. The scrap value is \$5 per unit. Chicago: Minimum demand of 200 units. An additional 100 units could be sold if available with revenue of \$20 per unit. LosAngeles: A firm demand of 200 units that must be received. Dallas: Contracted demand for 300 units. An additional 100 units may be sold with a revenue of \$20 each. Atlanta: Demand for 150 units that must be met. New York: One hundred units left over from previous shipments. No firm demand, but up to 250 units can be sold at \$25 each. Austin: Maximum production of 300 units with manufacturing cost of \$10 per unit. Gainesville: Work rules require that all regular time production of 200 units be shipped. An additional 100 units can be produced using overtime at a cost of \$14 per unit.

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.