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
- 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
- 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 Excel model
constructed for this situation is shown below. The added arcs
are numbered 18 through 23. The add-in allows the elimination
of data items that take only default values. The lower bounds
are not shown with the default value of 0. The gains are not
shown and have the default value 1.
Because arcs 18 through 23 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