INTERNATIONAL SHIPPING
Response to the following problem:
The Takahashi Transport Company (TTC) leases excess space on commercial vessels to the United States at a reduced rate of $10 per square foot. The only condition is that goods must be packaged in standard 30-inch-high crates. TTC ships items in two standard 30-inch-high crates, one 8 -square-foot crate (2 feet by 4 feet) and one 4-square-foot (2 feet by 2 feet) specially insulated crate. It charges customers $160 to ship an 8 -square-foot crate and $100 to ship the insulated 4-square-foot crate. Thus, allowing for the $10 per square foot cost, TTC makes a profit of $80 per standard 8 -foot crate and $60 on the 4-foot crate. TTC stores the crates until space becomes available on a cargo ship, at which time TTC receives payment from its customers. TTC has been able to lease 1200 square feet of cargo space on the Formosa Frigate cargo ship, which leaves for the United States in two days. As of this date, TTC has 140 8 -square-foot crates and 100 insulated 4-square-foot crates awaiting shipment to the United States. It has 48 hours to finish loading the crates, and it estimates the average loading time to be 12 minutes (0 .2 hour) per 8 -square-foot crate and 24 minutes (0.4 hour) per 4-square-foot crate (owing to the special handling of the insulated crates).
a. Formulate and solve a linear program for TTC to optimize its profit on the upcoming sailing of the Formosa Frigate. What are the optimal values of the slack on each constraint in the optimal solution? Express this result in words.
b. Determine the shadow price and the range of feasibility for the number of square feet available. What problem would you have interpreting the shadow prices and the range of feasibility?
c. Suppose that, at the last second, the Formosa Frigate decided to raise its charge per square foot from $ 1 0 to $12. Note how this change would affect the objective function coefficients. Show that the optimal solution would not change.
How does this $2 per square foot increase in leasing charges to TTC affect _ the shadow price for a square foot of space? Does this make sense?