Tucker Inc. needs to produce 1000 Tucker automobiles. The company has four production plants. Due to differing workforces, technological advances, and so on, the plants differ in the cost of producing each car. They also use a different amount of labor and raw material at each. This is summarized in the following table:

The labor contract signed requires at least 400 cars to be produced at plant 3; there are 3300 hours of labor and 4000 units of material that can be allocated to the four plants.

This leads to the following formulation:

MIN 15 X1 + 10 X2 + 9 X3 + 7 X4 SUBJECT TO 2 X1 + 3 X2 + 4 X3 + 5 X4 <= 3300 3 X1 + 4 X2 + 5 X3 + 6 X4 <= 4000 X3 >= 400 X1 + X2 + X3 + X4 = 1000 X1, X2, X3, X4 >= 0

Attached is the Solver output for this problem.

Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$11 X1 400 0 15 1E+30 3.5 $B$12 X2 200 0 10 2 1E+30 $B$13 X3 400 0 9 1E+30 4 $B$14 X4 0 7 7 1E+30 7 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $B$18 Labor 3000 0 3300 1E+30 300 $B$19 Material 4000 -5 4000 300 200 $B$20 Plant3Min 400 4 400 100 400 $B$21 Total 1000 30 1000 66.6667 100

Here are some questions to be answered:

- What are the current production quantities? What is the current
cost of production?
- How much will it cost to produce one more vehicle? How much
will we save by producing one less?
- How would our solution change if it cost only $8,000 to produce
at plant 2? For what ranges of costs is our solution (except for
the objective value) valid for plant 2?
- How much are we willing to pay for a labor hour?
- How much is our union contract costing us? What would be the
value of reducing the 400 car limit down to 200 cars? To 0 cars?
What would be the cost of increasing it by 100 cars? by 200 cars?
- How much is our raw material worth (to get one more unit)? How
many units are we willing to buy at that price? What will happen if
we want more?
- A new plant is being designed that will use only one unit of
workers and 4 units of raw material. What is the maximum cost it
can have in order for us to consider using it?
- By how much can the costs at plant 1 increase before we would
not produce there?

Mon Aug 24 16:30:59 EDT 1998