Tucker Automobiles

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:

1. What are the current production quantities? What is the current cost of production?

2. How much will it cost to produce one more vehicle? How much will we save by producing one less?

3. 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?

4. How much are we willing to pay for a labor hour?

5. 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?

6. 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?

7. 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?

8. By how much can the costs at plant 1 increase before we would not produce there?