The Porsche Club of America sponsors driver education events that provide high-performance driving instruction on actual race tracks. Because safety is a primary consideration at such events, many owners elect to install roll bars in their cars. Deegan Industries manufactures two types of roll bars for Porsches. Model DRB is bolted to the car using existing holes in the car's frame. Model DRW is a heavier roll bar that must be welded to the car's frame. Model DRB requires 20 pounds of a special high alloy steel, 40 minutes of manufacturing time, and 60 minutes of assembly time. Model DRW requires 25 pounds of the special high alloy steel, 100 minutes of manufacturing time, and 40 minutes of assembly time. Deegan's steel supplier indicated that at most 36,000 pounds of the high-alloy steel will be available next quarter. In addition, Deegan estimates that 2,000 hours of manufacturing time and 1,800 hours of assembly time will be available next quarter. The profit contributions are $200 per unit for model DRB and $280 per unit for model DRW. The linear programming model for this problem is as follows:
Max
|
200DRB
|
+
|
280DRW
|
|
|
|
s.t.
|
|
|
|
|
20DRB
|
+
|
25DRW
|
≤
|
36,000
|
Steel available
|
|
40DRB
|
+
|
100DRW
|
≤
|
120,000
|
Manufacturing minutes
|
|
60DRB
|
+
|
40DRW
|
≤
|
108,000
|
Assembly minutes
|
|
|
|
DRB,DRW
|
≥
|
0
|
|
|
The computer solution is shown below.
Optimal Objective Value = 388800.00000
Variable
|
Value
|
Reduced Cost
|
DRB
|
600.00000
|
0.00000
|
DRW
|
960.00000
|
0.00000
|
Constraint
|
Slack/Surplus
|
Dual Value
|
1
|
0.00000
|
8.80000
|
2
|
0.00000
|
0.60000
|
3
|
33600.00000
|
0.00000
|
Variable
|
Objective
Coefficient
|
Allowable
Increase
|
Allowable
Decrease
|
DRB
|
200.00000
|
24.00000
|
88.00000
|
DRW
|
280.00000
|
220.00000
|
30.00000
|
Constraint
|
RHS
Value
|
Allowable
Increase
|
Allowable
Decrease
|
1
|
36000.00000
|
7636.36364
|
6000.00000
|
2
|
120000.00000
|
24000.00000
|
48000.00000
|
3
|
108000.00000
|
Infinite
|
33600.00000
|
(a) What is the optimal solution and the total profit contribution (in $)?
DRB=600, DRW=960, total profit= 388800
(b)
Another supplier offered to provide Deegan Industries with an additional 500 pounds of the steel alloy at $2 per pound. Should Deegan purchase the additional pounds of the steel alloy? Explain.
Yes, the dual value for steel available is 8.8. Each pound of steel will increase profits more than the $2 per pound that the supplier is offering
(c) Deegan is considering using overtime to increase the available assembly time. What would you advise Deegan to do regarding this option? Explain.
Constraint 3 has a slack. Increasing the number of hours of assembly time will not improve profits.
(d) Because of increased competition, Deegan is considering reducing the price of model DRB such that the new contribution to profit is $175 per unit. How would this change in price affect the optimal solution? Explain.
The objective coefficient range for model DRB shows a lower limit of $112. Thus, the optimal solution will not change and the new value will be
$________?
(e) If the available manufacturing time is increased by 500 hours, will the dual value for the manufacturing time constraint change? Explain.
The allowable increase is
________ minutes, so the value for the constraints will change.
** In d I have found the lower limit but the upper limit value (224) is incorrect with my calculations and in e I used 30000 (500 hours=30000 minutes) however, this is not the correct output.