1. Consider the following linear program:
Max 3A + 2B
s.t.
lA + 1B 10
3A + 1B 24
I A + 2B -s 16 A, B
a. Use the graphical solution procedure to find the optimal solution.
b. Assume that the objective function coefficient for A changes from 3 to 5. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution.
c. Assume that the objective function coefficient for A remains 3, but the objective function coefficient for B changes from 2 to 4.
Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution.
d. The sensitivity report for the linear program in part (a) provides the following objective coefficient range information:
|
Variable
|
Objective Coefficient
|
Allowable Increase
|
Allowable Decrease
|
|
A
|
3
|
3
|
1
|
|
B
|
2
|
1
|
1
|
Use this objective coefficient range information to answer parts (b) and (c).
2. Consider the linear program in Problem 1. The value of the optimal solution is 27. Suppose that the right-hand side for constraint I is increased from 10 to 11.
a. Use the graphical solution procedure to find the new optimal solution.
b. Use the solution to part (a) to determine the shadow price for constraint 1.
c. The sensitivity report for the linear program in Problem 1 provides the following righthand-side range information:
|
Constraint
|
Constraint R.H.Side
|
Allowable Increase
|
Allowable Decrease
|
|
1
|
10
|
1.20
|
2
|
|
2
|
24
|
6
|
6
|
|
3
|
16
|
Infinite
|
3.0(X)
|
What does the right-hand-side range information for constraint 1 tell you about the shadow price for constraint I?
d. The shadow price for constraint 2 is 0.5. Using this shadow price and the right-hand-side range information in part (c), what conclusion can you draw about the effect of changes to the right-hand side of constraint 2?
3. Consider the following linear program:
Min 8X + 12Y
s.t.
1X+ 9
2X+ 217a-10
6X+ 217 a- 18
X, 0
a. Use the graphical solution procedure to find the optimal solution.
b. Assume that the objective function coefficient for X changes from 8 to 6. Does the optimal solution change?
Use the graphical solution procedure to find the new optimal solution.
c. Assume that the objective function coefficient for X remains 8, but the objective function coefficient for IT changes from 12 to 6.
Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution.
d. The sensitivity report for the linear program in part (a) provides the following objective coefficient range information:
|
Constraint
|
Objective Coefficient
|
Allowable Increase
|
Allowable Decrease
|
|
X
|
8.000
|
4,000
|
4.000
|
|
V
|
12.000
|
12.000
|
4.000
|
How would this objective coefficient range information help you answer parts (b) and (c) prior to resolving the problem?
4. Consider the linear program in Problem 3. The value of the optimal solution is 48. Suppose that the right-hand side for constraint 1 is increased from 9 to 10.
a. Use the graphical solution procedure to find the new optimal solution.
b. Use the solution to part (a) to determine the shadow price for constraint 1.
c. The sensitivity report for the linear program in Problem 3 provides the following righthand-side range information:
|
Constraint
|
Constraint R.H.Side
|
Allowable Increase
|
Allowable Decrease
|
|
1
|
9.000
|
2.000
|
4.000
|
|
2
|
10.000
|
8.000
|
1.000
|
|
3
|
18.000
|
4.000
|
Infinite
|
What does the right-hand-side range information for constraint 1 tell you about the shadow price for constraint 1?
d. The shadow price for constraint 2 is 3. Using this shadow price and the right-hand-side range information in part (c), what conclusion can be drawn about the effect of changes to the right-hand side of constraint 2?
14. Digital Controls, Inc. (DCI) manufactures two models of a radar gun used by police to monitor the speed of automobiles. Model A has an accuracy of plus or minus 1 mile per hour, whereas the smaller model B has an accuracy of plus or minus 3 miles per hour.
For the next week, the company has orders for 100 units of model A and 150 units of model B.
Although DCI purchases all the electronic components used in both models, the plastic cases for both models are manufactured at a DCI plant in Newark, New Jersey. Each model A case requires 4 minutes of injection-molding time and 6 minutes of assembly time.
Each model B case requires 3 minutes of injection-molding time and 8 minutes of assembly time. For next week the Newark plant has 600 minutes of injection-molding time available and 1080 minutes of assembly time available.
The manufacturing cost is $10 per case for model A and $6 per case for model B. Depending upon demand and the time available at the Newark plant, DCI occasionally purchases cases for one or both models from an outside supplier in order to fill customer orders that could not be filled otherwise.
The purchase cost is $14 for each model A case and $9 for each model B case.
Management wants to develop a minimum cost plan that will determine how many cases of each model should be produced at the Newark plant and how many cases of each model should be purchased. The following decision variables were used to formulate a linear programming model for this problem:
AM = number of cases of model A manufactured
BM = number of cases of model B manufactured
AP = number of cases of model A purchased
BP = number of cases of model B purchased
The linear programming model that can be used to solve this problem is as follows:
The linear programming model that can be used to solve this problem is as follows:
Min 1 OA + 68M + 14AP + 9BP
s_t_
1AM + + 1AP + = 100 Demand for model A
1BM + IBP = 150 Demand for model B
4AM + 3BM c 600 Injection molding time
6AM + 8BM 5. 1080 Assembly time
AM, BM, AP, BP 0
The sensitivity report is shown in Figure 8.19.
a. What is the optimal solution and what is the optimal value of the objective function?
b. Which constraints are binding?
c. What are the shadow prices? Interpret each.
d. If you could change the right-hand side of one constraint by one unit, which one would you choose? Why?
17. The Porsche Club of America sponsors driver education events that provide high-performance driving instruction on actual racetracks. 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 40,000 pounds of the high-alloy steel will be available next quarter. In addition, Deegan estimates that 2000 hours of manufacturing time and 1600 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:
The linear programming model that can be used to solve this problem is as follows:
Min 1 OA + 68M + 14AP + 9BP
s_t_
1AM + + 1AP + = 100 Demand for model A
1BM + IBP = 150 Demand for model B
4AM + 3BM c 600 Injection molding time
6AM + 8BM 5. 1080 Assembly time
AM, BM, AP, BP 0
The sensitivity report is shown in Figure 8.21.
a. What are the optimal solution and the total profit contribution?
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.
c. Deegan is considering using overtime to increase the available assembly time. What would you advise Deegan to do regarding this option? Explain.
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.
e. If the available manufacturing time is increased by 500 hours, will the shadow price for the manufacturing time constraint change? Explain.
|
FIGURE 8.21 SENSITIVITY REPORT FOR THE DEEGAN INDUSTRIES PROBLEM
|
|
|
Variable Cells
|
|
|
|
|
|
|
|
Model
Variable
|
Name
|
Final
Value
|
Reduced
Cost
|
Objective
Coefficient
|
Allowable
Increase
|
Allowable
Decrease
|
|
DRB
|
Model DRB
|
1000.000
|
0.000
|
200.000
|
24.000
|
88.000
|
|
DRW
|
Model DRW
|
800.000
|
0.000
|
280.000
|
220.000
|
30.000
|
|
Constraints
|
|
|
|
|
|
|
|
Constraint
Number
|
Name
|
Final
Value
|
Shadow
Price
|
Constraint
R.H. Side
|
Allowable
Increase
|
Allowable
Decrease
|
|
|
Steel available
|
40000.000
|
8.800
|
40000.000
|
909.091
|
10000.000
|
|
2
|
Manufacturing minutes
|
120000.000
|
0.600
|
120000.000
|
40000.000
|
5714.286
|
|
3
|
Assembly minutes
|
92000.000
|
0.000
|
96000.000
|
1E+30
|
4000.000
|