maths assignments
Homework 3 Supplemented Questions
1. Consider the following linear program:
. .
3 + 2
1 + 1 ? 10
3 + 1 ? 24
1 + 2 ? 16 , ? 0
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 computer solution for the linear program in part (a) provides the following objective coefficient range information:
Objective 
Allowable 
Allowable 

Variable 
Coefficient 
Increase 
Decrease 
A 
3.00000 
3.00000 
1.00000 
B 
2.00000 
1.00000 
1.00000 
Use this objective coefficient range information to answer parts (b) and (c)
Quantitative Analysis BA 452 Homework 3 Questions
2. Consider the linear program in Problem 1. The value of the optimal solution is 27. Suppose that the righthand side for constraint 1 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 dual value or constraint 1.
c. The computer solution for the linear program in Problem 1 provides the following righthandside range information:
RHS 
Allowable 
Allowable 

Constraint 
Value 
Increase 
Decrease 
1 
10.00000 
1.20000 
2.00000 
2 
24.00000 
6.00000 
6.00000 
3 
16.00000 
Infinite 
3.00000 
d. The dual value for constraint 2 is 0.5. Using this dual value and the righthandside range information in part (c), what conclusion can be drawn about the effect of changes to the righthand side of constraint 2?
Quantitative Analysis BA 452 Homework 3 Questions
3. Consider the following linear program:
Min 
8X + 12Y 

s.t. 
1X 
+ 3Y 
? 9 

2X 
+ 2Y 
? 10 

6X 
+ 2Y 
? 18 

A, B ? 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 S remains 8, but the objective function coefficient for Y changes from 12 to 6. Does the optimal solution change? Use the graphical solution procedure to find the new optimal solution.
d. The computer solution for the linear program in part (a) provides the following objective coefficient range information:
Objective 
Allowable 
Allowable 

Variable 
Coefficient 
Increase 
Decrease 
X 
8.00000 
4.00000 
4.00000 
Y 
12.00000 
12.00000 
4.00000 
How would this objective coefficient range information help you answer parts (b) and (c) prior to resolving the problem?
Quantitative Analysis BA 452 Homework 3 Questions
4. Consider the linear program in Problem 3. The value of the optimal solution is 48. Suppose that the righthand 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 dual value for constraint 1.
c. The computer solution for the linear program in Problem 3 provides the following righthandside range information:
RHS 
Allowable 
Allowable 

Constraint 
Value 
Increase 
Decrease 
1 
9.00000 
2.00000 
4.00000 
2 
10.00000 
8.00000 
1.00000 
3 
18.00000 
4.00000 
Infinite 
What does the righthandside range information for constraint 1 tell you about the dual value for constraint 1?
d. The dual value for constraint 2 is 3. Using this dual value and the righthandside range information in part (c), what conclusion can be drawn about the effect of changes to the righthand side of constraint 2?
Quantitative Analysis BA 452 Homework 3 Questions
5. Refer to the Kelson Sporting Equipment problem (Chapter 2, Problem 24). Letting
R=number of regular gloves C=number of catcherâ€™s mitts
Leads to the following formulation:
5 + 8 . .
+?^{3}2? ? 900?^{1}2?+?^{1}3? ?300 ?
?^{1}^{8}? +?^{1}^{4}? ? 100 ? ,? 0
The computer solution is shown I Figure 3.13.
Quantitative Analysis BA 452 Homework 3 Questions
a. What is the optimal solution, and what is the value of the total profit contribution?
b. Which constraints are binding?
c. What are the dual values for the resources? Interpret each.
d. If overtime can be scheduled in one of the departments, where would you recommend doing so?
6. Refer to the computer solution of the Kelson Sporting Equipment problem in Figure 3.13 (see Problem 5).
a. Determine the objective coefficient ranges.
b. Interpret the ranges in part (a).
c. Interpret the righthandsides ranges.
d. How much will the value of the optimal solution improve if 20 extra hours of packaging and shipping time are made available?
Quantitative Analysis BA 452 Homework 3 Questions
7. Investment Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A particular portfolio consists of U shares of U.S. Oil and H shares of Huber Steel. The annual return for U.S. Oil is $3 per share and the annual return for Huber Steel is $5 per share. U.S. Oil sells for $25 per share and Huber Steel sells for $50 per share. The portfolio has $80,000 to be invested. The
portfolio risk index (.50 per share of U.S. Oil and 0.25 per share for Huber Steel) has a maximum of 700. In addition, the portfolio is limited to a maximum of 1000 shares of U.S. Oil. The linear
programming formulation that will maximize the total annual return of the portfolio is as follows: 

3 + 
5 

. . 
25 + 
50 ? 80,000 

0.50 + 0.25 ? 
700 

1 
? 0 
? 
1000 
. . 

, 
The computer solution of this problem is shown in Figure 3.14.
Quantitative Analysis BA 452 Homework 3 Questions
a. What is the optimal solution, and what is the value of the total annual return?
b. Which constraints are binding? What is your interpretation of these constraints in terms of the problem?
c. What are the dual values for the constraints? Interpret each.
d. Would it be beneficial to increase the maximum amount invested in U.S. Oil? Why or why not?

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