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
.jpg">
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.
.jpg">
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?