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 right-hand 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 right-hand-side 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 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?

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 re-solving 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 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 dual value for constraint 1.

c. The computer solution for the linear program in Problem 3 provides the following right-hand-side 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 right-hand-side 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 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?

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 catcher’s mitts

Leads to the following formulation:

5 + 8 . .

+?³2? ? 900?¹2?+?¹3? ?300 ?

?¹⁸? +?¹⁴? ? 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 right-hand-sides 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.