Assignment
I. Formulating the Production Planning Model
A company manufactures two electric products, air conditioners and fans. The assembly process is similar in that both of them must pass through two stages of production: wiring and drilling. Each air conditioner takes 7 hours of wiring and 5 hours of drilling, while each fan must go through 3 hours of wiring and 2 hours of drilling. During the next period, 260 hours of wiring time are available and 160 hours of drilling time may be used. Each air conditioner sold yields a profit of
$27, and each fan sold yields a profit of $17.
1. Formulate the objective function algebraically.
2. Write the resource constraints. Would demand constraints or non-negativity constraints be more appropriate for this system of equations?
3. Upon looking over previous years demand, management has decided that to ensure an ad- equate supply of air conditioners for a contract, at least 24 units should be manufactured. Additionally, because they incurred an oversupply in the previous period, management has also requested that no more than 50 fans be produced during this period. Assist the company by reformulating the necessary constraints from the above question.
The corresponding sensitivity report for the original problem is shown below:
Variable cells
|
Cell
|
Name
|
Final Value
|
Reduced Price
|
Objective Coefficient
|
Allowable Increase
|
Allowable Decrease
|
|
$B$6
|
Air Conditioners
|
0
|
A
|
27
|
15.5
|
1E+30
|
|
$C$6
|
Fans
|
80
|
B
|
17
|
1E+30
|
6.2
|
Constraints
|
Cell
|
Name
|
Final Value
|
Shadow Price
|
Constraint R.H. Side
|
Allowable Increase
|
Allowable Decrease
|
|
$D$4
|
Wiring
|
240
|
0
|
260
|
1E+30
|
20
|
|
$D$5
|
Drilling
|
160
|
8.5
|
160
|
13.3
|
160
|
4. Calculate the missing values "A" and "B" in the above report.
5. Suppose management has decided to sell some of it's drilling capacity for $6 per hour. If 15 hours were sold, what is the impact on profit? What is the new objective function value?
6. The previous page mentions management requiring at least 24 units of air conditioners to be produced. If this policy were to be implemented, how will this affect contribution?
II. Understanding the Sensitivity Report
A manufacturing company produces two products X and Y with contributions to profit per unit of $10 and $9, respectively. Each product must pass through 4 departments during the manufac- turing process. The company has formulated a linear programming model to develop a production strategy to maximize profit, and satisfy the demand of 300 and 500 units for product X and Y , respectively. The corresponding sensitivity report for the problem is shown below:
Variable cells
|
Cell
|
Name
|
Final Value
|
Reduced Cost
|
Objective Coefficient
|
Allowable Increase
|
Allowable Decrease
|
|
$B$16
|
Product X
|
300
|
0
|
10
|
17
|
1E+30
|
|
$B$17
|
Product Y
|
500
|
0
|
9
|
1E+30
|
5.6666667
|
Constraints
|
Cell
|
Name
|
Final Value
|
Shadow Price
|
Constraint R.H. Side
|
Allowable Increase
|
Allowable Decrease
|
|
$B$21
|
Department 1
|
1300
|
0
|
6500
|
1E+30
|
5200
|
|
$B$22
|
Department 2
|
2100
|
0
|
6000
|
1E+30
|
3900
|
|
$B$23
|
Department 3
|
2600
|
0
|
7000
|
1E+30
|
4400
|
|
$B$24
|
Department 4
|
1400
|
9
|
1400
|
1100
|
0
|
|
$B$25
|
Demand for X
|
300
|
-17
|
300
|
0
|
300
|
|
$B$26
|
Demand for Y
|
500
|
0
|
500
|
0
|
1E+30
|
7. If the profit per unit of product X were increased to $25 per unit, would the optimal solution change? Why or why not?
8. Consider the following statement made by the manufacturing manager:
• "I am approving 1,000 extra hours of overtime in Department 4 at a projected cost level of $8.50 per hour."
As the operations manager, is this decision in the best interest of the company for the up- coming period? Why or why not? If so, comment on the objective function value (additional gain from accepting the offer).
III. Production Planning and Sensitivity Analysis
A wood furniture manufacturing firm asked your help in analyzing their production schedule. The firm can produce four types of furniture: Chairs, Desks, Tables, and Bookcases. Each item requires a certain number of man-hours in three departments: Cutting, Sanding, and Finishing. A sensitiv- ity report is given the table below where profit is expressed in dollars.
Variable cells
|
Cell
|
Name
|
Final Value
|
Reduced Cost
|
Objective Coefficient
|
Allowable Increase
|
Allowable Decrease
|
|
$C$2
|
Chairs
|
0
|
A
|
40
|
B
|
1E+30
|
|
$D$2
|
Desks
|
0
|
-10
|
90
|
10
|
1E+30
|
|
$E$2
|
Tables
|
140
|
0
|
100
|
100
|
10
|
|
$F$2
|
Bookcases
|
25
|
0
|
160
|
240
|
80
|
Constraints
|
Cell
|
Name
|
Final Value
|
Shadow Price
|
Constraint R.H. Side
|
Allowable Increase
|
Allowable Decrease
|
|
$C$15
|
Cutting
|
480
|
12.5
|
480
|
1120
|
160
|
|
$C$16
|
Sanding
|
800
|
15
|
800
|
100
|
560
|
|
$C$17
|
Finishing
|
800
|
0
|
900
|
1E+30
|
100
|
9. Express the objective function algebraically.
10. The resource coefficients for Chairs are as follows: 4 hours of cutting, 8 hours of sanding, and 5 hours of finishing. By how much should the profit for selling one unit of Chairs increase, before Chairs could be considered for production in the optimal production mix?
11. Suppose we are required to produce 75 Desks. What is the new overall profit?
12. The firm is considering the possibility of buying equipment to increase the current Sanding capacity. What is the maximum amount that the firm should be willing to pay to increase the number of available Sanding hours from 800 to 850?
13. Suppose the profit of Tables decreased from 100 to 95. What would be the new optimal solution? What would be the new profit? (If there is not enough information in the problem to give the answer, please state so.)
14. To cut costs, the firm wants to reduce the number of hours in the Finishing department from 900 to 700. Which of the following statements is the most accurate?
(a) reducing the number of Finishing hours by 200 will not affect total profit
(b) the number of Finishing hours by 200 will decrease profit by $100
(c) reducing the number of Finishing hours by 200 will increase profit by $100
(d) reducing the number of Finishing hours by 200 could affect total profit, but we do not have enough information to compute the magnitude of the change.
Format your assignment according to the following formatting requirements:
1. The answer should be typed, double spaced, using Times New Roman font (size 12), with one-inch margins on all sides.
2. The response also includes a cover page containing the title of the assignment, the student's name, the course title, and the date. The cover page is not included in the required page length.
3. Also include a reference page. The Citations and references should follow APA format. The reference page is not included in the required page length.