The following linear programming model is used for maximizing the profit of producing four products (x1, x2, x3, x4), with four resources (two manufacturing processes and two material requirements): maxZ=$50x1+$58x2+$52x3+$62x4 subject to 4x1+3.5x2+4.6x3+3.9x4≤500 hr. (available time for process 1) 2.1x1+2.6x2+2.3x3+1.9x4≤350 hr. (available time for process 2) 15x1+23x2+18x3+25x4≤3600 lbs. (material A availability) 8x1+12.6x2+9.7x3+10.5x4≤1500 lb. (material B availability) x1+x2≥0.50 (x1+x2+x3+x4) x1,x2,x3,x4≥0 Sensitivity Analysis for the optimal solution of this problem is shown below: Variable Cells Final Reduced Objective Allowable Allowable Name Value Cost Coefficient Increase Decrease X1 = 9.8706 0 50 11.5335 15.8961 X2 = 57.0301 0 58 13.2315 15.0886 X3 = 0 -13.5464 52 13.5464 1E+30 X4 = 66.9006 0 62 1E+30 6.3795 Constraints Final Shadow Constraint Allowable Allowable Name Value Price R.H. Side Increase Decrease Process 1 Usage 500 8.0544 500 140.5405 19.4805 Process 2 Usage 296.1176 0 350 1E+30 53.8824 Material A Usage 3132.2658 0 3600 1E+30 467.7342 Material B Usage 1500 2.6146 1500 60.8108 329.1139 Ratio 66.9006 -6.2689 0 68.0804 18.1598 Answer the following questions based on the sensitivity analysis provided above, and include them in your Word/pdf file. If you were given the following two options, which one would you select, why? i.Purchase an additional machine for process 1. The price for the machine is $750 and it can provide an additional 100 hours of process 1. Increase the availability of raw material B up to 1560 lbs. Additional material B can be purchased for $0.5/lb. Supplier of Material B has recently informed that they can provide at most 1400 lbs. of Material B. Since you have already paid for the cost of the material, they will refund you the payment for the amount they are not able to fulfill, which is $0.5/lb. Can you make a conclusion on the impact of the decrease in Material B on the profit? If yes, how much will the total profit change? If not, explain why.
The following linear programming model is used for maximizing the profit of producing four products (x1, x2, x3, x4), with four resources (two manufacturing processes and two material requirements): maxZ=$50x1+$58x2+$52x3+$62x4 subject to 4x1+3.5x2+4.6x3+3.9x4≤500 hr. (available time for process 1) 2.1x1+2.6x2+2.3x3+1.9x4≤350 hr. (available time for process 2) 15x1+23x2+18x3+25x4≤3600 lbs. (material A availability) 8x1+12.6x2+9.7x3+10.5x4≤1500 lb. (material B availability) x1+x2≥0.50 (x1+x2+x3+x4) x1,x2,x3,x4≥0 Sensitivity Analysis for the optimal solution of this problem is shown below: Variable Cells Final Reduced Objective Allowable Allowable Name Value Cost Coefficient Increase Decrease X1 = 9.8706 0 50 11.5335 15.8961 X2 = 57.0301 0 58 13.2315 15.0886 X3 = 0 -13.5464 52 13.5464 1E+30 X4 = 66.9006 0 62 1E+30 6.3795 Constraints Final Shadow Constraint Allowable Allowable Name Value Price R.H. Side Increase Decrease Process 1 Usage 500 8.0544 500 140.5405 19.4805 Process 2 Usage 296.1176 0 350 1E+30 53.8824 Material A Usage 3132.2658 0 3600 1E+30 467.7342 Material B Usage 1500 2.6146 1500 60.8108 329.1139 Ratio 66.9006 -6.2689 0 68.0804 18.1598 Answer the following questions based on the sensitivity analysis provided above, and include them in your Word/pdf file. If you were given the following two options, which one would you select, why? i.Purchase an additional machine for process 1. The price for the machine is $750 and it can provide an additional 100 hours of process 1. Increase the availability of raw material B up to 1560 lbs. Additional material B can be purchased for $0.5/lb. Supplier of Material B has recently informed that they can provide at most 1400 lbs. of Material B. Since you have already paid for the cost of the material, they will refund you the payment for the amount they are not able to fulfill, which is $0.5/lb. Can you make a conclusion on the impact of the decrease in Material B on the profit? If yes, how much will the total profit change? If not, explain why.
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter4: Linear Programming Models
Section: Chapter Questions
Problem 73P
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Question
The following linear programming model is used for maximizing the profit of producing four products (x1, x2, x3, x4), with four resources (two manufacturing processes and two material requirements):
maxZ=$50x1+$58x2+$52x3+$62x4
subject to
4x1+3.5x2+4.6x3+3.9x4≤500 hr. (available time for process 1)
2.1x1+2.6x2+2.3x3+1.9x4≤350 hr. (available time for process 2)
15x1+23x2+18x3+25x4≤3600 lbs. (material A availability)
8x1+12.6x2+9.7x3+10.5x4≤1500 lb. (material B availability)
x1+x2≥0.50 (x1+x2+x3+x4)
x1,x2,x3,x4≥0
Sensitivity Analysis for the optimal solution of this problem is shown below:
|
Variable Cells |
Final |
Reduced |
Objective |
Allowable |
Allowable |
|
Name |
Value |
Cost |
Coefficient |
Increase |
Decrease |
|
X1 = |
9.8706 |
0 |
50 |
11.5335 |
15.8961 |
|
X2 = |
57.0301 |
0 |
58 |
13.2315 |
15.0886 |
|
X3 = |
0 |
-13.5464 |
52 |
13.5464 |
1E+30 |
|
X4 = |
66.9006 |
0 |
62 |
1E+30 |
6.3795 |
|
Constraints |
Final |
Shadow |
Constraint |
Allowable |
Allowable |
|
Name |
Value |
Price |
R.H. Side |
Increase |
Decrease |
|
Process 1 Usage |
500 |
8.0544 |
500 |
140.5405 |
19.4805 |
|
Process 2 Usage |
296.1176 |
0 |
350 |
1E+30 |
53.8824 |
|
Material A Usage |
3132.2658 |
0 |
3600 |
1E+30 |
467.7342 |
|
Material B Usage |
1500 |
2.6146 |
1500 |
60.8108 |
329.1139 |
|
Ratio |
66.9006 |
-6.2689 |
0 |
68.0804 |
18.1598 |
Answer the following questions based on the sensitivity analysis provided above, and include them in your Word/pdf file.
- If you were given the following two options, which one would you select, why? i.Purchase an additional machine for process 1. The price for the machine is $750 and it can provide an additional 100 hours of process 1.
- Increase the availability of raw material B up to 1560 lbs. Additional material B can be purchased for $0.5/lb.
- Supplier of Material B has recently informed that they can provide at most 1400 lbs. of Material B. Since you have already paid for the cost of the material, they will refund you the payment for the amount they are not able to fulfill, which is $0.5/lb. Can you make a conclusion on the impact of the decrease in Material B on the profit? If yes, how much will the total profit change? If not, explain why.
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