1. (30 pts.) A company that manufactures three products, A, B, and C, using three machines, M1, M2, and M3, wants to determine the optimal production schedule that maximizes the total profit. Product has to be processed by machines M1, M2, and M3, product B requires M1 and M3, while product Crequires M1 and M2. The unit profits on the three products are $4, $2, and $5, respectively. The following linear program is formulated to determine the optimal product mix: Maximize Z=4x+2x2+5x3 5.L x+2x2+ x ≤430 (Machine 1) 3x + 2x ≤ 460 (Machine 2) x+4x2 ≤450 (Machine 3) and 11.12.20 where x1, x2, and xs are the amounts of products A. B, and C and the constraints reflect the available capacities of M1, M2, and M3. The optimum solution for the problem is as follows: optimal solution: -0, -100, x-230 optimal value: maximum profit, Z=1350 shadow prices: 1.0, 2.0, and 0.0 for constraints 1, 2 and 3 respectively reduced costs: for x 3.0, for x = 0, for x)=0 RANGES ON RIGHT-HAND-SIDE (RHS) CONSTRAINTS Row Lower Limit Present Value Upper Limit 1 230 430 455 2 410 460 860 3 400 450 8 RANGES ON OBJECTIVE FUNCTION COEFFICIENTS Variable Lower Limit Present Value Upper Limit F 88 4.0 7.0 32 0 2.0 10.0 F 3.0 5.0 8 Using the above information answer the following questions: a. Because of an increase in the cost of raw material used for product C, its unit profit drops to $4. Determine the new optimal solution and the maximum profit. b. Suppose it is possible to increase the capacity of one of the machines. Which one would you recommend for expansion and why? C Due to an improvement in product design, the unit profit on product A can be increased to $6. Is it worthwhile producing product.A now? Explain. d. Suppose that the capacity of machine 2 can be increased by another 200 minutes at a cost of $250. Is it economical to do so? Explain. c. Because of an increase in the cost of energy for operating the machines, the unit profits of 4, B, and C decrease by $2.0, $0.5, and $1.0, respectively. How will it affect the optimal solution and maximum profit?
1. (30 pts.) A company that manufactures three products, A, B, and C, using three machines, M1, M2, and M3, wants to determine the optimal production schedule that maximizes the total profit. Product has to be processed by machines M1, M2, and M3, product B requires M1 and M3, while product Crequires M1 and M2. The unit profits on the three products are $4, $2, and $5, respectively. The following linear program is formulated to determine the optimal product mix: Maximize Z=4x+2x2+5x3 5.L x+2x2+ x ≤430 (Machine 1) 3x + 2x ≤ 460 (Machine 2) x+4x2 ≤450 (Machine 3) and 11.12.20 where x1, x2, and xs are the amounts of products A. B, and C and the constraints reflect the available capacities of M1, M2, and M3. The optimum solution for the problem is as follows: optimal solution: -0, -100, x-230 optimal value: maximum profit, Z=1350 shadow prices: 1.0, 2.0, and 0.0 for constraints 1, 2 and 3 respectively reduced costs: for x 3.0, for x = 0, for x)=0 RANGES ON RIGHT-HAND-SIDE (RHS) CONSTRAINTS Row Lower Limit Present Value Upper Limit 1 230 430 455 2 410 460 860 3 400 450 8 RANGES ON OBJECTIVE FUNCTION COEFFICIENTS Variable Lower Limit Present Value Upper Limit F 88 4.0 7.0 32 0 2.0 10.0 F 3.0 5.0 8 Using the above information answer the following questions: a. Because of an increase in the cost of raw material used for product C, its unit profit drops to $4. Determine the new optimal solution and the maximum profit. b. Suppose it is possible to increase the capacity of one of the machines. Which one would you recommend for expansion and why? C Due to an improvement in product design, the unit profit on product A can be increased to $6. Is it worthwhile producing product.A now? Explain. d. Suppose that the capacity of machine 2 can be increased by another 200 minutes at a cost of $250. Is it economical to do so? Explain. c. Because of an increase in the cost of energy for operating the machines, the unit profits of 4, B, and C decrease by $2.0, $0.5, and $1.0, respectively. How will it affect the optimal solution and maximum profit?
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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Transcribed Image Text:1. (30 pts.) A company that manufactures three products, A, B, and C, using three machines, M1,
M2, and M3, wants to determine the optimal production schedule that maximizes the total profit.
Product has to be processed by machines M1, M2, and M3, product B requires M1 and M3,
while product Crequires M1 and M2. The unit profits on the three products are $4, $2, and $5,
respectively. The following linear program is formulated to determine the optimal product mix:
Maximize
Z=4x+2x2+5x3
5.L
x+2x2+ x ≤430
(Machine 1)
3x + 2x ≤ 460
(Machine 2)
x+4x2
≤450
(Machine 3)
and
11.12.20
where x1, x2, and xs are the amounts of products A. B, and C and the constraints reflect the
available capacities of M1, M2, and M3. The optimum solution for the problem is as follows:
optimal solution: -0, -100, x-230
optimal value:
maximum profit, Z=1350
shadow prices:
1.0, 2.0, and 0.0 for constraints 1, 2 and 3 respectively
reduced costs:
for x 3.0, for x = 0, for x)=0
RANGES ON RIGHT-HAND-SIDE (RHS) CONSTRAINTS
Row
Lower Limit
Present Value
Upper Limit
1
230
430
455
2
410
460
860
3
400
450
8
RANGES ON OBJECTIVE FUNCTION COEFFICIENTS
Variable
Lower Limit
Present Value
Upper Limit
F
88
4.0
7.0
32
0
2.0
10.0
F
3.0
5.0
8
Using the above information answer the following questions:
a. Because of an increase in the cost of raw material used for product C, its unit profit drops to
$4. Determine the new optimal solution and the maximum profit.
b. Suppose it is possible to increase the capacity of one of the machines. Which one would you
recommend for expansion and why?
C Due to an improvement in product design, the unit profit on product A can be increased to $6.
Is it worthwhile producing product.A now? Explain.
d. Suppose that the capacity of machine 2 can be increased by another 200 minutes at a cost of
$250. Is it economical to do so? Explain.
c. Because of an increase in the cost of energy for operating the machines, the unit profits of 4,
B, and C decrease by $2.0, $0.5, and $1.0, respectively. How will it affect the optimal
solution and maximum profit?
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