A company that manufactures three products, A, B, and C, using three machines, M1,M2 and M3 wants
Question:
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 profit. Product A has to be processed by machines M1 , M2, and M3, product B requires M1 and M3, while product C requires M1 and M2. The unit profits on the three products are $4, $2, and $5, respectively. The following linear program is formu-lated to determine the optimal product-mix:
Maximize: Subject to:
Z = 4x, + 2x2 + 5x3
x1 + 2x2 + x3 <=430 (Machine 1)
3x1 + 2x3 <=460 (Machine 2)
X1+ 4x2 <=450 (Machine 3)
x1, x2, x3 >=0
where x1, x2, and x3 are the amounts of products A, B, and C and the constraints reflect the available capacities of M1, M2, and M3. The computer prints out the following so-lution:
optimal solution: X1= 0, x2= 100, x3= 230
optimal value:max Z =1350
shadow prices:1.0, 2.0 and 0.0 for constraints 1, 2 and 3, respectively
opportunity costs: 3.0, 0 and 0 for variables X1, x2, x3, respectively
RANGES ON OBJECTIVE FUNCTION COEFFICIENTS
| variable | lower limits | present value | upper limit |
|---|---|---|---|
| x1 |  | 4.9 | 7.0 |
| x2 | 0 | 2.0 | 10.0 |
| x3 | 3.0 | 5.0 |  |
RANGES ON RHS CONSTANTS
| Row | Lower Limit | Present Value | Upper Limit |
|---|---|---|---|
| 1 | 230 | 430 | 455 |
| 2 | 410 | 460 | 860 |
| 3 | 400 | 450 |  |
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 the capacity of machine 2 can be increased by another 200 minutes at a cost of $250. Is it economical to do so? Explain.(e) Because of an increase in the cost of energy for operating the machines, the units profits of A, B, and C decrease by $2.0, $0.50, and $1.0, respectively. How will it affect the optimal solution and maximum profit?
Expert Answer:
a The revised profit of C still remains in between the upper and lower limit So the optima... View the full answer
