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 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 formulated to determine the optimal product mix:

Maximize Z = 4x1 + 2x2 + 5x3

s.t. 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 pr 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 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 $5, $4, and $6, respectively. The following linear program is formulated to determine the optimal product mix:

Maximize Z = 5x1 + 4x2 + 6x3

s.t. 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 optimum solution for the problem is as follows:

(following figures are hypothetical)

optimal solution : x1 = 0, x2 = 120, x3 = 260

optimal value : maximum profit, Z = 1450

shadow prices: 2.0, 4.0, and 0.0 for constraints 1, 2 and 3 respectively

reduced costs : for x1 = 4.0, for x2 = 0, for x3 = 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

RANGES ON OBJECTIVE FUNCTION COEFFICIENTS

Variable Lower Limit Present Value Upper Limit

x1 - 5.0 7.0

x2 0 4.0 10.0

x3 3.0 6.0

Using the above information answer the following question:

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? Select the correct answer.

Your answer: An increase in machine 2 capacity will change Z by 4200 = 800 dollars. Since the cost of $250 is less than $800, it is economical to increase machine 2 capacity.

An increase in machine 2 capacity will change Z by 2200 = 400 dollars. Since the cost of $250 is less than $400, it is economical to increase machine 2 capacity.

An increase in machine 2 capacity will change Z by 0200 = 0 dollars. Therefore there is no need to change machine 2 capacity.

An increase in machine 2 capacity will change Z by 120200 = 24,000 dollars. Since the cost of $250 is less than $24,000, it is economical to increase machine 2 capacity.