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Project MAnagement, Operation Management

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A firm uses three machines in the manufacturing of three products:

Each unit of product 1 requires three hours on machine 1, two hours on machine 2 and one hour on machine 3.
Each unit of product 2 requires four hours on machine 1, one hour on machine 2 and three hours on machine 3.
Each unit of product 3 requires two hours on machine 1, two hours on machine 2 and two hours on machine 3.

The contribution margin of the three products is £30, £40 and £35 per unit, respectively.

The following are available for scheduling:

90 hours of machine 1 time
54 hours of machine 2 time
93 hours of machine 3 time

The linear programming formulation of this problem is as follows:

Maximise Z = 30X1 + 40X2 + 35X3

3X1 + 4X2 + 2X3 <= 90

2X1 + 1X2 + 2X3 <=54

X1 + 3X2 + 2X3 <=93

With X1, X2, X3 >= 0

To answer this question:

Answer each question and explain your reasoning or show your calculations.

What is the optimal production schedule for this firm? What is the profit contribution of each of these products?
What is the marginal value of an additional hour of time on machine 1? Over what range of time is this marginal value valid?
What is the opportunity cost associated with product 1? What interpretation should be given to this opportunity cost?
How many hours are used for machine 3 with the optimal solution?
How much can the contribution margin for product 2 change before the current optimal solution is no longer optimal?

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