Question

The profit function for two products is:

    Profit = −3x ₁² + 42x ₁ − 3x ₂² + 48x ₂ + 700,

where x ₁ represents units of production of product 1, and x ₂ represents units of production of product 2. Producing one unit of product 1 requires 5 labor-hours, and producing one unit of product 2 requires 6 labor-hours. Currently, 24 labor-hours are available. The cost of labor-hours is already factored into the profit function, but it is possible to schedule overtime at a premium of $5 per hour.

Formulate an optimization problem that can be used to find the optimal production quantity of products 1 and 2 and the optimal number of overtime hours to schedule.

Solve the optimization model you formulated. How much should be produced and how many overtime hours should be scheduled? If needed, round your answers to two decimal digits.

	Amount
Product 1		units
Product 2		units
Overtime Used		hours

Answer 1

Answer:

Given that,

The profit function for the two products is:

Where represents units of production of product 1, and represents units of production of product 2.

Producing one unit of product 1 requires 5 labor-hours, and producing one unit of product 2 requires 6 labor-hours.

Currently, 24 labor-hours are available. The cost of labor-hours is already factored into the profit function, but it is possible to schedule overtime at a premium of $5 per hour.

(a).

Formulate an optimization problem that can be used to find the optimal production quantity of products 1 and 2 and the optimal number of overtime hours to schedule:

Let and be the units of Product 1 and Product 2 respectively.

Profit Function (Maximization Function):

(b).

Solve the optimization model you formulated. How much should be produced and how many overtime hours should be scheduled:

I am using solver to solve the problem.

We are not allowed to upload excel. Hence uploading the screenshot of the same.

We are Maximising Profit, so our target cell would be Total Profit.

The Screenshot below shows the constraints that were used.

Solver Output:

Hence to maximize profit,

X1 = 3

And X2 = 2