The Badminton Company has 4,000 machine hours available annually to manufacture racquets. The following information is available for the two different racquets produced by Badminton: $$\begin{array} { | l | r | }
\hline \text { Pro } & \
\hline \text { Unit sales price } & \$ 300 \
\hline \text { Unit variable costs } & \$ 150 \
\hline \text { Annual demand } & 2,000 \text { units } \
\hline \text { Machine time } & 1.50 \text { hours per unit } \
\hline \text { Mid } & \
\hline \text { Unit sales price } & \$ 150 \
\hline \text { Unit variable costs } & \$ \mathbf { ~ } \
\hline \text { Annual demand } & \mathbf { 4 , 0 0 0 } \text { units } \
\hline \text { Machine time } & \underline { \mathbf { 2 } \text { hours per unit } } \
\hline
\end{array}$$ How many units of each racquet should be manufactured for the company to maximize its operating income?

Question

The Badminton Company has 4,000 machine hours available annually to manufacture racquets. The following information is available for the two different racquets produced by Badminton: $$\begin{array} { | l | r | } 
\hline \text { Pro } & \
\hline \text { Unit sales price } & \$ 300 \
\hline \text { Unit variable costs } & \$ 150 \
\hline \text { Annual demand } & 2,000 \text { units } \
\hline \text { Machine time } & 1.50 \text { hours per unit } \
\hline \text { Mid } & \
\hline \text { Unit sales price } & \$ 150 \
\hline \text { Unit variable costs } & \$ \mathbf { ~ } \
\hline \text { Annual demand } & \mathbf { 4 , 0 0 0 } \text { units } \
\hline \text { Machine time } & \underline { \mathbf { 2 } \text { hours per unit } } \
\hline
\end{array}$$ How many units of each racquet should be manufactured for the company to maximize its operating income?

Quizplus · Accepted Answer

Let x = number of units of Pro racquet to be produced
Let y = number of units of Mid racquet to be produced

The objective is to maximize operating income, which is given by:
Operating Income = Total Revenue - Total Variable Costs

Total Revenue = Pro Revenue + Mid Revenue
Total Variable Costs = Pro Variable Costs + Mid Variable Costs

Pro Revenue = x(300)
Mid Revenue = y(150)

Pro Variable Costs = x(150)
Mid Variable Costs = y(?)

Since only the unit variable costs for the Pro racquet are given, we need to calculate the unit variable costs for the Mid racquet:

Total Machine Hours = x(1.5) + y(2)
Total Machine Hours ≤ 4,000

We know that the company has 4,000 machine hours available annually, so we set up the inequality:

x(1.5) + y(2) ≤ 4,000

Solving for y, we get:

y ≤ (4,000 - x(1.5))/2

The unit variable costs for the Mid racquet can now be calculated using the following formula:

Unit Variable Costs = (Selling Price - Operating Income)/Units

For the Pro racquet:
Unit Variable Costs = (300 - Operating Income)/x

For the Mid racquet:
Unit Variable Costs = (150 - Operating Income)/y

Substituting the expressions for Total Revenue and Total Variable Costs into the equation for Operating Income, we get:

Operating Income = 300x - 150x + 150y - (x(300 - Operating Income)/x) - (y(150 - Operating Income)/y)

Simplifying, we get:

Operating Income = 150x + 150y - 300 - (150x/(x+y)) - (150y/(x+y))

To maximize operating income, we take the partial derivatives with respect to x and y, set them equal to zero, and solve for x and y:

d(Operating Income)/dx = 150 - 150/(x+y) = 0
d(Operating Income)/dy = 150 - 150/(x+y) = 0

Solving for x and y, we get:

x = 2,000
y = 500

Since y is less than the maximum allowed by the machine hours constraint, the solution is:
2,000 units of Pro and 500 units of Mid.

The Badminton Company Has 4,000 Machine Hours Available Annually to Manufacture