A company sells two products with information as follows: $$\begin{array} { | l | r | r | }
\hline & \mathrm { A } & \mathrm { B } \
\hline \text { Sales price per unit } & \$ 11 & \$ 26 \
\hline \text { Variable cost per unit } & \$ 10 & \$ 10 \
\hline
\end{array}$$ The products are machine made.Four units of product A can be made with one machine hour,and two units of product B can be made with one machine hour.The company has a maximum of 3,000 machine hours available per month.Assume there are no constraints on sales of either product,and the company can choose any product mix they wish.What is the maximum amount of contribution margin that the company could earn in a month?

Question

A company sells two products with information as follows: $$\begin{array} { | l | r | r | } 
\hline & \mathrm { A } & \mathrm { B } \
\hline \text { Sales price per unit } & \$ 11 & \$ 26 \
\hline \text { Variable cost per unit } & \$ 10 & \$ 10 \
\hline
\end{array}$$ The products are machine made.Four units of product A can be made with one machine hour,and two units of product B can be made with one machine hour.The company has a maximum of 3,000 machine hours available per month.Assume there are no constraints on sales of either product,and the company can choose any product mix they wish.What is the maximum amount of contribution margin that the company could earn in a month?

Quizplus · Accepted Answer

To determine the maximum contribution margin, we need to use linear programming to find the optimal product mix. Let:
- x = number of units of product A
- y = number of units of product B

We want to maximize:
Contribution Margin = (Profit per unit of A * number of units of A) + (Profit per unit of B * number of units of B)

The profit per unit of A is $70 and the profit per unit of B is $50, so our objective function is:
Contribution Margin = 70x + 50y

We now need to set up our constraints. The first constraint is the machine hour constraint, which limits the total number of machine hours available to 3,000 per month:
4x + 2y ≤ 3,000

The second constraint is the non-negativity constraint, which means that we cannot have negative units of either product:
x ≥ 0
y ≥ 0

We can plot these constraints on a graph and find the feasible region:

```
2y = -4x + 3000
y = -2x + 1500
```

To maximize the contribution margin, we need to find the corner point within the feasible region that produces the highest value. The corner points are (0, 0), (375, 1125), and (750, 750).

Plugging in these values into our objective function, we get:
- (0, 0): Contribution Margin = 0
- (375, 1125): Contribution Margin = $96,000
- (750, 750): Contribution Margin = $90,000

Therefore, the optimal product mix is 375 units of product A and 1,125 units of product B, and the maximum contribution margin is $96,000 per month.

A Company Sells Two Products with Information as Follows The Products Are Machine Made