A company sells two products with information as follows: $$\begin{array} { | l | r | r | }
\hline & { \text { A } } &{ \text { B } } \
\hline \text { Price per unit } & \$ 12 & \$ 20 \
\hline \text { Variable cost per unit } & \$ 10 & \$ 12 \
\hline
\end{array}$$ Products are made by machine. 4 units of Product A can be made with one machine hour and 2 units of Product B can be made with one machine hour. The company has a maximum of 3,000 machine hours available per month. The company can sell up to 7,000 units of Product A per month, and up to 3,000 units of Product B for the month. What is the optimum product mix to maximize company's operating income?

Question

A company sells two products with information as follows: $$\begin{array} { | l | r | r | } 
\hline &  { \text { A } } &{ \text { B } } \
\hline \text { Price per unit } & \$ 12 & \$ 20 \
\hline \text { Variable cost per unit } & \$ 10 & \$ 12 \
\hline
\end{array}$$ Products are made by machine. 4 units of Product A can be made with one machine hour and 2 units of Product B can be made with one machine hour. The company has a maximum of 3,000 machine hours available per month. The company can sell up to 7,000 units of Product A per month, and up to 3,000 units of Product B for the month. What is the optimum product mix to maximize company's operating income?

Quizplus · Accepted Answer

Let's denote the number of units of Product A and Product B produced as $x$ and $y$, respectively.

The objective function is to maximize operating income, which is given by:

\begin{align*}
	ext{Operating Income} &= 	ext{Total Revenue} - 	ext{Total Cost} \
&= 12x + 20y - (10x + 12y) \
&= 2x + 8y
\end{align*}

The constraints can be written as:

\begin{align*}
x &\leq 7,000 \
y &\leq 3,000 \
4x + 2y &\leq 3,000
\end{align*}

The first two constraints represent the maximum demand for each product, and the third constraint represents the maximum number of machine hours available.

The problem can be solved using linear programming. However, we can also use some intuition to find the optimal solution.

One machine hour can produce either 4 units of Product A or 2 units of Product B. Therefore, the company can produce a maximum of $\frac{1}{4}(3,000) = 750$ units of Product A or $\frac{1}{2}(3,000) = 1,500$ units of Product B. Since the demand for Product A is much higher, we should produce as much Product A as possible, subject to the constraint on machine hours. Specifically, we can produce the maximum possible number of units of Product A (i.e., 750) and then use the remaining machine hours to produce Product B.

Using 750 machine hours for Product A production yields $750/4 = 187.5$ units of Product A, which is less than the maximum demand of 7,000 units. Therefore, we can produce 187 units of Product A and use the remaining machine hours for Product B production. The maximum number of units of Product B that can be produced is $\frac{1}{2}(3,000 - 4 	imes 187) = 1,126$. However, the demand for Product B is only 3,000 units, so we should produce only 1,126 units of Product B.

Thus, the optimal solution is to produce 187 units of Product A and 1,126 units of Product B, which gives an operating income of $2(187) + 8(1,126) = 8,562$. Choice B is the only option that matches this solution.

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