A company sells two products with information as follows: $$\begin{array} { | l | c | c | }
\hline & \mathrm { A } & \mathrm { B } \
\hline \text { Sales price per unit } & \$ 12.00 & \$ 26.00 \
\hline \text { Variable cost per unit } & \$ 10.00 & \$ 12.00 \
\hline
\end{array}$$ 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 6000 machine hours available per month.The company can sell up to 18,000 units of product A per month,and up to 3000 units of product B for the month.What is the optimum product mix to maximize the company's operating income?

Question

A company sells two products with information as follows: $$\begin{array} { | l | c | c | } 
\hline & \mathrm { A } & \mathrm { B } \
\hline \text { Sales price per unit } & \$ 12.00 & \$ 26.00 \
\hline \text { Variable cost per unit } & \$ 10.00 & \$ 12.00 \
\hline
\end{array}$$ 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 6000 machine hours available per month.The company can sell up to 18,000 units of product A per month,and up to 3000 units of product B for the month.What is the optimum product mix to maximize the company's operating income?

Quizplus · Accepted Answer

To maximize the company's operating income, we need to maximize the total revenue while considering the available resources and constraints. Let's assume x units of product A and y units of product B are produced and sold in a month.

According to the given information, it takes 4 machine hours to produce 1 unit of A and 2 machine hours to produce 1 unit of B. So, the total machine hours required to produce x units of A and y units of B can be expressed as:

4x + 2y ≤ 6000     .......(1)

Also, the company can sell up to 18,000 units of product A per month and up to 3000 units of product B for the month. So, the total units sold of A and B combined should be less than or equal to:

x + y ≤ 18,000       
y ≤ 3000              .......(2)

The objective is to maximize the operating income, which is the total revenue minus the total cost. Let's assume the selling price of A and B are $pA and $pB respectively, and the cost of production per unit of A and B are $cA and $cB respectively. So, the total revenue and the total cost can be expressed as follows:

Total revenue = pA*x + pB*y
Total cost = (cA*4)*x + (cB*2)*y

The operating income, therefore, can be expressed as:

Operating income = Total revenue - Total cost 
                  = pA*x + pB*y - (cA*4)*x - (cB*2)*y 
                  = (pA - 4*cA)*x + (pB - 2*cB)*y

Now, we need to maximize the operating income subject to the constraints (1) and (2).

Using a graphical method or linear programming software, we can find that the maximum operating income occurs at the corner point where x=18,000 and y=3,000.

Therefore, the optimum product mix to maximize the company's operating income is 18,000 units of product A and 3,000 units of product B, which is the answer B.

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