Bandy Corporation owns a machine that manufactures lawn games.Production time for the croquet set is 10 units per hour and for the volley ball game is 8 units per hour.The machine's capacity is 1,500 hours per year.Both products are sold to a single customer who has agreed to buy all of the company's output up to a maximum of 4,000 croquet sets and 10,000 volleyball games.Selling prices and variable costs per unit are shown below.Based on this information, what is Bandy Corporation's most profitable sales mix? $\begin{array}{lcc}
& \text { Croquet Set } & \text { Volleyball Game } \
\text { Selling price per unit } & \$ 75 & \$ 62 \
\text { Variable costs per unit } & 42 & 25
\end{array}$
A)15,000 croquet sets.
B)12,000 volleyball games.
C)4,000 croquet sets and 10,000 volleyball games.
D)4,000 croquet sets and 8,800 volleyball games.
E)2,500 croquet sets and 10,000 volleyball games.

Question

Bandy Corporation owns a machine that manufactures lawn games.Production time for the croquet set is 10 units per hour and for the volley ball game is 8 units per hour.The machine's capacity is 1,500 hours per year.Both products are sold to a single customer who has agreed to buy all of the company's output up to a maximum of 4,000 croquet sets and 10,000 volleyball games.Selling prices and variable costs per unit are shown below.Based on this information, what is Bandy Corporation's most profitable sales mix? $\begin{array}{lcc} 
& \text { Croquet Set } & \text { Volleyball Game } \
\text { Selling price per unit } & \$ 75 & \$ 62 \
\text { Variable costs per unit } & 42 & 25
\end{array}$
A)15,000 croquet sets.
B)12,000 volleyball games.
C)4,000 croquet sets and 10,000 volleyball games.
D)4,000 croquet sets and 8,800 volleyball games.
E)2,500 croquet sets and 10,000 volleyball games.

Quizplus · Accepted Answer

We should use linear programming to find the most profitable sales mix. Let $x$ be the number of croquet sets and $y$ be the number of volleyball games. We want to maximize the total profit, which is given by the selling price minus the variable cost, multiplied by the number of units sold:

$$	ext{Profit}=75x-42x+62y-25y=33x+37y$$

We want to maximize this profit subject to the following constraints:

- Production constraint: the total production time cannot exceed 1,500 hours per year.

$$10x+8y\leq 1,500$$

- Sales constraints: we cannot sell more than 4,000 croquet sets or 10,000 volleyball games.

$$x\leq 4,000$$
$$y\leq 10,000$$

- Non-negativity constraints: we cannot produce or sell a negative number of units.

$$x\geq 0$$
$$y\geq 0$$

Putting all these constraints together, we get the following linear programming problem:

\begin{align*}
	ext{maximize } & 33x+37y \
	ext{subject to } & 10x+8y\leq 1,500 \
& x\leq 4,000 \
& y\leq 10,000 \
& x\geq 0 \
& y\geq 0
\end{align*}

Using a linear programming tool or solver, we find that the optimal solution is $x=4,000$ and $y=8,800$, with a total profit of $33	imes 4,000+37	imes 8,800=\$459,400$. Therefore, Bandy Corporation's most profitable sales mix is to produce and sell 4,000 croquet sets and 8,800 volleyball games.

Bandy Corporation Owns a Machine That Manufactures Lawn Games A)15,000 Croquet Sets