Question

# Furnco manufactures desks and chairs. Each desk uses 4 units of wood, and each chair uses 3. A desk contributes $40 to profit, and a chair contributes$25. Marketing restrictions require that the number of chairs produced be at least twice the number of desks produced. If 20 units of wood are available, formulate an LP to maximize Furnco’s profit. Then graphically solve the LP.

Solution

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$\text{\underline{\color{#4257b2}Decision variables}}$

We will use $x_1 =$ number of desks, $x_2 =$ number of chairs.

$\text{\underline{\color{#4257b2}Objective function}}$

We must maximize the profit. This means that the objective function is

$\max \ z = 40x_1 + 25x_2$

$\text{\underline{\color{#4257b2}Constraints}}$

Constraint 1. The number of chairs must be at least twice the number of desks. Thus, we have $x_2 \geqslant 2x_1$, which can be written as $2x_1 - x_2 \leqslant 0$. Constraint 2. Notice that $x_1$ desks and $x_2$ chairs require $4x_1 + 3x_2$ units of wood. Therefore, $4x_1 + 3x_2 \leqslant 20$.

$\text{\underline{\color{#4257b2}Sign restrictions}}$

Clearly $x_1 \geqslant 0$ and $x_2 \geqslant 0$.

$\text{\underline{\color{#4257b2}Conclusion}}$

\begin{align*} \max \ z &= 40x_1 + 25x_2 \tag{Objective function}\\ 2x_1 - x_2 &\leqslant 0 \tag{Marketing restriction}\\ 4x_1 + 3x_2 &\leqslant 20 \tag{Wood restriction}\\ x_1 \phantom{+.x_2}&\geqslant 0 \tag{Sign restriction} \\ x_2 & \geqslant 0 \tag{Sign restriction} \end{align*}

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