Related questions with answers

Leary Chemical manufactures three chemicals: A, B, and C. These chemicals are produced via two production processes: 1 and 2. Running process 1 for an hour costs $4 and yields 3 units of A, 1 of B, and 1 of C. Running process 2 for an hour costs$1 and produces 1 unit of A and 1 of B. To meet customer demands, at least 10 units of A, 5 of B. and 3 of C must be produced daily. Graphically determine a daily production plan that minimizes the cost of meeting Leary Chemical’s daily demands.

Farmer Jane owns 45 acres of land. She is going to plant each with wheat or corn. Each acre planted with wheat yields $200 profit; each with corn yields$300 profit. The labor and fertilizer used for each acre are given in Table 1. One hundred workers and 120 tons of fertilizer arc available. Use linear programming to determine how Jane can maximize profits from her land. TABLE 1:

\begin{matrix} \text{ } & \text{Wheat} & \text{Corn}\\ \text{Labor} & \text{3 workers} & \text{2 workers}\\ \text{Fertilizer} & \text{2 tons} & \text{4 tons}\\ \end{matrix}

Truckco manufactures two types of trucks: 1 and 2. Each truck must go through the painting shop and assembly shop. If the painting shop were completely devoted to painting Type I trucks, then 800 per day could be painted; if the painting shop were completely devoted to painting Type 2 trucks, then 700 per day could be painted. If the assembly shop were completely devoted to assembling truck 1 engines, then 1,500 per day could be assembled; if the assembly shop were completely devoted to assembling truck 2 engines, then 1,200 per day could be assembled. Each Type 1 truck contributes $300 to profit; each Type 2 truck contributes$500. formulate an LP that will maximize Truckco’s profit.

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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Step 1

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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, $4x_1 + 3x_2 \leqslant 20$ .