Solutions to Lp Problems
Solutions to LP Practice Problems[1]
1. Furnco manufactures desks and chairs. Each desk uses 4 units of wood, and each chair uses 3 units of wood. 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. There are 20 units of wood available.
Using the graph below, determine a production plan that maximizes Furnco’s profit. a) Draw isoprofit lines where the total profit equals 125, 150, 175, and 200.
Here are the points where the isoprofit lines cross the axes:
| |X1 |X2 |
|z = 125 |0 |5 |
| |3.125 |0 |
|z = 150 |0 |6 |
| |3.75 |0 |
|z = 175 |0 |7 |
| |4.375 |0 |
|z = 200 |0 |8 |
| |5 |0 |
Your isoprofit lines ought to look like this:
[pic]
Shade in the feasible region.
[pic]
b) Determine a daily production plan that maximizes total profit.
There are three critical points: Point A (at the origin), Point B (where the wood constraint crosses the non-negativity constraint on desks), and Point C (where the wood constraint line crosses the marketing constraint).
The best solution is at Point C (2, 4). Produce 2 desks and 4 chairs.
c) What is the optimal total profit?
Point C is (2, 4), where the total profit is $180.
2. A farmer in Iowa owns 45 acres of land. She is going to plant each acre 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 the table below. 100 workers and 120 tons of fertilizer are available.
| |Wheat |Corn |
|Labor |3 workers |2 workers |
|Fertilizer |2 tons |4 tons |
Using the graph below, determine the planting scheme that will maximize profit for the
1. Furnco manufactures desks and chairs. Each desk uses 4 units of wood, and each chair uses 3 units of wood. 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. There are 20 units of wood available.
Using the graph below, determine a production plan that maximizes Furnco’s profit. a) Draw isoprofit lines where the total profit equals 125, 150, 175, and 200.
Here are the points where the isoprofit lines cross the axes:
| |X1 |X2 |
|z = 125 |0 |5 |
| |3.125 |0 |
|z = 150 |0 |6 |
| |3.75 |0 |
|z = 175 |0 |7 |
| |4.375 |0 |
|z = 200 |0 |8 |
| |5 |0 |
Your isoprofit lines ought to look like this:
[pic]
Shade in the feasible region.
[pic]
b) Determine a daily production plan that maximizes total profit.
There are three critical points: Point A (at the origin), Point B (where the wood constraint crosses the non-negativity constraint on desks), and Point C (where the wood constraint line crosses the marketing constraint).
The best solution is at Point C (2, 4). Produce 2 desks and 4 chairs.
c) What is the optimal total profit?
Point C is (2, 4), where the total profit is $180.
2. A farmer in Iowa owns 45 acres of land. She is going to plant each acre 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 the table below. 100 workers and 120 tons of fertilizer are available.
| |Wheat |Corn |
|Labor |3 workers |2 workers |
|Fertilizer |2 tons |4 tons |
Using the graph below, determine the planting scheme that will maximize profit for the