Solutions to Lp Problems

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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 farmer. a) Draw isoprofit lines where the total profit equals $6,000, $8,000, $10,000, and $12,000. Here are the points where the isoprofit lines cross the axes: | |X1 |X2 |

|z = 6,000 |0 |20 |
| |30 |0 |
|z = 8,000 |0 |26.67 |
| |40 |0 |
|z = 10,000 |0 |33.33 |
| |50 |0 |
|z = 12,000 |0 |40 |
| |60 |0 |

Your isoprofit lines ought to look like this:
[pic]

b) Shade in the feasible region.
[pic]

c) Determine the planting scheme that maximizes total profit. There are four critical points: Point A (at the origin), Point B (where the fertilizer constraint crosses the non-negativity constraint on wheat), Point C (where the fertilizer constraint line crosses the labor constraint), and Point D, where the labor constraint crosses the non-negativity constraint on corn). The best solution is at Point C (20, 20). Plant 20 acres of wheat and 20 acres of corn.

d) What is the optimal total profit?
The optimal total profit is $10,000.

3. A bank is attempting to determine where its assets should be invested during the current year. At present, $500,000 is available for investment in bonds, home loans, auto loans, and personal loans. The annual rates of return on each type of investment are known to be the following: bonds, 10%; home loans, 16%; auto loans, 13%; and personal loans, 20%. To ensure that the bank’s portfolio is not too risky, the bank’s investment manager has placed the following three restrictions on the bank’s portfolio: • The amount invested in personal loans cannot exceed the amount invested in bonds. • The amount invested in home loans cannot exceed the amount invested in auto loans. • No more than 25% of the total amount invested may be in personal loans. Below are various elements of the Excel model used to solve the problem: the spreadsheet model, the Solver parameters, the Solver...
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