A farmer is planning for the coming year. Three crops are being considered. These are wheat, soybeans, and corn. The farm has 250 acres for crops, and his family can commit to 600 days of labor to the farm. They also have only $25,000 to cover input costs. Table 1 gives the time requirement, input cost, income per bushel, and yield per acre for each of the three crops. Note: to keep their farm supplement, they have to plant as least 10 acres of corn.
|
Table
|
Income per bushel
|
Days per acre
|
Bushels per acre
|
Input Cost per acre
|
|
Corn
|
$2.50
|
3
|
140
|
$150
|
|
Soybeans
|
$5.00
|
4
|
80
|
$100
|
|
Wheat
|
$3.00
|
2
|
10
|
$140
|
Let C=number of acres of corn, S=number of acres of soybeans, W= number of acres of wheat.
Note: For C thru I, refer back to the optimal solution from B as the start point for the question.
- Formulate and LP model to choose how much of each crop to grow so as to maximize income.
- What is the optimal solution?
- A neighbor has offered to rent all or part of a 50-acre section of land for $100 per acre. Describe the impact on the optimal solution to this optional available acreage.
- If additional labor can be hired for $50 per day, what is the impact on the solution?
- Sugar beets can be planted in addition to those above. Sugar beets provides $10 income per bushel, requires 4 days per acre for labor, produces 50 bushels per acre, and has an input cost of $50. Should sugar beets be included in the plan?
- Due to an uncertain marketplace, income per bushel for corn could be $3.50 per bushel. If this happens, what should they do?
- After solving the problem, the farmer discovers that he forgot to include a requirement for chemicals. He has 5000 pounds available. The usage is 10 pounds per acre for corn, 8 pounds per acre soybeans, and 7 pounds per acre for wheat. What is the impact of this additional constraint on the optimal solution?
- Explain the shadow price for each constraint.
- Explain the reduced cost for each decision variable.