A farmer in Kentucky has 3,000 acres of land on which she intends to plant corn, oats, soybeans, and wheat, and this total acreage cannot be exceeded. Each acre of corn costs $125 for preparation, requires 7 worker-days of labor, and yields a profit of $285. Each acre of oats costs $90 for preparation, requires 8 worker-days of labor, and yields a profit of $185. An acre of soybeans costs $90 to prepare, requires 9 worker-days of labor, and yields a profit of $180. An acre of wheat costs $80 to prepare, requires 10 worker-days of labor, and yields a profit of $175. The farmer has taken out a loan of $250,000 for crop preparation and has contracted with a union for 25,000 worker-days of labor. A granary has agreed to purchase 800 acres of corn, 1000 acres of oats, 600 acres of wheat, and 600 acres of soybeans. The farmer has established the following goals, in order of their importance:
The farmer cannot plant more acres (3,000) than she owns.
The farmer desires a profit of at least $300,000 to remain in good financial condition.
The farmer would like to meet the sales agreement with the granary. However, the goal should be weighted according to the profit returned by each crop.
To maintain good relations with the union, the labor contract must be honored; that is, the full 25,000 worker-days of labor contracted for must be used.
Contracting for excess labor should be avoided.
The farmer would like to use as much of the available acreage as possible.
Preparation costs should not exceed the loan amount so that additional loans will not have to be secured.
Formulate a goal programming model to determine the number of acres of each crop the farmer should plant to satisfy the goals in the best possible way.
Solve this model using the computer.