A farmer in the Midwest has 1,000 acres of land on which she intends to plant corn, wheat, and soybeans. Each acre of corn costs $100 for preparation, requires 7 worker-days of labor, and yields a profit of $30. An acre of wheat costs $120 to prepare, requires 10 worker-days of labor, and yields $40 profit. An acre of soybeans costs $70 to prepare, requires 8 worker-days, and yields $20 profit. The farmer has taken out a loan of $80,000 for crop preparation and has contracted with a union for 6,000 worker-days of labor. A midwestern granary has agreed to purchase 200 acres of corn, 500 acres of wheat, and 300 acres of soybeans. The farmer has established the following goals, in order of their importance:
(1) To maintain good relations with the union, the labor contract must be honored; that is, the full 6,000 worker-days of labor contracted for must be used.
(2) Preparation costs should not exceed the loan amount so that additional loans will not have to be secured.
(3) The farmer desires a profit of at least $105,000 to remain in good financial condition.
(4) Contracting for excess labor should be avoided.
(5) The farmer would like to use as much of the available acreage as possible.
(6) 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.
a. 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.
b. Solve this model by using the computer.