A farmer has 100 acres that can be used for growing wheat or corn. The yield is 60 bushels per acer per year of wheat or 95 bushels of corn. Any fraction of the 100 acres can be devoted to growing wheat or corn. Labor requirements are four hours per acre per year, plus 0.15 hours per bushel of wheat and 0.70 hour per bushel of corn. The cost of seed, fertilizer, etc., is 20 and 12 cents per bushel of wheat and corn, respectively. Wheat can be sold for $1.75 per bushel, and corn for $0.95 per bushel. Wheat can be bought for $2.50 per bushel, and corn for $1.50 per bushel.
In addition the farmer may raise pigs and poultry. The farmer sells the livestock when they reach the age of one year. A pig sells for $40. He measures the poultry in terms of one ``pig equivalent" (the number of chickens needed to bring in $40 at the time of sale). One pig requires 25 bushels of wheat or 20 bushels of corn, plus 25 hours of labor and 25 square feet of floor space. An equivalent amount of poultry requires 25 bushels of corn or 10 bushels of wheat, 40 hours of labor, and 15 square feet of floor space.
The farmer has 10,000 square feet of floor space. This year he has available 2000 hours of his own time, 20000 hours from his family, and can hire labor at $1.50 per hour. However, for each hour of hired labor, 0.15 hour of the farmer's time is required for supervision.
Formulate a linear program to determine how much land should be devoted to corn and wheat,
and how many pigs and poultry (if any) should be raised to maximize the farmer's profits.