A farmer has 200 acres and 18,000 man-hours available. He wishes to determine th

Question

A farmer has 200 acres and 18,000 man-hours available. He wishes to determine the acreage allocated to the following products: corn, wheat, okra, tomatoes, and green beans. The farmer must produce at least 250 tons of corn to feed his hogs and cattle, and he must produce at least 80 tons of wheat, which he has precontracted. The tonnage and labor in man-hours per acre of the different products are summarized below:

CORN WHEAT OKRA TOMATOES BEANS
Tons/acre 10 4 4 8 6
Man-hours/acre 120 150 100 80 120

The corn, wheat, okra, tomatoes, and beans can be sold for $120.00, $150.00,
$80.00, $60.00, and $75.00 per ton.

Explanation / Answer

Linear programming (LP, or linear optimization) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships. CORN..x1 WHEAT..x2 OKRA..x3 TOMATOES..x4 BEANS..x5 S...Selling Price Maximize: S1x1 + S2x2 + S3x3 + S4x4 + S5x5 (maximize the revenue—revenue is the "objective function") Subject to: 0 = x1+ x2 +x3 +x4 +x5= L (limit on total area) x1 = 0, x2 = 0,x3 = 0, x4 = 0,x5 = 0 (cannot plant a negative area).

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