Answer to Question #237046 in Operations Research for opr

Question

A farmer has 10 acres to plant in wheat and rye. He has to plant at least 7 acres. However, he has only $1200 to spend and each acre of wheat costs $200 to plant and each acre of rye costs $100 to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to plant an acre of wheat and 2 hours to plant an acre of rye. If the profit is $500 per acre of wheat and $300 per acre of rye how many acres of each should be planted to maximize profits?

Answer 1

Let W= acres of Wheat planted

R= acres of Rye planted

Constraint on the total land available

W + R ≤ 10

Constraint on the least that can be planted

W + R ≥ 7

Max profit

f(W,R) = 500W + 300R

Constraint on cost

200W + 100R ≤ 1200

Constraint on Time

W + 2R ≤ 12

Taking coordinates (2,5), (5,2) and (4,4) from the diagram;

(W,R) f = 500W + 300R

(2,5). 500(2) + 300(5) = 2500

(5,2). 500(5) + 300(2) = 3100

(4,4). 500(4) + 300(4) = 3200

From our results, it can be deduced that for a maximum profit of $3200, the farmer should plant 4 acres of Wheat and 4 acres of Rye.