Question

A farmer in Addison County has 600 acres of land that can be used to grow corn

and/or soybeans. The net profit per acre is $60 for soybeans and $55 for corn. In addition to   

the land constraint, the farmer has limited labor resources: 200 hours of labor available for

planting, and 300 hours of labor available for cultivation. Labor required for planting is 0.6

hour per acre for corn and 0.7 hour per acre for soybeans. Cultivation is only necessary for

corn production which requires 0.5 hour per acre. Also, corn yield is known to be 70

bushels per acre and the farmer does not want to produce more than 21,000 bushels of corn

due to the limited storage space. Develop a linear programming (LP) model for the farmer  

to maximize the total profit (you do not need to solve your LP model) in three steps

(1)

(2)

(3)

Answer 1

1. Decision Variables:

Let the number of acres of Soyabean = x1

Number of acres of corn = x2

2. Objective Function:

We have to maximize the profit.

Profit per acre of Soyabean = $60

Profit per acre of corn = $55

Maximize 60*x1 + 55*x2

3. Constraints:

Labor required for planting is 0.6 Hour per acre for corn and 0.7 hour per acre for soybeans. Cultivation is only necessary for Corn production which requires 0.5 hour per acre.

Available time of planting 200 hours

Available time of cultivation 300 hours

The used time should not exceed the available time.

For planting:

0.7*x1 + 0.6*x2 <= 200

For cultivation:

0.5*x2 <= 300

Corn yield is known to be 70 bushels per acre and the farmer does not want to produce more than 21,000 bushels of corn due to the limited storage space.

70*x2 <= 21000

Total 600 acres available. hence used acres should not exceed 600 acres.

x1 + x2 <= 600

None of the acres can be negative.

x1 >= 0

x2 >= 0

.

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