Question

A farmer has a 40-acre farm in Georgia. The farmer is trying to determine how many acres of corn, peanuts, and cotton to plant. Each crop requires labor, fertilizer, and insecticide. The farmer has developed the following linear programming model to determine the number of acres of corn (X1), peanuts (X2), and cotton (X3) to plant in order to maximize profit. Max 550 X1 350 X2 450 X3 s.t. Constraint 1: 2 X1 3 X2 2 X3

Answer 1

z (max) = 17461.54 $

x₁ = 29.23 acres

x₂ = 0

x₃ = 3.07

Step-by-step explanation: INCOMPLETE QUESTION.

As the problem statement establishes: "Each crop requires labor, fertilizer, and insecticide" and information about quantities and availability does not exist. To build a model and that such model would be feasible I copy from the internet the following data. We assume the problem is to maximize the number of acres to plant with a maximum of profit ( we will use as profit the numbers 550 ; 350 ; 450 for acres of corn, peanuts, and cotton)

Then z = 550*x₁ + 350*x₂ + 450*x₃ to maximize

labor (h) fertilizer ( tn) Insecticide (Tn)

acres of corn (x₁) 2 4 3

acres of peanut (x₂) 3 3 2

acres of cotton (x₃) 2 1 4

Availability acres 40 120 120 100

Constraints:

1) Size of the farm 120 acres

x₁ + x₂ + x₃ ≤ 40

2) labor 120 h

2*x₁ + 3*x₂ + 2*x₃ ≤ 120

3) Fertilizer 120 Tn

4*x₁ + 3*x₂ + 1*x₃ ≤ 120

4) Insecticide 100 Tn

3*x₁ + 2*x₂ + 4*x₃ ≤ 100

The Model is:

z = 550*x₁ + 350*x₂ + 450*x₃ to maximize

Subject to:

x₁ + x₂ + x₃ ≤ 40

2*x₁ + 3*x₂ + 2*x₃ ≤ 120

4*x₁ + 3*x₂ + 1*x₃ ≤ 120

3*x₁ + 2*x₂ + 4*x₃ ≤ 100

x₁ ≥ 0 ; x₂ ≥ 0 ; x₃ ≥ 0

After 3 iterations using an on-line solver optimal solution is:

z (max) = 17461.54 $

x₁ = 29.23 acres

x₂ = 0

x₃ = 3.07