linear programming

Question

A farmer owns 500 acres of farmland in North Carolina on which he grows wheat, corn and tobacco. Each acre of wheat costs $100, each acre of corn costs $125 and each acre of tobacco costs $200 to plant, cultivate, and harvest the crop. The farmer has a budget of $60,000 for next year. The government limits the number of acres of tobacco that can be planted to 120. The profit from each acre of wheat is $280, the profit from each acre of corn is $350 and the profit from each acre of tobacco is $560. The farmer wants to know how many acres of each crop to plant in order to maximize his profit.

Formulate a linear programming model for this problem by determining
a. The decision variables.
b. The objective function.
c. All the constraints.

Answer 1

how many acres of wheat to plant (x) costing 100$/acre, profit 280$/acre
how many acres of corn to plant (y) costing 125$/acre, profit 350$/acre
how many acres of tobacco to plant (z) costing 200$/acre, profit 560$/acre

 

Decision Variables: how many acres of wheat to plant (x) 

how many acres of corn to plant (y) 

how many acres of tobacco to plant (z)

 

Objective function: Want to maximize profit P=280x+350y+560z

 

Constraints:

x>=0, y>=0, 120>=z>=0

100x+125y+200z<=60,000 total money to spend on crops

x+y+z<=500 total available land to plant crops on

 

Answer 2

Anna

 

The decision variables are acres of corn, wheat, and tobacco.

The objective function is the profit of growing all three crops

The constraints are his budget and acreage and the government limit on tobacco.

 

Z profit is 280x+350y+560z

Constraints:

X,y,z>=0

100x+125y+200z<=60000

X+y+z<=500

Z<=120

 

A general solution wold maximize the tobacco profit at 120 acres and then distribute the remaining acreage for growing coRn or wheat.