A farmer is planning his spring planting He has 20 acres on

A farmer is planning his spring planting. He has 20 acres on which he can plant a combination Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each crop. Each crop also requires fertilizer and irrigation water which are is short supply. The following table summarizes the data for the problem.

Solution

Linear programming model

let x=acres of corn,

y=acres of pumpkin

z=acres of beans

We want to maximize the profit

Max p: 2100x +900y+ 10500z

subject to:

21x <=200

10y <=180

3.5 <=80

2x+3y+z <= 50

5x+4y+3z <=80

x,y,z =>0,

To solve this we use simplex method

Tableau #1

x y z s1 s2 s3 s4 s5 p   
21 0 0 1 0 0 0 0 0 200
0 10 0 0 1 0 0 0 0 180
0 0 7/2 0 0 1 0 0 0 80   
2 3 1 0 0 0 1 0 0 50   
5 4 3 0 0 0 0 1 0 80   

-2100 -900 -10500 0 0 0 0 0 1 0

Tableau #2
x y z s1 s2 s3 s4 s5 p   
21 0 0 1 0 0 0 0 0 200
0 10 0 0 1 0 0 0 0 180
0 0 1 0 0 2/7 0 0 0 160/7
2 3 0 0 0 -2/7 1 0 0 190/7
5 4 0 0 0 -6/7 0 1 0 80/7   
-2100 -900 0 0 0 3000 0 0 1 240000

Tableau #3
x y z s1 s2 s3 s4 s5 p   
0 -84/5 0 1 0 18/5 0 -21/5 0 152
0 10 0 0 1 0 0 0 0 180
0 0 1 0 0 2/7 0 0 0 160/7
0 7/5 0 0 0 2/35 1 -2/5 0 158/7
1 4/5 0 0 0 -6/35 0 1/5 0 16/7   
0 780 0 0 0 2640 0 420 1 244800

Optimal Solution: p = 244800; x = 16/7, y = 0, z = 160/7