Use the simplex method to maximize the objective function z

Use the simplex method to maximize the objective function z = 9x_1 + 30x_2 subject to the following constraints. Assume that x_1 greaterthanorequalto 0, x_2 greaterthanorequalto 0 {2x_1 + 10x_2 lessthanorequalto 50 2x_1 + x_2 lessthanorequalto 20.

Solution

Using Simplex method, rewriting the given equations in tableau form, we get :

Tableau #1

x1 x2 s1 s2 s3 s4 z     
2 10 1 0 0 0 0 50   
2 1 0 1 0 0 0 20   
1 0 0 0 -1 0 0 0
0 1 0 0 0 -1 0 0
-9 -30 0 0 0 0 1 0

Tableau #2
x1 x2 s1 s2 s3 s4 z   
2 10 1 0 0 0 0 50   
2 1 0 1 0 0 0 20   
-1 0 0 0 1 0 0 0
0 1 0 0 0 -1 0 0
-9 -30 0 0 0 0 1 0

Tableau #3
x1 x2 s1 s2 s3 s4 z   
2 10 1 0 0 0 0 50   
2 1 0 1 0 0 0 20   
-1 0 0 0 1 0 0 0
0 -1 0 0 0 1 0 0
-9 -30 0 0 0 0 1 0

Tableau #4
x1 x2 s1 s2 s3 s4 z     
1/5 1 1/10 0 0 0 0 5
9/5 0 -1/10 1 0 0 0 15   
-1 0 0 0 1 0 0 0
1/5 0 1/10 0 0 1 0 5
-3 0 3 0 0 0 1 150

Tableau #5
x1 x2 s1 s2 s3 s4 z   
0 1 1/9 -1/9 0 0 0 10/3   
1 0 -1/18 5/9 0 0 0 25/3   
0 0 -1/18 5/9 1 0 0 25/3   
0 0 1/9 -1/9 0 1 0 10/3   
0 0 17/6 5/3 0 0 1 175

So the Optimal Solution is:

z = 175;

x1 = 25/3,

x2 = 10/3