Consider the following linear programming model Min Z 8x1

Consider the following linear programming model Min Z = 8x_1 + 5x_2 subject to -3x_1 + 2x_2 lessthanorequalto 30 2x_1 + x_2 Greaterthanorequalto 50 x_1 + x_2 Greaterthanorequalto 30 and x_1 Greaterthanorequalto 0, x_1 Greaterthanorequalto 0 Solve the above problem using the Simplex method.

Solution

Optimise : Z = 8x1 +5x2

subject to : x1>=0 ; x2>=0

-3x1 +2x2 <= 30

2x1 +x2 >=50

x1+x2 >=30

Form the tableaus from the inequalties above:

initial tableau of Simplex method consists of all the coefficients of the decision variables of the original problem and the slack, surplus and artificial variables added in second step

Tableau #1
x1 x2 s1 s2 s3 s4 s5 -p
1 0 -1 0 0 0 0 0 0
0 1 0 -1 0 0 0 0 0
-3 2 0 0 1 0 0 0 30   
2 1 0 0 0 -1 0 0 50   
1 1 0 0 0 0 -1 0 30   
8 5 0 0 0 0 0 1 0

Tableau #2
x1 x2 s1 s2 s3 s4 s5 -p
-1 0 1 0 0 0 0 0 0
0 1 0 -1 0 0 0 0 0
-3 2 0 0 1 0 0 0 30   
2 1 0 0 0 -1 0 0 50   
1 1 0 0 0 0 -1 0 30   
8 5 0 0 0 0 0 1 0

Tableau #3
x1 x2 s1 s2 s3 s4 s5 -p
-1 0 1 0 0 0 0 0 0
0 -1 0 1 0 0 0 0 0
-3 2 0 0 1 0 0 0 30   
2 1 0 0 0 -1 0 0 50   
1 1 0 0 0 0 -1 0 30   
8 5 0 0 0 0 0 1 0

Tableau #4
x1 x2 s1 s2 s3 s4 s5 -p
0 0.5 1 0 0 -0.5 0 0 25   
0 -1 0 1 0 0 0 0 0
0 3.5 0 0 1 -1.5 0 0 105
1 0.5 0 0 0 -0.5 0 0 25   
0 0.5 0 0 0 0.5 -1 0 5
0 1 0 0 0 4 0 1 -200   

Tableau #5
x1 x2 s1 s2 s3 s4 s5 -p
0 0 1 0 0 -1 1 0 20   
0 0 0 1 0 1 -2 0 10   
0 0 0 0 1 -5 7 0 70   
1 0 0 0 0 -1 1 0 20   
0 1 0 0 0 1 -2 0 10   
0 0 0 0 0 3 2 1 -210   

Optimal Solution: Z = 210 with x1 = 20, x2 = 10