Use the simplex method to solve Minimize Z3x1 x2 x3 x4 Subj

Use the simplex method to solve: Minimize: Z=3x_1 +x_2 + x_3 +x_4 Subject to: -2x_1 +2x_2 + x_3 =4 3x_1 +x_2 +x_4 =6 x_1, x_2, x_3, x_4 greaterthanoverequalto 0 Find an alternative optimal solution if one exists.

Solution

Tableau #1
x1 x2 x3 x4 s1 s2 s3 s4 s5 s6 s7 s8 p   
-2 2 1 0 1 0 0 0 0 0 0 0 0 4
3 1 0 1 0 1 0 0 0 0 0 0 0 6
1 0 0 0 0 0 -1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 -1 0 0 0 0 0 0
0 0 1 0 0 0 0 0 -1 0 0 0 0 0
0 0 0 1 0 0 0 0 0 -1 0 0 0 0
-2 2 1 0 0 0 0 0 0 0 -1 0 0 4
3 1 0 1 0 0 0 0 0 0 0 -1 0 6
-3 -1 -1 -1 0 0 0 0 0 0 0 0 1 0

Tableau #2
x1 x2 x3 x4 s1 s2 s3 s4 s5 s6 s7 s8 p     
-2 2 1 0 1 0 0 0 0 0 0 0 0 4
3 1 0 1 0 1 0 0 0 0 0 0 0 6
-1 0 0 0 0 0 1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 -1 0 0 0 0 0 0
0 0 1 0 0 0 0 0 -1 0 0 0 0 0
0 0 0 1 0 0 0 0 0 -1 0 0 0 0
-2 2 1 0 0 0 0 0 0 0 -1 0 0 4
3 1 0 1 0 0 0 0 0 0 0 -1 0 6
-3 -1 -1 -1 0 0 0 0 0 0 0 0 1 0

Tableau #3
x1 x2 x3 x4 s1 s2 s3 s4 s5 s6 s7 s8 p   
-2 2 1 0 1 0 0 0 0 0 0 0 0 4
3 1 0 1 0 1 0 0 0 0 0 0 0 6
-1 0 0 0 0 0 1 0 0 0 0 0 0 0
0 -1 0 0 0 0 0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0 -1 0 0 0 0 0
0 0 0 1 0 0 0 0 0 -1 0 0 0 0
-2 2 1 0 0 0 0 0 0 0 -1 0 0 4
3 1 0 1 0 0 0 0 0 0 0 -1 0 6
-3 -1 -1 -1 0 0 0 0 0 0 0 0 1 0

Tableau #4
x1 x2 x3 x4 s1 s2 s3 s4 s5 s6 s7 s8 p   
-2 2 1 0 1 0 0 0 0 0 0 0 0 4
3 1 0 1 0 1 0 0 0 0 0 0 0 6
-1 0 0 0 0 0 1 0 0 0 0 0 0 0
0 -1 0 0 0 0 0 1 0 0 0 0 0 0
0 0 -1 0 0 0 0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0 0 -1 0 0 0 0
-2 2 1 0 0 0 0 0 0 0 -1 0 0 4
3 1 0 1 0 0 0 0 0 0 0 -1 0 6
-3 -1 -1 -1 0 0 0 0 0 0 0 0 1 0

Tableau #5
x1 x2 x3 x4 s1 s2 s3 s4 s5 s6 s7 s8 p   
-2 2 1 0 1 0 0 0 0 0 0 0 0 4
3 1 0 1 0 1 0 0 0 0 0 0 0 6
-1 0 0 0 0 0 1 0 0 0 0 0 0 0
0 -1 0 0 0 0 0 1 0 0 0 0 0 0
0 0 -1 0 0 0 0 0 1 0 0 0 0 0
0 0 0 -1 0 0 0 0 0 1 0 0 0 0
-2 2 1 0 0 0 0 0 0 0 -1 0 0 4
3 1 0 1 0 0 0 0 0 0 0 -1 0 6
-3 -1 -1 -1 0 0 0 0 0 0 0 0 1 0

Tableau #6
x1 x2 x3 x4 s1 s2 s3 s4 s5 s6 s7 s8 p     
0 0 0 0 1 0 0 0 0 0 1 0 0 0
4 0 -0.5 1 0 1 0 0 0 0 0.5 0 0 4
-1 0 0 0 0 0 1 0 0 0 0 0 0 0
-1 0 0.5 0 0 0 0 1 0 0 -0.5 0 0 2
0 0 -1 0 0 0 0 0 1 0 0 0 0 0
0 0 0 -1 0 0 0 0 0 1 0 0 0 0
-1 1 0.5 0 0 0 0 0 0 0 -0.5 0 0 2
4 0 -0.5 1 0 0 0 0 0 0 0.5 -1 0 4
-4 0 -0.5 -1 0 0 0 0 0 0 -0.5 0 1 2

Tableau #7
x1 x2 x3 x4 s1 s2 s3 s4 s5 s6 s7 s8 p   
0 0 0 0 1 0 0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0 0 0 0 1 0 0
0 0 -0.125 0.25 0 0 1 0 0 0 0.125 -0.25 0 1
0 0 0.375 0.25 0 0 0 1 0 0 -0.375 -0.25 0 3
0 0 -1 0 0 0 0 0 1 0 0 0 0 0
0 0 0 -1 0 0 0 0 0 1 0 0 0 0
0 1 0.375 0.25 0 0 0 0 0 0 -0.375 -0.25 0 3
1 0 -0.125 0.25 0 0 0 0 0 0 0.125 -0.25 0 1
0 0 -1 0 0 0 0 0 0 0 0 -1 1 6

Tableau #8
x1 x2 x3 x4 s1 s2 s3 s4 s5 s6 s7 s8 p   
0 0 0 0 1 0 0 0 0 0 1 0 0 0   
0 0 0 0 0 1 0 0 0 0 0 1 0 0   
0 0 0 0.333333 0 0 1 0.333333 0 0 0 -0.333333 0 2   
0 0 1 0.666667 0 0 0 2.66667 0 0 -1 -0.666667 0 8   
0 0 0 0.666667 0 0 0 2.66667 1 0 -1 -0.666667 0 8   
0 0 0 -1 0 0 0 0 0 1 0 0 0 0   
0 1 0 0 0 0 0 -1 0 0 0 0 0 0   
1 0 0 0.333333 0 0 0 0.333333 0 0 0 -0.333333 0 2   
0 0 0 0.666667 0 0 0 2.66667 0 0 -1 -1.66667 1 14

Tableau #9
x1 x2 x3 x4 s1 s2 s3 s4 s5 s6 s7 s8 p   
0 0 0 0 1 0 0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0 0 0 0 1 0 0
0 0 0 0.333333 0 0.333333 1 0.333333 0 0 0 0 0 2
0 0 1 0.666667 0 0.666667 0 2.66667 0 0 -1 0 0 8
0 0 0 0.666667 0 0.666667 0 2.66667 1 0 -1 0 0 8
0 0 0 -1 0 0 0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 -1 0 0 0 0 0 0
1 0 0 0.333333 0 0.333333 0 0.333333 0 0 0 0 0 2
0 0 0 0.666667 0 1.66667 0 2.66667 0 0 -1 0 1 14   

Tableau #10
x1 x2 x3 x4 s1 s2 s3 s4 s5 s6 s7 s8 p   
0 0 0 0 1 0 0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0 0 0 0 1 0 0
0 0 0 0.333333 0 0.333333 1 0.333333 0 0 0 0 0 2
0 0 1 0.666667 1 0.666667 0 2.66667 0 0 0 0 0 8
0 0 0 0.666667 1 0.666667 0 2.66667 1 0 0 0 0 8
0 0 0 -1 0 0 0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 -1 0 0 0 0 0 0
1 0 0 0.333333 0 0.333333 0 0.333333 0 0 0 0 0 2
0 0 0 0.666667 1 1.66667 0 2.66667 0 0 0 0 1 14   

so the solution is

Solution: z = 14; x1 = 2, x2 = 0, x3 = 8, x4 = 0