Homework Sourse1 Solve the following LP model using Simplex method HW 1 Q3
Solution
Maximize z = 8x1 + 2x2
Subject to
2x1 + 4x2 <= 22
-x1 + 4x2 <= 10
4x1 - 2x2 <= 14
x1 - 3x2 <=1
x1 , x2 >= 0
Optimal Solution: z = 46; x1 = 5, x2 = 3
Tableau #1
x1 x2 s1 s2 s3 s4 z
2 4 1 0 0 0 0 22
-1 4 0 1 0 0 0 10
4 -2 0 0 1 0 0 14
1 -3 0 0 0 1 0 1
-8 -2 0 0 0 0 1 0
Tableau #2
x1 x2 s1 s2 s3 s4 z
0 10 1 0 0 -2 0 20
0 1 0 1 0 1 0 11
0 10 0 0 1 -4 0 10
1 -3 0 0 0 1 0 1
0 -26 0 0 0 8 1 8
Tableau #3
x1 x2 s1 s2 s3 s4 z
0 0 1 0 -1 2 0 10
0 0 0 1 -0.1 1.4 0 10
0 1 0 0 0.1 -0.4 0 1
1 0 0 0 0.3 -0.2 0 4
0 0 0 0 2.6 -2.4 1 34
Tableau #4
x1 x2 s1 s2 s3 s4 z
0 0 0.5 0 -0.5 1 0 5
0 0 -0.7 1 0.6 0 0 3
0 1 0.2 0 -0.1 0 0 3
1 0 0.1 0 0.2 0 0 5
0 0 1.2 0 1.4 0 1 46
Maximize z = 8x1 + 2x2
subject to
2x1 - 6x2 <= 12
5x1 + 4x2 >= 40
x1 + 2x2 >= 14
x2 <= 6
x1 , x2 >= 0
Optimal Solution: z = 204; x1 = 24, x2 = 6
Tableau #1
x1 x2 s1 s2 s3 s4 z
2 -6 1 0 0 0 0 12
5 4 0 -1 0 0 0 40
1 2 0 0 -1 0 0 14
0 1 0 0 0 1 0 6
-8 -2 0 0 0 0 1 0
Tableau #2
x1 x2 s1 s2 s3 s4 z
1 -3 0.5 0 0 0 0 6
0 19 -2.5 -1 0 0 0 10
0 5 -0.5 0 -1 0 0 8
0 1 0 0 0 1 0 6
0 -26 4 0 0 0 1 48
Tableau #3
x1 x2 s1 s2 s3 s4 z
1 0 0.105263 -0.157895 0 0 0 7.57895
0 1 -0.131579 -0.0526316 0 0 0 0.526316
0 0 0.157895 0.263158 -1 0 0 5.36842
0 0 0.131579 0.0526316 0 1 0 5.47368
0 0 0.578947 -1.36842 0 0 1 61.6842
Tableau #4
x1 x2 s1 s2 s3 s4 z
1 0 0.2 0 -0.6 0 0 10.8
0 1 -0.1 0 -0.2 0 0 1.6
0 0 0.6 1 -3.8 0 0 20.4
0 0 0.1 0 0.2 1 0 4.4
0 0 1.4 0 -5.2 0 1 89.6
Tableau #5
x1 x2 s1 s2 s3 s4 z
1 0 0.5 0 0 3 0 24
0 1 0 0 0 1 0 6
0 0 2.5 1 0 19 0 104
0 0 0.5 0 1 5 0 22
0 0 4 0 0 26 1 204
Maximize z = 60x1 + 90x2
subject to
60x1 + 30x2 <= 1500
100x1 + 100x2 >= 6000
x2 >= 30
x1 , x2 >= 0
No optimal solution exists for this problem.
Tableau #1
x1 x2 s1 s2 s3 z
60 30 1 0 0 0 1500
100 100 0 -1 0 0 6000
0 1 0 0 -1 0 30
-60 -90 0 0 0 1 0
Tableau #2
x1 x2 s1 s2 s3 z
1 0.5 0.0166667 0 0 0 25
0 50 -1.66667 -1 0 0 3500
0 1 0 0 -1 0 30
0 -60 1 0 0 1 1500
Tableau #3
x1 x2 s1 s2 s3 z
1 0 0.0166667 0 0.5 0 10
0 0 -1.66667 -1 50 0 2000
0 1 0 0 -1 0 30
0 0 1 0 -60 1 3300
Tableau #4
x1 x2 s1 s2 s3 z
2 0 0.0333333 0 1 0 20
-100 0 -3.33333 -1 0 0 1000
2 1 0.0333333 0 0 0 50
120 0 3 0 0 1 4500
