Please answer all the questions including the showing the st

Please answer all the questions including the showing the steps.

Find the minimum value of

Z = 2x + 10y + 8z S.T.

x + y + z >= 6

y + 2z >= 8

-x + 2y + 2z >= 4

where x,y,z >=0.

2). minimize Z = 1a - 3b + 2c S.T.

3a - 1b + 3c <= 7

-2a + 4b + 0c <= 12

-4a + 3b + 8c <= 10

a,b,c >= 0.

3). minimize Z = 2x + 9y + 1z S.T.

1x + 4y + 2z >= 5

3x + 1y + 2z >= 4

x,y,z >=0.

4). minimize Z = 3a + 2b + 1c S.T.

2a + 5b + 1c = 12

3a + 4b + 0c = 11

and a is unrestricted and b and c are >=0

5). Maximize Z = 22x + 30y + 25z S.T

2x + 2y + 0z <= 100

2x + 1y + 1z <= 100

1x + 2y + 2z <= 100

x,y,z all >=0.

Solution

1)

consider the minimize Z = 2x + 10y + 8z

subject to

x + y + z >= 6

y + 2z >= 8

-x + 2y + 2z >= 4

where x,y,z >=0.

use simplex method to solve the given problem is as follows:

iteration1:

x y z s1 s2 s3 -Z
1 1 1 -1 0 0 0 6
0 1 2 0 -1 0 0 8
-1 2 2 0 0 -1 0 4
2 10 8 0 0 0 1 0

iteration 2:
x y z s1 s2 s3 -Z
1 1 1 -1 0 0 0 6
0 1 2 0 -1 0 0 8
0 3 3 -1 0 -1 0 10   
0 8 6 2 0 0 1 -12

iteration 3:
x y z s1 s2 s3 -Z
1 0 0 -0.666667 0 0.333333 0 2.66667   
0 -1 0 0.666667 -1 0.666667 0 1.33333   
0 1 1 -0.333333 0 -0.333333 0 3.33333   
0 2 0 4 0 2 1 -32   

iteration 4:
x y z s1 s2 s3 -Z
1 -1 0 0 -1 1 0 4
0 -1.5 0 1 -1.5 1 0 2
0 0.5 1 0 -0.5 0 0 4
0 8 0 0 6 -2 1 -40

iteration 5:
x y z s1 s2 s3 -Z
1 0.5 0 -1 0.5 0 0 2
0 -1.5 0 1 -1.5 1 0 2
0 0.5 1 0 -0.5 0 0 4
0 5 0 2 3 0 1 -36

therefore, optimum solution is arrived with values of variables as

x=2, y=0,z=4,

and minimize Z=36.