Use the simplex method to solve method to solve the followin

Use the simplex method to solve method to solve the following problem. Find y_1 greaterthanorequalto 0, y_2 greaterthanorequalto 0, and y_3 greaterthanorequalto 0 such that 3y_1 + 2y_2 + 7y_3 lessthanorequalto 12, 3y_1 + 4y_2 + 5y_3 lessthanorequalto 9, and w = 24y_1 + 20y_2 + 43y_3 is minimized. The minimum value w = occurs when y_1 =, y_2 =, and y_3 =. (Simplify your answers.)

Solution

we are given

Minimize w = 24y1 + 4y2 + 5y3 subject to
3y1+2y2+7y3<=12
3y1+4y2+5y3>=9
y1>=0
y2>=0
y3>=0

Use simplex method

Firstly , we will make tables

Table#1
y1 y2 y3 s1 s2 s3 s4 s5 -w
3 2 7 1 0 0 0 0 0 12   
3 4 5 0 -1 0 0 0 0 9
1 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 -1 0 0 0
0 0 1 0 0 0 0 -1 0 0
24 4 5 0 0 0 0 0 1 0

Table #2
y1 y2 y3 s1 s2 s3 s4 s5 -w
3 2 0 1 0 0 0 7 0 12   
3 4 0 0 -1 0 0 5 0 9
1 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 -1 0 0 0
0 0 1 0 0 0 0 -1 0 0
24 4 0 0 0 0 0 5 1 0

Table #3
y1 y2 y3 s1 s2 s3 s4 s5 -w
0.43 0.29 0 0.14 0 0 0 1 0 1.7
0.86 2.6 0 -0.71 -1 0 0 0 0 0.43   
1 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 -1 0 0 0
0.43 0.29 1 0.14 0 0 0 0 0 1.7
22 2.6 0 -0.71 0 0 0 0 1 -8.6   

Table #4
y1 y2 y3 s1 s2 s3 s4 s5 -w
0.43 0 0 0.14 0 0 0.29 1 0 1.7
0.86 0 0 -0.71 -1 0 2.6 0 0 0.43   
1 0 0 0 0 -1 0 0 0 0
0 1 0 0 0 0 -1 0 0 0
0.43 0 1 0.14 0 0 0.29 0 0 1.7
22 0 0 -0.71 0 0 2.6 0 1 -8.6

Table#5
y1 y2 y3 s1 s2 s3 s4 s5 -w
0.33 0 0 0.22 0.11 0 0 1 0 1.7
0.33 0 0 -0.28 -0.39 0 1 0 0 0.17   
1 0 0 0 0 -1 0 0 0 0
0.33 1 0 -0.28 -0.39 0 0 0 0 0.17   
0.33 0 1 0.22 0.11 0 0 0 0 1.7
21 0 0 0 1 0 0 0 1 -9

  

Table#6
y1 y2 y3 s1 s2 s3 s4 s5 -w
0.33 0 0 0.22 0.11 0 0 1 0 1.7
0.33 0 0 -0.28 -0.39 0 1 0 0 0.17   
-1 0 0 0 0 1 0 0 0 0
0.33 1 0 -0.28 -0.39 0 0 0 0 0.17   
0.33 0 1 0.22 0.11 0 0 0 0 1.7
21 0 0 0 1 0 0 0 1 -9   

so, we will get optimal solution as

y1 = 0, y2 = 0.17, y3 = 1.7

minimum value is w=9...........Answer