Simplex Method, Duality Problem

[IsaacStats]

Hello everyone, I have the following question:

Show without using the simplex method that
x1=5/26, x2=5/2, x3=27/26
is an optimal solution to the following LPP.

Maximize z=9x1+14x2+7x3 subject to
2x1+x2+3x3<= 6
5x1+4x2+x3<= 12
12x2 <= 5
x1,x2,x3 unrestricted.

=>
Dual is the following:

Minimize z'=6w1+12w2+6w3 subject to
2w1+5w2 >= 9
w1+4w2+2w3>= 14
3w1+w2 >= 7
w1,w2,w3 >= 0

I am lost regarding where I should proceed next. Looking for your guidance.

 

[Mark44]

As this appears to be a homework question, I have moved it to the Homework & Coursework section.

 

[Ray Vickson]

Hello everyone, I have the following question:

Show without using the simplex method that
x1=5/26, x2=5/2, x3=27/26
is an optimal solution to the following LPP.

Maximize z=9x1+14x2+7x3 subject to
2x1+x2+3x3<= 6
5x1+4x2+x3<= 12
12x2 <= 5
x1,x2,x3 unrestricted.

=>
Dual is the following:

Minimize z'=6w1+12w2+6w3 subject to
2w1+5w2 >= 9
w1+4w2+2w3>= 14
3w1+w2 >= 7
w1,w2,w3 >= 0

I am lost regarding where I should proceed next. Looking for your guidance.

If the third primal right-hand-side is 5 (as written) the third dual objective coefficient is wrong. If the coefficient of x2 on the left of the third primal constraint is 12 (as written) the coefficient of w3 in the second dual constraint is wrong.

After deciding on correct statements of both the primal and dual problems, use the known properties of the relation between the primal and dual solution at optimality. For example, if a primal variable $x_j$ is $> 0$, what can you say about the $j$th dual constraint, etc.?