Write the Dual the following L.P.P

Max. $Z = 2x_1 -x_2 + 4x_3 ,$

$ \text{subject to} \hspace{0.2cm} x_1 +2 x_2 - x_3 ≤ 5 ; 2 x1 -x2 + x3 ≤6$

$x_1 +x_2 + 3x_3 ≤10 ; 4x_1 +x_3≤12$

$x_1 , x_2 , x_3 ≥ 0$

To write the dual , if z is maximization type, all the constraint must be less or equal to type.

The given L.P.P. can be written as:

$Max. \ z = 2x_1 -x_2 + 4x_3$

Subject to, $x_1 +2 x_2 - x_3 ≤ 5$

$2 x_1 -x_2 + x_3 ≤6$

$x_1 +x_2 + 3x_3 ≤10$

$4x_1 +0x_2 +x_3≤12$

Thus, the dual of the given problem becomes

$Min.\ w = 5y_1 + 6y_2 + 10y_3 +12y_4$

Subject to, $y_1 +2y_2 +y_3 +4y_4 ≥2$

$2y_1 – y_2 +y_3 ≥ -1$

$-y_1 +y_2 + 3y_3 +y_4 ≥4$

$y_1, y_2 , y_3 , y_4 ≥ 0$