Solve the following LPP using simplex method:

Maximise: $z=6x_{1}-2x_{2}+3x_{3}$

subject to,

$2x_{1}-x_{2}+2x_{3} \le 2$

$x_{1}+4x_{3} \le 4$

$x_{1}, \ x_{2}, \ x_{3} \ge 0$

In standard form,

$z-6x_{1}+2x_{2}-3x_{3} + 0s_{1} + 0s_{2}=0$

$2x_{1}-x_{2}+2x_{3}+s_{1}0s_{2}=2$

$x_{1}+0x_{2}+4x_{3}+0s_{1}+s_{2}=4$

$x_{1}, \ x_{2}, \ x_{3} \ge 0$

In table form,

$\therefore x_{1}=4, \ x_{2}=6, \ x_{3}=0, z_{max}=12$