Solve the following model with simplex method and find the optimum values.

Max $z=4x_1+3x_2+6x_3$

$st$; $3x_1+x_2+3x_{3}30$

$2x_1+2x_2+3x_{3}40$

$x_1,x_2,x_{3}0$

Find the dual model of following linear programming models.

a$.$

Max $z=2x_1+5x_2+3x_3$

$st$; $x_1-x_{2}20$

$2x_1+4x_2+x_3=50$

$x_1,x_2,x_{3}0$

b$.$

Maxz$=-x_1+2x_2+x_3$

$st$: $3x_2+x_{3}120$

$x_1-x_2-4x_{3}80$

$-3x_1+x_2-2x_{3}100$

$x_1,x_2,x_3$ unrestricted

Maxz $=4x_1+x_2$

$st$; $x_1+2x_2=30$

$x_1-x_{2}3$

$2x_1+x_{2}10$

$x_1,x_{2}0$

a$.$ Find the dual of this model.

b$.$ Solve this model using simplex method.

c$.$ Write the dual model optimum solutions from the optimum solution table that you have found from b$.$ option.