$3$ Question

Consider the following LP$.$

max$2x_1+x_2+x_3$

$s.t.$

$x_1+x_{3}1$

$x_2+x_{3}2$

$x_1+x_{2}3$

$x_1,x_2,x_{3}0$

a$.$ Consider the basis with basic variables $x_1,x_2$ and $x_3$ What are the $c_B,B,B^{-1}$ and $b?$

b$.$ What is the basic solution corresponding to this basis?

c$.$ Construct the table corresponding to the basic solution above.

d$.$ Can you conclude that this table is optimal? Why or why not?

e$.$ Write down the dual of this LP$.$

f$.$ Find $c_B{B}^{-1}$ from the table. Show that it is feasible for the above dual LP$.$

g$.$ Is $w$ in part $($f$)$ optimal for the dual LP in part $($e$)?$ If so$,$ show via CS theorem.