Consider the following LP and its optimal tableau:

Maximize $Z=3x_1+x_2-x_3$

subject $to$

$2x_1+x_2+x_{3}8$

$4x_1+x_2-x_{3}10$

$x_1,x_2,x_{3}0$

a$.$ For what values of $c_1$ does the current basis remain optimal?

b$.$ For what values of $c_3$ does the current basis remain optimal?

c$.$ For what values of $b_2$ does the current basis remain optimal?

d$.$ Currently for the third decision variable: $c_3=-1,a_{13}=1,a_{23}=-1.$ Suppose then: $c_3=$

$2,a_{11}=2,a_{12}=-2.$ Would this change the optimal solution to the problem?

e$.$ If we add a new constraint: $2x_1+2x_2-2x_{3}20$