Question $2(25$ points$)$:

The initial and final tableau for a LP problem is given as follows:

a$)$ What is the maximum value of $c_1$ for the optimal basis to remain optimal?

b$)$ Determine the range of values on $c_2$ for which the optimal basis remains optimal?

c$)$ Determine the range of values on $b_1$ for which the optimal basis remains optimal?

d$)$ If the vector $a_1$ is changed to ${\left[\begin{array}{ccc}1& 1& 2\end{array}\right]}^T,$ does the optimal solution change? If so$,$ find the new

optimal solution.

e$)$ If a new constraint is added such that

$3x_1+3x_2+x_{3}500$

Determine whether the current solution stays feasible. If the current solution is infeasible,

find the optimal solution with this additional constraint.