Consider the following problem:

Maximize: $Z=3^{**}x1+5^{**}x2+2**x3$;

Subject to:

$-2^{**}1+2^{**}2+x35$

$3^{**}1+x2-x310$

$x10,x20$ and $x30.$

Use $x4$ and $x5$ as the slack variables for the constraints $1$ and $2$ respectively. Answer the following:

$($i$)$ Can $x_{1}3$ form the optimal basis? Validate using the revised simplex method.

$($ii$)$ Can $23$ form a feasible basis? Explain and show your work.

$($iii$)$ Use the optimal tableau and carry out a sensitivity analysis with respect to the non$-$basic variable $x2.$

$($iv$)$ Use the optimal tableau and perform a sensitivity study on the level of the first resource$($first constraint$).$ Report the range of value for the first resource that will keep the current $\left[\begin{array}{cc}x1& x3\end{array}\right]$ optimal.