Question marks$]$: A factory manufactures three products using two resources. The following linear program is formulated to maximize the total profit.

Maximize $Z=2x_1+3x_2+x_3$

Subject $to\frac{1}{2}{x}_1+\frac{1}{2}{x}_2+\frac{1}{2}{x}_3\frac{3}{2}(Resource1)$

$\frac{1}{2}{x}_1+\frac{3}{2}{x}_2+\frac{5}{2}{x}_3\frac{7}{2}(Resource2)$

$x_1,x_2,x_{3}0$

where $x_1,x_2,$ and $x_3$ are the amount of products $1,2,$ and $3.$ The optimal solution is given by the following set of equations, where $x_4$ and $x_5$ are the slack variables corresponding to Resource $1$ and Resource $2$ constraints, respectively.

$Z+3x_3+3x_4+x_5=8$

$x_1-x_3+3x_4-x_5=1$

$x_2+2x_3-x_4+x_5=2$

Using the analytical procedure of sensitivity analysis, answer the following questions.

$($a$)$ What is the range of profit per unit of product $1$ so that the current solution is still optimal?

$($b$)$ What should be the profit per unit of product $3$ before it becomes worthwhile to manufacture?

$($c$)$ Find the range of $$ for which the given basis is still feasible and optimal if the original RHS vector $b$ is replaced by $b+b^{**}$ where $b^{**}=\left[\begin{array}{c}1\\ -1\end{array}\right]$ and

$($d$)$ Find the optimal solution when a new constraint $x_1+x_{2}2$ is added to the original problem.

$($e$)$ Due to technological breakthrough the resource requirement of product $3$ for the second resource is reduced to $1$ unit. Does this affect the optimal solution? Why or why not?