Maximize $,Z=2x_1+7x_2-3x_3,$

subject to

$x_1+3x_2+4x_{3}30$

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

and

$x_{1}0,x_{2}0,x_{3}0.$

By letting $x_4$ and $x_5$ be the slack variables for the respective constraints, the simplex method yields the following final set of equations:

$(0)Z,+x_2+x_3+2x_5,=20$

$,-x_2+5x_3+x_4-x_5,=20$

$(1)x_1,+4x_2-x_3+x_5,=10.$

Now you are to conduct sensitivity analysis by independently investigating each of the following seven changes in the original model. For each change, use the sensitivity analysis procedure to revise this set of equations $($in tableau form$)$ and convert it to proper form from Gaussian elimination for identifying and evaluating the current basic solution. Then test this solution for feasibility and for optimality. If either test fails, reoptimize to find a new optimal solution.

$($a$)$ Change the right$-$hand sides to

$\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]=\left[\begin{array}{c}20\\ 30\end{array}\right].$

$($b$)$ Change the coefficients of $x_3$ to

$\left[\begin{array}{c}{c}_3\\ a_{13}\\ a_{23}\end{array}\right]=\left[\begin{array}{c}-2\\ 3\\ -2\end{array}\right].$

$($c$)$ Change the coefficients of $x_1$ to

$\left[\begin{array}{c}{c}_1\\ a_{11}\\ a_{21}\end{array}\right]=\left[\begin{array}{c}4\\ 3\\ 2\end{array}\right].$

$($d$)$ Introduce a new variable $x_6$ with coefficients

$\left[\begin{array}{c}{c}_6\\ a_{16}\\ a_{26}\end{array}\right]=\left[\begin{array}{c}-3\\ 1\\ 2\end{array}\right].$

$($e$)$ Change the objective function to $Z=x_1+5x_2-2x_3.$

$($f$)$ Introduce a new constraint $3x_1+2x_2+3x_{3}25.$

$($g$)$ Change constraint $2$ to $x_1+2x_2+2x_{3}35.$