Question $2$

$25$ points

Consider the following problem.

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

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

Using slack variables $x_4$ and $x_5$ for the respective constraints, the optimal simplex tableau yields the

following final set of equations: $($Hint: you can create the optimal tableau by extracting the coefficients$)$

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

$(1),-x_2+5x_3+x_4-x_5=20$

$(2)x_1+4x_2-x_3+x_5=10.$

Conduct sensitivity analysis by independently investigating each of the following changes in the original

model.

a$.$ Change the right$-$hand sides to $\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]=\left[\begin{array}{c}20\\ 30\end{array}\right]$ in the original problem. Without re$-$

optimizing from scratch, compute the new optimal solution for the new problem.

b$.$ Change the coefficients of $x_3$ to $\left[\begin{array}{c}{c}_3\\ a_{13}\\ a_{23}\end{array}\right]=\left[\begin{array}{c}-2\\ 3\\ -2\end{array}\right]$ in the original problem. Without re$-$

optimizing from scratch, compute the new optimal solution for the new problem.

c$.$ Introduce a new variable $x_6$ with coefficients $\left[\begin{array}{c}{c}_6\\ a_{16}\\ a_{26}\end{array}\right]=\left[\begin{array}{c}-3\\ 1\\ 2\end{array}\right]$ to the original model. Check

if the optimal solution found $($for the original problem$)$ is still optimal for the new

problem.