Consider the following LP:

maximize $5$x$1+8$x$2+6$x$3$

subject to

x$1+$x$2+$x$3<=10$

$$x$1+2$x$2+4$x$3<=10$

$4$x$1+4$x$2+2$x$3<=8$

x$1,$x$2,$x$3>=0$

Answer the following questions $$manually$$ without using CPLEX.

$($a$)$ Apply Simplex method to solve. What is the optimal basis? What are the basic and non$-$basic variables?

$($b$)$ By how much can the objective coefficient for x$1$ change while keeping the same optimal basis $($in other words, what is $)?$

$($c$)$ Verify your answer in CPLEX by performing a sensitivity analysis.

$($d$)$ What is the shadow price of x$1?$

$($e$)$ By how much can the objective coefficient for x$2$ change while keeping the same optimal basis $($in other words, what is $)?$

$($f$)$ Verify your answer in CPLEX by performing a sensitivity analysis.

$($g$)$ What is the shadow price of x$2?$

$($h$)$ By how much can the rhs of the $2$nd constraint change while keeping the same optimal basis $($in other words, what is the allowable $)?$

$($i$)$ Verify your answer in CPLEX by performing a sensitivity analysis.

$($j$)$ If the $3$rd constraint changed to $2$x$1+4$x$2+2$x$3<=8$ will there be a change to the optimal basis?

$($k$)$ If the $3$rd constraint changed to $4$x$1+5$x$2+2$x$3<=8$ will there be a change to the optimal basis?

$($l$)$ Assume we add an activity corresponding to the decision variable x$4,$ with objective coefficient c$4=10,$ and constraint coefficients $2,4,$ and $1$ corresponding to the $1$st$,2$nd$,$ and $3$rd constraints. Will there be a change to the optimal basis?