Consider the following problem.

Maximize Z $=2$x$1+7$x$23$x$3,$

subject to

x$1+3$x$2+4$x$330$

x$1+4$x$2$ x$310$

and

x$10,$ x$20,$ x$30.$

By letting x$4$ and x$5$ be the slack variables for the respective con

straints, the simplex method yields the following final set of

equations:

$(0)$

Z $+$ x$2+$ x$3$

$+2$x$5=20,$

$(1)$

$$ x$2+5$x$3+$ x$4$ x$5=20,$

$(2)$

x$1+4$x$2$ x$3$

$+$ x$5=10.$

Now you are to conduct sensitivity analysis by independently in

vestigating 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 evalu

ating the current basic solution. Then test this solution for feasibil

ity and for optimality. If either test fails, reoptimize to find a new

optimal solution.

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

$[$ b

$1$

b

$2$

$]=[20$

$30$

$].$

$($b$)$ Change the coefficients of x$3$ to

$$

$$

$$

c$3$

a

$13$

a$23$

$$

$$

$$

$=$

$$

$$

$$

$2$

$3$

$2$

$$

$$

$.$

$($c$)$ Change the coefficients of x$1$ to

$$

$$

$$

c$1$

a

$11$

a$21$

$$

$$

$$

$=$

$$

$$

$$

$4$

$3$

$2$

$$

$.$

$($d$)$

Introduce a new variable x

$6$ with coefficients

$$

$$

$$

c$6$

a

$16$

a$26$

$$

$$

$$

$=$

$$

$$

$$

$31$

$2$

$$

$$

$.$

$($e$)$ Change the objective function to Z $=$ x$1+5$x$22$x$3.$

$($f$)$

Introduce a new constraint $3$x$1+2$x$2+3$x$325.$

$($g$)$ Change constraint $2$ to x$1+2$x$2+2$x$335.$