Consider the linear program below and answer the following.

Max $3$A $+2$B

s$.$t$.$

$1$A $+1$B $<=10$

$3$A $+1$B $<=22$

$1$A $+2$B $<=18$

A$,$ B $>=0$

$($a$)$

Use the graphical solution procedure to find the optimal solution.

What is the value of the objective function at the optimal solution?

at

$($A$,$ B$)=$

$($b$)$

Assume that the objective function coefficient for A changes from $3$ to $5.$ Use the graphical solution procedure to find the new optimal solution.

Does the optimal solution change?

The extreme point

$---$Select$---$

optimal. The value of the objective function becomes

$.$

$($c$)$

Assume that the objective function coefficient for A remains $3,$ but the objective function coefficient for B changes from $2$ to $4.$ Use the graphical solution procedure to find the new optimal solution.

Does the optimal solution change?

The extreme point

$---$Select$---$

optimal. The value of the objective function becomes

$.$

$($d$)$

The computer solution for the linear program in part $($a$)$ provides the following objective coefficient range information.

Variable Objective

Coefficient Allowable

Increase Allowable

Decrease

A $3\mathrm{.}000003\mathrm{.}000001\mathrm{.}00000$

B $2\mathrm{.}000001\mathrm{.}000001\mathrm{.}00000$

Use this objective coefficient range information to answer parts $($b$)$ and $($c$).$

The objective coefficient range for variable A is

to

$.$ Since the change in part $($b$)$ is

$---$Select$---$

this range, we know the optimal solution

$---$Select$---$

change. The objective coefficient range for variable B is

to

$.$ Since the change in part $($c$)$ is

$---$Select$---$

this range, we know the optimal solution

$---$Select$---$

change.