Consider the following linear program:

Max $3$A $+2$B

s$.$t$.$

$1$A $+1$B

$3$A $+1$B

$1$A $+2$B

A$,$ B $>=0$

a$.$ Use the graphical solution procedure to find the optimal solution.

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

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

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

$3\mathrm{.}00000$

$3\mathrm{.}00000$

$1\mathrm{.}00000$

B

$2\mathrm{.}00000$

$1\mathrm{.}00000$

$1\mathrm{.}00000$

Use this objective coefficient range information to answer parts $($b$)$ and $($c$).$ Consider the following linear program:

Max $3$A$+2$B

s$.$t$.$

$1A+1B10$

$3A+1B24$

$1A+2B16$

$A,B0$

a$.$ Use the graphical solution procedure to find the optimal solution.

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

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

find the new optimal solution.

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

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