Consider the following linear program:

Max $3$A $+2$B

s$.$t$.$

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

$3$A $+1$B $<=24$

$1$A $+2$B $<=16$

A$,$ B $>=0$

The value of the optimal solution is $27.$ Suppose that the right$-$hand side of the constraint $1$ is increased from $10$ to $11.$

Use the graphical solution procedure to find the new optimal solution.

$($i$)$

$($ii$)$

$($iii$)$

$($iv$)$

Use the solution to part $($a$)$ to determine the dual value for constraint $1.$ If required, round your answer to $1$ decimal place.

Dual Value: fill in the blank $2$

The computer solution for the linear program in Problem $1$ provides the following right$-$hand$-$side range information:

Constraint RHS

Value Allowable

Increase Allowable

Decrease

$110\mathrm{.}000001\mathrm{.}200002\mathrm{.}00000$

$224\mathrm{.}000006\mathrm{.}000006\mathrm{.}00000$

$316\mathrm{.}00000$ Infinite $3\mathrm{.}00000$

What does the right$-$hand$-$side range information for constraint $1$ tell you about the dual value for constraint $1?$

The right$-$hand$-$side range for constraint $1$ is fill in the blank $3$

to fill in the blank $4$

$.$ As long as the right$-$hand side stays within this range, the dual value

$.$

The dual value for constraint $2$ is $0\mathrm{.}5.$ Using this dual value and the right$-$hand$-$side range information in part $($c$),$ what conclusion can be drawn about the effect of changes to the right$-$hand side of constraint $2?$ If required, round your answers to $1$ decimal place.

The improvement in the value of the optimal solution will be fill in the blank $6$

for every unit increase in the right$-$hand side of constraint $2$ as long as the right$-$hand side is between fill in the blank $7$

and fill in the blank $8$

$.$