$2-($worth $20$ points$)$ The Good$-$to$-$Go Suitcase Company makes three kinds of suitcases: $(1)$

Standard, $(2)$ Deluxe, and $(3)$ Luxury styles. Each suitcase goes through four production

stages: $(1)$ cutting and coloring, $(2)$ assembly, $(3)$ finishing, and $(4)$ quality and packaging.

The total number of hours available in each of these departments is $630,600,708,$ and $135,$

respectively. Each Standard suitcase requires $0\mathrm{.}7$ hours of cutting and coloring, $0\mathrm{.}5$ hours of

assembly, $1$ hour of finishing, and $0\mathrm{.}1$ hours of quality and packaging. The corresponding

numbers for each Deluxe suitcase are $1$ hour, $\frac{5}{6}$ hours, $\frac{2}{3}$ hours, and $0\mathrm{.}25$ hours,

respectively. Likewise, the corresponding numbers for each Luxury suitcase are $1$ hour, $\frac{2}{3}$

hours, $0\mathrm{.}9$ hours, and $0\mathrm{.}4$ hours, respectively.

The sales revenue for each type of suitcase is as follows: Standard $$36\mathrm{.}05,$ Deluxe $$39\mathrm{.}50,$

and Luxury $$43\mathrm{.}30.$ The material coits are Standard $$6\mathrm{.}25,$ Deluxe $$7\mathrm{.}50,$ and Luxury $$8\mathrm{.}50.$

The hourly cost of labor for each department is cutting and coloring $$10,$ assembly $$6,$

finishing $$9,$ and quality and packaging $$8.$

The Excel layout and LP Sensitivity Report of Good$-$to$-$Go's problem are shown in

Screenshots $4-11$A and $4-11$B$,$ respectively. Each of the following questions is independent

of the others.

a$.$ What is the optimal production plan? Which of the resources are binding?

b$.$ Suppose Good$-$to$-$Go is considering including a polishing process, the cost of which

would be added directly to the price. Each Standard suitcase would require $10$ minutes

of time in this treatment, each Deluxe suitcase would need $15$ minutes, and each Luxury

suitcase would need $20$ minutes. Would the current production plan change as a result

of this additional process if $170$ hours of polishing time were available? the polishing

hourly cost is $$5.$ Explain your answer. $($tip: this is a new constraint that you should add

to the original problem$)$

c$.$ Now consider the addition of a waterproofing process where each Standard suitcase

would use $1$ hour of time in the process, each Deluxe suitcase would need $1\mathrm{.}5$ hours,

and each Luxury suitcase would require $1\mathrm{.}75$ hours. the waterproofing hourly cost is $$5.$

Would this change the production plan if $900$ hours were available? Why or why not?

$($tip: this is a new constraint that you should add to the original problem$)$