$($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, $5/6$ hours, $2/3$ hours, and $0\mathrm{.}25$ hours,

respectively. Likewise, the corresponding numbers for each Luxury suitcase are $1$ hour, $2/3$

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