The Furniture City $($FC$)$ company makes $3$ kinds of couches: $(1)$ Standard, $(2)$ Deluxe, and $(3)$ Luxury styles. Each couch goes through $3$ production departments: $(1)$ Cutting and Coloring, $(2)$ Assembly, and $(3)$ Finishing. The total number of hours available each month in each of the above departments is $200,300$ and $400,$ respectively. The time $($in hours$)$ required by $3$ departments for making each couch and the profit per couch are listed below. The company has a contract of supplying at least a total of $350$ Standard and Luxury couches per month.