Shale Oil, located on the island of Aruba, has a capacity of $1,500,000bbl$ of crude oil per day. The final products from the refincry include three types of unleaded gasoline with different octane numbers $(ON)$ : regular with $ON=87,$ premium with $ON=89,$ and super with $ON=92.$ The refining process encompasses three stages: $(1)$ a distillation tower that produces feedstock $)$ at the rate of $\mathrm{.}2bbl$ per bbl of crudo oil, $(2)$ a crack$$r unit that produces gasoline stock $)=(98$ by using a portion of the feedstock produced from the distillation tower at the rate of:$5$ bbl per bbl of feedstock, and $(3)$ a blender unit that blends the gasoline stock from the cracker unit and the feedstock from the distillation tower. The company estimates the net profit per barrel of the three types of gasoline to be $$6\mathrm{.}70,$$$7\mathrm{.}20,$ and $$8\mathrm{.}10$; respectively. The input capacity of the cracker unit is $200,000$ barrels of feedstock a day. The demand limits for regular, premium, and supor gasoline are $50,000,30,000,$ and $40,000$ barrels, respectively, per day. Develop a model for determining the optimum production schedule for the refinery.