Example $2\mathrm{.}3-7($Crude Oil Refining and Gasoline Blending$)$

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 refinery 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 $)=(82$ at the rate of $\mathrm{.}2bbl$ per bbl of crude oil, $(2)$ a cracker unit that produces gasoline stock $)=(98$ by using a portion of the feedstock produced from the distillation tower at the rate of $\mathrm{.}5bbl$ 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 super gasoline are $50,000,30,000,$ and $40,000$ barrels per day. Show the model $($contraints and equations step by step$).$ Then plug in excel and show optimal solution and objective value.