A factory manufactures custom$-$blended whiskey, consisting of rye whiskey and bourbon. The company is open to orders for a minimum of $400$ gallons of the custom blend. As per the customer's specifications, the order must contain at least $40\%$ rye and no more than $250$ gallons of bourbon. Additionally, the blend should have a ratio of two parts rye to one part bourbon. The distillery has a production capacity of $500$ gallons per week, regardless of the blend. The production manager aims to fulfill the order within one week. The blend is priced at $$12$ per gallon, while the distillery incurs costs of $$4$ per gallon for rye and $$2$ per gallon for bourbon. The company seeks to optimize the blend mix for maximum profit. Inquired: a$)$ Formulate with linier optimization b$)$ Calculate the maximum profit.