Exercise $4\mathrm{.}1.$ An oil company is about to decide which crude oils to buy in order to fulfill the booked gas oil orders for next month. The company has booked orders for a total of $120,000$ gallons of gas oil type A$,$ and another $90,000$ gallons of gas oil type B$.$ These gas oil products can be made using crude oils of types $1$ and $2.$ From each gallon of crude oil of type $1,$ the refirtery extracts $0\mathrm{.}6$ gallons of type A gas oil and $0\mathrm{.}3$ gallons of type B gas oil $($the remaining $0\mathrm{.}1$ gallon constitutes a valueless waste product$).$ Each gallon of crude oil of type $2$ gives $0\mathrm{.}4$ gallons of gas oil A and $0\mathrm{.}6$ gallons of gas oil B$.$ The prices of the crude oils are $$0\mathrm{.}12$ per gallon of type $1$ and $$0\mathrm{.}18$ per gallon of type $2.$ The production manager is considering alternative production plans that lead to production quantities that fulfill the booked orders, and he is wondering which one to select.

a$)$ Write down the mathematical representation of the LP model at hand.

b$)$ Use Excel's Solver to find the optimal solution. The associated sensitivity report