(Solved): 3. Linear programming was already used in the oil industry in the 1950 's to work out the best oper ...

3. Linear programming was already used in the oil industry in the 1950 's to work out the best operating plan. Real-life problems that oil industry tries to solve are complex and interwoven. Here we consider a simple problem. Suppose there are two oil fields, A and B. Oil field A produces crude oil at the cost of \( \$ 80 \) per barrel and oil field B produces crude oil at \( \$ 90 \) per barrel. A refinery then process the crude oil to produce diesel, gasoline and jet fuel. Each barrel of the crude from A can be refined to produce 20 gallons of diesel, 15 gallons of gasoline, and 12 gallons of jet fuel. Each barrel of the crude from B can produce 15 gallons of each diesel, gasoline, and jet fuel. The refinery has to produce 160 gallons of diesel, 100 gallons of gasoline, and 180 gallons of jet fuel to meet the demand. (a) Formulate an LP problem to determine how many barrels of crude from each oil field should the oil refinery use to minimize the cost while meeting the demand. (b) Obtain the dual to the problem in (a). (c) Give solution to the oil refinery's question by solving the dual problem.