Quantitative Analysis for Management

Raptor Fuels produces three grades of gasoline Regular, Premium, and Super. All of these are produced by blending two types of crude oil Crude A and Crude B. The two types of crude contain specific ingredients which help in determining the octane rating of gasoline. The important ingredients and the costs are contained in the following table: [pic] .:. In order to achieve the desired octane ratings, at least 41% of Regular gasoline should be Ingredient 1; at least 44% of Premium gasoline must be Ingredient 1, and at least 48% of Super gasoline must be Ingredient 1.
Due to current contract commitments, Raptor Fuels must produce as least 20,000 gallons of Regular, at least 15,000 gallons of Premium, and at least 10,000 gallons of Super. Formulate a linear program that could be used to determine how much of Crude A and Crude B should be used in each of the gasolines to meet the demands at the minimum cost. What is the minimum cost? How much of Crude A and Crude B are used in each gallon of the different types of gasoline?
SOLUTION: Let A1 = gallons of crude A used in Regular A2 = gallons of crude A used in Premium A3 = gallons of crude A used in Super B1 = gallons of crude B used in Regular B2 = gallons of crude B used in Premium B3 = gallons of crude B used in Super Minimize cost = 0. 42A1 + 0. 42A2 + 0. 42A3 + 0. 47B1 + 0. 47B2 + 0. 47B3 Subject to 0. 40A1 + 0. 52B1 ( 0. 41(A1 + B1) 0. 40A2 + 0. 52B2 ( 0. 44(A2 + B2) 0. 40A3 + 0. 52B3 ( 0. 8(A3 + B3) A1 + B1 ( 20,000 A2 + B2 ( 15,000 A3 + B3 ( 10,000 A1, A2, A3, B1, B2, B3 ( 0 The solution is A1 = 18,333. 33 gallons of crude A used in Regular; A2 = 10,000 gallons of crude A used in Premium; A3 = 3,333. 33 gallons of crude A used in Super; B1 = 1. 666. 67 gallons of crude B used in Regular, B2 = 5,000 gallons of crude B used in Premium; B3 = 6,666. 67 gallons of crude B used in Super; total cost = $19,566. 67.