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Unformatted text preview: Chapter 2 Deterministic Optimization Models in Operations Research EXAMPLE 2.1: Two Crude Petroleum Two Crude Petroleum runs a small refinery on the Texas coast. The refinery distills crude petroleum from two sources, Saudi Arabia and Venezuela, into the three main products: gasoline, jet fuel and lubricants. The two crudes differ in chemical composition and yield different product mixes. Each barrel of Saudi crude yields 0.3 barrel of gasoline, 0.4 barrel of jet fuel, and 0.2 barrel of lubricants. Each barrel of Venezuelan crude yields 0.4 barrel of gasoline, 0.2 barrel of jet fuel and 0.3 barrel of lubricants. The remaining 10% is lost to refining. EXAMPLE 2.1: Two Crude Petroleum The crudes differ in cost and availability. Two Crude can purchase up to 9000 barrels per day from Saudi Arabia at $20 per barrel. Up to 6000 barrels per day of Venezuelan petroleum are available at the lower cost of $15 per barrel. Two contracts require it to produce 2000 barrels per day of gasoline,1500 barrels per day of jet fuel and 500 barrels per day of lubricants. How can these requirements be fulfilled most efficiently? EXAMPLE 2.1: Two Crude Petroleum Saudi Arabia Venezuela Requirements (barrels / day) Yields /barrel gasoline 0.3 barrel 0.4 barrel 2000 jet fuel 0.4 barrel 0.2 barrel 1500 lubricant 0.2 barrel 0.3 barrel 500 lost to refining 0.1 barrel 0.1 barrel Availability barrels / day 9000 6000 Purchase cost per barrel $20 $15 2.1 Decision Variables, Constraints, and Objective Functions • Decision Variables: Variables in optimization models represent the decisions to be taken. [2.1] • Input parameters: fixed information – Yields, Cost, Availability, Requirements • Decision Variables: x1 barrels of Saudi crude refined /day (in 1000s) x2 barrels of Venezuelan crude refined /day (in 1000s) (2.1) Constraints • Variabletype Constraints specify the domain of definition for decision variables: the set of values for which the variables have meaning. [2.2] Nonnegativity: x1 , x2 (2.2) Constraints • Main Constraints of optimization models specify the restrictions and interactions, other than variabletype, that limit decision variable values. [2.3] 0.3 x1 + 0.4 x2 2.0 (gasoline) 0.4 x1 + 0.2 x2 1.5 (jet fuel) 0.2 x1 + 0.3 x2 0.5 (lubricants) x1 9 (Saudi) x2 6 (Venezuelan) (2.3) (2.4) Objective Functions • Objective Functions in optimization models quantity the decision consequences to be maximized or minimized. [2.4] min 20 x1 + 15 x2 (2.5) Standard Model The standard statement of an optimization model has the form max or min (objective function(s)) s.t. (main constraints) (variabletype constraints) • min 20 x1 + 15 x2 (total cost) s.t....
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 Summer '11
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 Optimization, objective function, X1, optimization model, Optimal Solutions, DuPage Land Use

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