One of the earliest and most successful applications of linear optimization was in the oil industry...

  

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One of the earliest and most successful applications of linear optimization was in the oil industry in the 1950s. According to Bill Drew, the former manager of research for Exxon Mo- bil, the company used linear optimization to "schedule our tanker fleets, design port facilities, blend gasoline, create financial models, you name it." In this problem, we will use optimization to blend gasoline. (While the application of this problem is real, the data that we will be using here has been created for this problem.) Gasoline blending occurs in oil refineries, where crude oil is processed and refined into more useful products, such as gasoline and diesel fuel. We will consider three products: super gaso- line, regular gasoline, and diesel fuel. These can be made by mixing three different types of crude oil: crude 1, crude 2, and crude 3. Each product is distinguished by its octane rating, which measures the quality of the fuel, and its iron content, which is a contaminant in the gas. The crude oils each have an octane rating and iron content as well. Table 1 shows the required octane ratings and iron contents for each of the products, as well as the known octane ratings and iron contents of each of the crude oils. The gasoline produced must meet these standards for octane ratings and iron content. The octane rating and iron content of a product is the weighted average of the octane rating and iron content of the crude oils used to produce it. For example, if we produce regular gasoline using 20 barrels of Crude 1, 5 barrels of Crude 2, and 10 barrels of Crude 3, the Octane Rating of the regular gasoline would be: (20 x 11) + (5 x 6) + (10 x 8) 9.42 35 The numerator is the number of barrels of Crude 1 used times the octane rating of Crude 1, plus the number of barrels of Crude 2 used times the octane rating of Crude 2, plus the number of barrels of Crude 3 used times the octane rating of Crude 3. The denominator is the total number of barrels used. Similarly, the iron content of the regular gasoline would be: (20 x 0.5) + (5 x 2.0) + (10 x 2.8) = 1.37 35 The objective of the oil company is to marimize profit. In this case, profit is the total revenue from selling gasoline and fuel, minus the total cost of the crude oils. The revenue from one barrel of super gasoline is $70; the revenue from one barrel of regular gasoline is $60; and the revenue from one barrel of diesel fuel is $50. The cost of one barrel of Crude 1 is $45, the cost of one barrel of Crude 2 is $35, and the cost of one barrel of Crude 3 is $25. Lastly, the company can only buy 5,000 barrels of each type of crude oil, and can process no more than 14,000 barrels total of crude oil. Assume that one barrel of crude oil makes exactly one barrel of gasoline or fuel (nothing is lost in the conversion). Table 1: Octane Ratings and Iron Content for the Gasoline Blending problem Product or Oil Octane Rating Iron Content Super Gasoline Regular Gasoline Diesel Fuel no more than 1 no more than 2 at least 10 at least 8 at least 6 no more than 1 Crude 1 11 0.5 Crude 2 Crude 3 6. 2.0 2.8 a) (20 points) Formulate this problem as a linear optimization problem. i) (5 points) What are the decision variables? ii) (5 points) What is the objective function? iii) (10 points) What are the constraints? (Recall: constraints must be linear.) b) (5 points) Solve this problem using Gurobi. Describe the optimal solution. What are the values of the decision variables? What is the optimal objective. One of the earliest and most successful applications of linear optimization was in the oil industry in the 1950s. According to Bill Drew, the former manager of research for Exxon Mo- bil, the company used linear optimization to "schedule our tanker fleets, design port facilities, blend gasoline, create financial models, you name it." In this problem, we will use optimization to blend gasoline. (While the application of this problem is real, the data that we will be using here has been created for this problem.) Gasoline blending occurs in oil refineries, where crude oil is processed and refined into more useful products, such as gasoline and diesel fuel. We will consider three products: super gaso- line, regular gasoline, and diesel fuel. These can be made by mixing three different types of crude oil: crude 1, crude 2, and crude 3. Each product is distinguished by its octane rating, which measures the quality of the fuel, and its iron content, which is a contaminant in the gas. The crude oils each have an octane rating and iron content as well. Table 1 shows the required octane ratings and iron contents for each of the products, as well as the known octane ratings and iron contents of each of the crude oils. The gasoline produced must meet these standards for octane ratings and iron content. The octane rating and iron content of a product is the weighted average of the octane rating and iron content of the crude oils used to produce it. For example, if we produce regular gasoline using 20 barrels of Crude 1, 5 barrels of Crude 2, and 10 barrels of Crude 3, the Octane Rating of the regular gasoline would be: (20 x 11) + (5 x 6) + (10 x 8) 9.42 35 The numerator is the number of barrels of Crude 1 used times the octane rating of Crude 1, plus the number of barrels of Crude 2 used times the octane rating of Crude 2, plus the number of barrels of Crude 3 used times the octane rating of Crude 3. The denominator is the total number of barrels used. Similarly, the iron content of the regular gasoline would be: (20 x 0.5) + (5 x 2.0) + (10 x 2.8) = 1.37 35 The objective of the oil company is to marimize profit. In this case, profit is the total revenue from selling gasoline and fuel, minus the total cost of the crude oils. The revenue from one barrel of super gasoline is $70; the revenue from one barrel of regular gasoline is $60; and the revenue from one barrel of diesel fuel is $50. The cost of one barrel of Crude 1 is $45, the cost of one barrel of Crude 2 is $35, and the cost of one barrel of Crude 3 is $25. Lastly, the company can only buy 5,000 barrels of each type of crude oil, and can process no more than 14,000 barrels total of crude oil. Assume that one barrel of crude oil makes exactly one barrel of gasoline or fuel (nothing is lost in the conversion). Table 1: Octane Ratings and Iron Content for the Gasoline Blending problem Product or Oil Octane Rating Iron Content Super Gasoline Regular Gasoline Diesel Fuel no more than 1 no more than 2 at least 10 at least 8 at least 6 no more than 1 Crude 1 11 0.5 Crude 2 Crude 3 6. 2.0 2.8 a) (20 points) Formulate this problem as a linear optimization problem. i) (5 points) What are the decision variables? ii) (5 points) What is the objective function? iii) (10 points) What are the constraints? (Recall: constraints must be linear.) b) (5 points) Solve this problem using Gurobi. Describe the optimal solution. What are the values of the decision variables? What is the optimal objective.

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a iThere is one decision variable We have 9 decision variable in the optimisation model one for each ... View the full answer