10. Major oil companies use linear programming to model many phases of their ope

Question

10. Major oil companies use linear programming to model many phases of their operations. Consider the following simplified version of part of a refinery operation. Two kinds of aviation gasoline, high octane and low octane, are made by blending four components from the refinery output. For the low octane gasoline the components are augmented with a small amount of tetraethyllead (TEL) to give the low-octane ratings shown in the accompanying table. The high-octane gasoline is made from the same components when these have been augmented with a larger amount of TEL, giving the high-octane ratings in the table. Assume that the octane rating (OR) of the mixture is the volumetric average of the octane ratings of the components. That is, letting V denote volume, we have OR compl compi ORmix - comp comp2 The vapor pressure (a measure of the tendency of the gasoline to evaporate) of both gasolines must be 7. Assume that the vapor pressure of a mixture is the volumetric average of the vapor pressures of the components. Vapor pressure and octane rating are the only two physical properties for which there are constraints. Data for the components and desired mixtures are given in the accompanying table OR Va pressure High Low Demand Supply Revenue Cost Component Alkylate Catalytic cracked Straight run Isopentane 6.5 4 18 108 98 94 87 8780 108 100 700 600 900 500 7.20 4.35 3.80 4.30 Mixture 1300 90 800 6.50 7.50 100 High octane Low octane 60 Chapter 1 Introduction to Linear Assume that the demands must be met exactly. Measure the profit by using revenue less cost. Set up a model of this situation that maximizes this measure of profit

Explanation / Answer

Let,

V11 = volume of Alkylate used in high octane; V12 = volume of Alkylate used in low octane
V21 = volume of Catalytic cracked used in high octane; V22 = volume of Catalytic cracked used in low octane
V31 = volume of Straight run used in high octane; V32 = volume of Straight run used in low octane
V41 = volume of Isopentane used in high octane; V42 = volume of Isopentane used in low octane

Objective Function: Maximize. Z = Total Profit = Revenue - Cost

or, Z = 6.5(V11+V21+V31+V41) + 7.5(V12+V22+V32+V42) - 7.2(V11+V12) - 4.35(V21+V22) - 3.80(V31+V32) - 4.3(V41+V42)

or, Z = -0.7V11 + 2.15V21 + 2.7V31 + 2.2V41 + 0.3V12 + 3.15V22 + 3.7V32 + 3.2V42

Subject to,

5V11 + 6.5V21 + 4V31 + 18V41 = 7(V11 + V21 + V31 + V41) (vapor pressure high octane)

or, 2V11 + 0.5V21 + 3V31 - 11V41 = 0

5V12 + 6.5V22 + 4V32 + 18V42 = 7(V12 + V22 + V32 + V42) (vapor pressure low octane)

or, 2V12 + 0.5V22 + 3V32 - 11V42 = 0

108V11 + 94V21 + 87V31 + 108V41 = 100(V11 + V21 + V31 + V41) (OR for high octane)

or, -8V11 + 6V21 + 13V31 - 8V41 = 0

98V12 + 87V22 + 80V32 + 100V42 = 90(V12 + V22 + V32 + V42) (OR for low octane)

or, -8V12 + 3V22 + 10V32 - 10V42 = 0

V11 + V21 + V31 + V41 = 1300 (demand for high octane)

V12 + V22 + V32 + V42 = 800 (demand for low octane)

V11 + V12 700 (supply of Alkylate)

V21 + V22 600 (supply of Catalytic cracked)

V31 + V32 900 (supply of Straight run)

V41 + V42 500 (supply of Isopentane)

Vij 0

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