Midterm solutions

# Midterm solutions - University of Toronto Department of...

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University of Toronto Department of Mechanical and Industrial Engineering MIE468: Facility Planning (Winter 2012) Page 1 of 6 Midterm solutions Problem 1: (13 points) a) 55,000 barrels of gasoline = 2,310,000 gallons of gasoline 25,000 barrels of diesel = 1,050,000 gallons of diesel To produce 55,000 barrels of gasoline, need 𝑰 𝑮 = 𝟐 , 𝟑𝟏𝟎 , 𝟎𝟎𝟎 𝟏𝟗 . 𝟔 = 𝟏𝟏𝟕 , 𝟖𝟓𝟕 . 𝟏𝟒 → 𝟏𝟏𝟕 , 𝟖𝟓𝟖 barrels of crude To produce 25,000 barrels of diesel, need 𝑰 𝑫 = 𝟏 , 𝟎𝟓𝟎 , 𝟎𝟎𝟎 𝟏𝟎 = 𝟏𝟎𝟓 , 𝟎𝟎𝟎 barrels of crude Since one barrel of crude produces both 19.6 gallons of gas and 10 gallons of diesel, then the total input is simply the max of the two amounts above: 𝑰 𝒕𝒐𝒕𝒂𝒍 = 𝐦𝐚𝐱 { 𝑰 𝑮 , 𝑰 𝑫 } = 𝟏𝟏𝟕 , 𝟖𝟓𝟖 b) Let x = amount of extra gasoline output = amount of diesel output reduction Adjusting the previous equations: 𝑰 𝑮 = 𝟐 , 𝟑𝟏𝟎 , 𝟎𝟎𝟎 𝟏𝟗 . 𝟔 + 𝒙 , 𝑰 𝑫 = 𝟏 , 𝟎𝟓𝟎 , 𝟎𝟎𝟎 𝟏𝟎 − 𝒙 To minimize the number of crude barrels needed, we want the above two quantities to be equal. Then we have to check to make sure x is between 0 and 1. Setting 𝑰 𝑮 = 𝑰 𝑫 , and solving for x, we get x = 0.75, which is between 0 and 1. Plugging this value into the equations for 𝑰 𝑮 and 𝑰 𝑫 , we get 𝑰 𝒕𝒐𝒕𝒂𝒍 = 𝐦𝐚𝐱 { 𝑰 𝑮 , 𝑰 𝑫 } = 𝐦𝐚𝐱 { 𝟏𝟏𝟑 , 𝟓𝟏𝟒 , 𝟏𝟏𝟑 , 𝟓𝟏𝟒 } = 𝟏𝟏𝟑 , 𝟓𝟏𝟒 Problem 2: (12 points) a) machine 1 2 3 4 5 part 1 1 1 2 1 1 3 1 1 4 1 5 1 1 6 1 1

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