hw4.sol[1]

hw4.sol[1] - IE 335 Operations Research Optimization...

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IE 335 Operations Research - Optimization Solutions to Homework 4 Fall 2011 Problem 15 Let x ijg represent the number of grade g students in neighborhood i assigned to school j for i = 1 ,...,I j = 1 ,...,J , and g = 1 ,...,G min G X g =1 J X j =1 I X i =1 d ij x ijg (1) s.t. I X i =1 x ijg C jg for j = 1 ,...,J and g = 1 ,...,G (2) J X j =1 x ijg = S ig for i = 1 ,...,I and g = 1 ,...,G (3) x ijg 0 for i = 1 ,...,I , j = 1 ,...,J and g = 1 ,...,G (4) The objective function (1) tries to minimize the weighted sum of the number of students of each grade, neighborhood and school weighted by the distance of the neighborhood from the school. Constraint (2) ensures that the total number of students do not exceed the capacity of a grade in a school and constraint (3) ensures that all students are assigned to schools. Problem 16 Let’s use the decision variables given in the hint: x i = fraction of item i allocated to Captain Hook i = 1 ,..., 4; y i = fraction of item i allocated to Captain Sparrow i = 1 ,..., 4 . If we allocate the items according to the decision variables above, then we can write the total value points of Captain Hook as 40 x 1 + 10 x 2 + 10 x 3 + 40 x 4 . Similarly, we can write the total value points of Captain Sparrow as 30 y 1 + 20 y 2 + 20 y 3 + 30 y 4 . Then, the following optimization model with a maximin objective determines how to allocate the treasure between Captain Hook and Captain Sparrow, in a way that maximizes the minimum total value points of any captain: max min { 40 x 1 + 10 x 2 + 10 x 3 + 40 x 4 , 30 y 1 + 20 y 2 + 20 y 3 + 30 y 4 } (5) s.t. x i + y i 1 for i = 1 , 2 , 3 , 4 , (6) x i 0 for i = 1 , 2 , 3 , 4 , (7) y i 0 for i = 1 , 2 , 3 , 4 . (8) 1

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Constraint (6) ensures that at most 100% of each item is allocated to the captains. Constraints (7) and (8) ensure that the fractions are nonnegative. We can convert this optimization model into a linear program, using the technique we discussed in Lecture 9. Let f be an auxiliary decision variable that represents the minimum total value points of any captain. Then the above optimization model is equivalent to the following linear program: max f s.t. f 40 x 1 + 10 x 2 + 10 x 3 + 40 x 4 , f 30 y 1 + 20 y 2 + 20 y 3 + 30 y 4 , x i + y i 1 for i = 1 , 2 , 3 , 4 , x i 0 for i = 1 , 2 , 3 , 4 , y i 0 for i = 1 , 2 , 3 , 4 . Problem 17
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This note was uploaded on 12/14/2011 for the course IE 335 taught by Professor Jean-philippe,r during the Fall '08 term at Purdue.

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hw4.sol[1] - IE 335 Operations Research Optimization...

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