chap9_for_real - Chapter 9 Integer Programming Outline...

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Chapter 9: Integer Programming Outline Outline For Fun Introduction BIP Applications Modeling Tricks with Binary Variables Some Perspectives on Solving IPs Branch and Bound Chapter 9: Integer Programming 1
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Chapter 9: Integer Programming For Fun Outline For Fun Introduction BIP Applications Modeling Tricks with Binary Variables Some Perspectives on Solving IPs Branch and Bound Chapter 9: Integer Programming 2
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Chapter 9: Integer Programming For Fun I Greg Hogan: I Class president, son of a Baptist minister, second cello in the Lehigh University orchestra, chaplain’s office assistant ... I bank robber! Chapter 9: Integer Programming 3
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Chapter 9: Integer Programming For Fun The Traveling Bank Robber Problem I My mission: rob 8 specific banks as quickly as possible. I In what order do I rob the banks? I Objective: to minimize the total distance traveled. I Constraints: visit every bank. I Start and end at my house. I Information I need: the distance between every bank (can be found using maps.google.com). Chapter 9: Integer Programming 4
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Chapter 9: Integer Programming For Fun My Daddy was a bank robber, but he never hurt nobody I One possible solution: Try every possible order. I How many different possible solutions are there? I n banks means n ! many possible solutions. Too hard. I I can use a heuristic method (try to find good, but maybe not optimal solutions). I Examples: “Nearest Neighbor” or “Cheapest-Intersection” Chapter 9: Integer Programming 5
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Chapter 9: Integer Programming For Fun An Integer Program I Good solutions are good, but every second is important (I am too pretty for jail). I Solution, use optimization techniques! I Step 1: Declare variables. I x ij = 1 means that rob bank i immediately before bank j . I Step 2: Objective Function. I Minimize i j c ij x ij Chapter 9: Integer Programming 6
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Chapter 9: Integer Programming For Fun Continued I What are some constraints? I I must walk into every bank and walk out of every bank once (and only once): N X i =1 x ij = 1 for all j N X j =1 x ij = 1 for all i . I This is not enough! I As is, I can have the following solution 1 2 3 1 and 4 5 4. I We call these subtours . Chapter 9: Integer Programming 7
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Chapter 9: Integer Programming For Fun Subtour Elimination I We can prohibit this type of solution by adding the inequality x 12 + x 13 + x 21 + x 23 + x 31 + x 32 2 I For every subset of banks S the constraint: X i S , j S x ij ≤ | S | - 1 prohibits “beaming” (as in “Beam me up Scotty”). Chapter 9: Integer Programming 8
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Chapter 9: Integer Programming For Fun The formulation min X i X j c ij x ij s.t. N X i =1 x ij = 1 j N X j =1 x ij = 1 i X i S , j S x ij ≤ | S | - 1 S N x ij = 0 or 1 Chapter 9: Integer Programming 9
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Chapter 9: Integer Programming For Fun Interesting facts I The previous problem is an Integer Program .
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