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4120_lecture-22-F10

4120_lecture-22-F10 - Computational Methods for Management...

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Lecture 22 Reading: 9.1-9.3 of text book. Computational Methods for Management and Economics Carla Gomes
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Outline Shortest Path
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Shortest Path Problem
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The Shortest Path Problem 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 What is the shortest path from a source node (often denoted as s) to a sink node, (often denoted as t)? What is the shortest path from node 1 to node 6? Assumptions for this lecture: 1. There is a path from the source to all other nodes. 2. All arc lengths are non-negative s t
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Shortest Path Problem Where does it arise in practice? Common applications shortest paths in a vehicle shortest paths in internet routing Less obvious: close connection to dynamic programming How will we solve the shortest path problem? Dijkstra’s algorithm
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Shortest Path Problem 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 Find the shortest paths by inspection.
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Shortest Path Problem 1 2 3 4 5 6 2 4 2 1 3 4 2 3 2 Shortest Path Problem Special case of Min. Cost Flow Problem. (more on this problem later) 1 -1 0 0 0 0 All arcs have capacity 1
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Representation as an integer program An integer program is a linear program in which some or all of the variables are required to be integer We will formulate the shortest path problem as an integer program. Find the shortest path from node 1 to node 6 Decision variables: – x ij = 1 if arc (i,j) is in the path. – x ij = 0 if arc (i,j) is not in the path
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The constraint matrix is the node arc incidence matrix 1 1 0 0 0 0 0 0 0 1 = 1 1 1 -1 0 0 0 0 0 0 = 1 -1 0 -1 0 0 0 0 0 0 = 1 -1 -1 0 0 0 0 0 0 0 = 1 1 -1 -1 0 0 0 0 0 0 = -1 -1 0 0 0 0 0 0 0 -1 = x 12 x 13 x 23 x 25 x 24 x 35 x 46 x 54 x 56 1 2 3 4 5 6 Constraint matrix of Shortest Path Problem
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On Incidence Matrices If the constraint matrix for a linear program is a node-arc incidence matrix (at most one 1 and at most one –1 per column), then the linear program solves in integer optima. Thus, we can solve the shortest path problem as an LP, and get the optimum path.
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On Incidence Matrices If the constraint matrix for a linear program is a node-arc incidence matrix (at most one 1 and at most one –1 per column), then the linear program solves in integer optima. Thus, we can solve the shortest path problem as an LP, and get the optimum path. Shortest Path Pivoting
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Littletown Fire Department Littletown is a small town in a rural area. Its fire department serves a relatively large geographical area that includes many farming communities. Since there are numerous roads throughout the area, many possible routes may be available for traveling to any given farming community. Question: Which route from the fire station to a certain farming community minimizes the total number of miles?
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The Littletown Road System   F ire S tation H G F E D C B A 3 6 4 2 1 7 5 4 6 8 6 4 3 4 6 7 5 2 3 Farm ing Community
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The Network Representation   T H G F E D B C A O (Destination) (O rig in) 3 6 1 2 6 4 3 4 7 8 6 5 4 2 3 4 6 7 5
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Spreadsheet Model 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 B C D E F G H I J K From To On Route Distance Nodes Net Flow Supply/Demand Fire St. A 1 3 Fire St. 1 = 1 Fire St. B 0 6 A 0 = 0 Fire St.
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