lab4 - ENGRI 115 Engineering Applications of OR Fall 2007...

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ENGRI 115 Engineering Applications of OR Fall 2007 The Maximum Flow Problem Lab 4 Name: Objectives: Introduce the maximum flow problem. Demonstrate how to solve the maximum flow problem by the Ford-Fulkerson algorithm. Key Ideas: sink and source nodes flow feasible flow value of flow cuts capacity of cut max-flow min-cut theorem integrality properties sensitivity analysis Prelab Exercise: For the following graph, compute the maximum flow from node 1 to node 3 by trial-and-error. Can you give a convincing argument that you have found an optimal solution? Can you find a cut of minimum capacity? What is the capacity of the minimum cut? What is the value of the maximum flow? 1 2 3 4 5 1 4 2 5 1 2 2 3 2 0 1
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#1 - Solving an example with the Ford-Fulkerson Algorithm In the first part of the lab we will solve an example using the algorithm you’ve learned in class. Instead of the labeling method you’ve seen, we will look for paths that can carry additional flow from the source to the sink using the so called residual graph . Take the following example. The source (where the flow comes from) is node 1 , the sink (where the flow goes to) is node 7 . The numbers on the arcs are the capacities. We would like to push as much flow as we can from node 1 to node 7 . So how do we start? 1 2 3 4 5 6 7 6 6 1 3 3 7 1 1 4 4 5 Let’s pick a path that goes from 1 to 7 (if there is any), and push as much flow as possible through this path. Say we pick the path 1-3-5-7 . How many units of flow can be pushed through this path? Is there an arc which got saturated (that is, no more flow can go through it)? Now we write down the flow on the arcs (the first number is the flow value, the second is the capacity of the arc as usual). 1 2 3 4 5 6 7 0, 3 0, 6 0, 1 4, 6 0, 1 0, 3 0, 1 4, 7 0, 5 4, 4 0, 4 What would be the next step? We might think that it’s a reasonable idea to decrease the capacities on the
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