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Chapter 7 - Network Flow Models The Maximal Flow Problem imal flow problem involves determining the mum amount of material that can flow from on t (the source) to another (the sink) in a ne of this type of problem include determining aximum number of cars that can flow through m, the maximum amount of a liquid that can f gh series of pipes, and the maximum amount o can flow through a computer network.
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Chap07 Maximal Flow

Nov 07, 2014

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Page 1: Chap07 Maximal Flow

Chapter 7 - Network Flow Models

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The Maximal Flow Problem

The maximal flow problem involves determining the maximum amount of material that can flow from one point (the source) to another (the sink) in a network.

Example of this type of problem include determining the maximum number of cars that can flow through highway system, the maximum amount of a liquid that can flow through series of pipes, and the maximum amount of data that can flow through a computer network.

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Chapter 7 - Network Flow Models

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An example of a maximal flow problem is illustrated by the network of a railway system between Omaha and St.Louis shown in figure 7.18.

The Scott Tractor Company ships tractor parts from Omaha and St.Louis by railroad. However, a contract limits the number of railroad cars the company secure on each branch during a week.

The number of railroad cars available to the tractor company on each rail branches indicated by the number on the branch to the immediate right of each node (which represents a rail junction).

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Chapter 7 - Network Flow Models

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From the figure 7.18, from node 1 (Omaha) to node 2 six cars are available. 8 cars are available from node 2 to node 5, 5 cars are available from node 4 to node 6 (St. Louis), and so forth.

The number on each branch to immediate left of each node is the number of cars available for shipping in the opposite direction.

For example, no cars are available from node 2 to node 1. The branch from node 1 to node 2 is referred to as a directed branch because flow is possible in only one direction (from node 1 to node 2, but not from 2 to 1)

Notice that flow is possible in both direction on the branches between nodes 2 and 4 and nodes 3 and 4. These are referred to as undirected branches.

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Figure 7.19 Maximal Flow for Path 1-2-5-6

The Maximal Flow Problem Solution Approach (1 of 5)

Step 1: Arbitrarily choose any path through the network from origin to destination and ship as much as possible.

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Chapter 7 - Network Flow Models

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•Notice that the remaining capacities of the branches from node 1 to node 2 and from node 2 to node 5 are 2 and 4 cars, respectively and no cars available from node 5 to node 6.

•These values were computed by subtracting the flow of four (4) cars from the original number available.

•The actual flow of four (4) cars along each branch is shown enclosed in a box.

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Chapter 7 - Network Flow Models

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•Notice that the present input of four (4) cars node 1 and ouput of four cars from node 6 are also designated.

•The final adjustment on this path is to add the designated flow of four cars to the values at the immediate left of each node on paths, 1-2-5-6. These are the flows in the opposite direction. Thus the value 4 is added to the zeros nodes 2, 5, and 6.

•The maximum flow along this path is four cars, which is subtracted at each of the nodes.

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Figure 7.20 Maximal Flow for Path 1-4-6

The Maximal Flow Problem Solution Approach (2 of 5)

Step 2: Re-compute branch flow in both directions

Step 3: Select other feasible paths arbitrarily and determine maximum flow along the paths until flow is no longer possible.

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Chapter 7 - Network Flow Models

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The maximal flow for path 1-4-6 is eight cars, because of the flow four along path 1-4-6 is added to the flow previously determined in figure 7.19.

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Figure 7.21 Maximal Flow for Path 1-3-6

The Maximal Flow Problem Solution Approach (3 of 5)

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Chapter 7 - Network Flow Models

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The figure 7.21 is the maximal flow for path 1-3-6 within maximum possible flow of six cars.

This flow of six is subtracted from the branches along path 1-3-6 and added to the branches in the opposite direction as shown in figure 7.21.

This flow of 6 for this path is added to the previous flow of 8, which results in a total flow of 14 railroad cars.

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Figure 7.22 Maximal Flow for Path 1-3-4-6

The Maximal Flow Problem Solution Approach (4 of 5)

Continue

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Chapter 7 - Network Flow Models

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The figure 7.22 is the maximal flow for path 1-3-4-6. These path the available flow capacity along the path is one car.

This increase the total flow from 14 cars to 15 cars.

Close observation of the network in figure 7.23 which shown there are no more paths with available flow capacity. All paths out of nodes 3,4 and 5 shows zero available capacity which prohibits any further paths through the network.

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Figure 7.23 Maximal Flow for Railway Network

The Maximal Flow Problem Solution Approach (5 of 5)

Optimal Solution

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The Maximal Flow Problem Solution Method Summary

1. Arbitrarily select any path in the network from origin to destination.

2. Adjust the capacities at each node by subtracting the maximal flow for the path selected in step 1.

3. Add the maximal flow along the path to the flow in the opposite direction at each node.

4. Repeat steps 1, 2, and 3 until there are no more paths with available flow capacity.