Network Optimization II Maximum Flow and Minimum Cost Flow

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Network Optimization IIMaximum Flow and Minimum
Cost Flow

• TMR4115
• Thursday, Nov 21 2003
• Prof. Stein Ove Erikstad

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Recalling Our North Sea Pipeline Network
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Vocabulary

• Nodes the platforms
• Arcs the pipelines
• Directed or undirected arc
• Directed and undirected networks
• Path - sequence of arcs connecting two nodes
• Directed and undirected path
• Cycle (directed or undirected)
• Connected/unconnected network
• Arc capacity
• Supply node, demand node, transshipment node

Directed arc
Cycle
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Problem types

• Shortest path
• Minimum spanning tree
• Maximum flow
• Minimum cost flow

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Maximum Flow Algorithm
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Network
Source
2
6
5
2
1
Transhipment nodes
7
4
4
3
Arc capacity
1
7
5
Sink
7

• Move capacity to start of arcs
• residual capacity
• Add current flow at the other end
• currently zero
• Identify an augmenting path
• e.g. 0?A ?D ?T
• Find the minimum residual capacity along this
  path
• min 2,7,5 2
• Assign this minimum flow to the path

2
6
0
5
0
2
7
1
4
0
0
0
4
3
0
0
0
0
0
1
7
5
0
0
8
0
6
2
5
0
2
5
1
4
0
0
0
4
3
0
0
0
2
0
1
7
3
0
2
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Minimum cut
2
6
0
5
0
2
7
1
4
0
0
0
4
i.e. Max Flow capacity min cut 12
3
0
0
0
0
0
1
7
5
0
0
13
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• Blackboard calculation
• Excel solution

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Minimum cost flow

• Directed, connected network
• At least one node is a supply node
• At least one node is a demand node
• All remaining nodes are transhipment nodes
• Each arc has a max capacity (might be 8)
• The arcs have sufficient capacity to enable all
  flow generated at supply nodes to reach all
  demand nodes
• The cost of flow at each arc is proportional to
  the amount
• The objective is to minimize the total cost of
  sending the available supply through the network
  to satisfy the given demand

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Min Cost Flow Network
Supply nodes
2
6
5
2
1
Transhipment nodes
7
T1
4
4
3
1
5
7
Demand nodes
13
2
4
5
2
1
7
4
4
3
1
7
5
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