Network Optimization Flow Decomposition The initial flow 4

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Network Optimization Flow Decomposition

Network Optimization Flow Decomposition

The initial flow 4 7 2 3 2 1 2 3 4 8 4

The initial flow 4 7 2 3 2 1 2 3 4 8 4 9 3 5 4 6 -2 5 A deficit node (more flow leaving than entering). An excess node (more flow entering than leaving) A balanced node (flow in = flow out) 2

Find a Path or Cycle W Select a node with deficit if there is

Find a Path or Cycle W Select a node with deficit if there is one. 4 2 Carry out a depth first search. Stop when a node with excess is reached or when there is a cycle. Determine the capacity of the walk W. 7 2 3 1 2 4 8 4 3 9 3 5 2 4 3 5 4 6 -2 5 1 The capacity of 1 -24 -5 -3 -1 is 2. 3

Updates Add the flow in W to the decomposition. 2 Subtract the flow in

Updates Add the flow in W to the decomposition. 2 Subtract the flow in W from the current flow. 4 2 7 5 2 3 1 2 0 3 4 8 6 4 9 7 3 5 4 6 -2 5 2 units around 1 -2 -4 -5 -3 -1 cycle flows path flows 4

Find a path or cycle W Select a node with deficit if there is

Find a path or cycle W Select a node with deficit if there is one. 2 5 2 3 2 1 3 3 6 4 Carry out a dfs. Determine the capacity of W. 4 5 7 2 4 3 5 The capacity of 2 -4 -5 -3 -2 is 3. 4 6 -2 5 5

Updates Add the cycle flow to the decomposition 2 3 0 2 1 update

Updates Add the cycle flow to the decomposition 2 3 0 2 1 update the current flow 5 2 2 3 4 6 3 4 7 4 3 5 4 6 -2 5 2 units around 1 -2 -4 -5 -3 -1 3 units around 2 -4 -5 -3 -2 cycle flows path flows 6

Find a path or cycle using dfs Select a node with deficit if there

Find a path or cycle using dfs Select a node with deficit if there is one. 2 2 1 4 3 4 Carry out a dfs. 3 Determine the capacity of W. 2 2 4 2 1 capacity of a path = min {arc capacity, excess, deficit} = 2 6 -2 4 5 4 2 3 5 3 6 -2 7

Updates Add the path flow to the decomposition 2 0 02 1 update the

Updates Add the path flow to the decomposition 2 0 02 1 update the current flow 2 units around 1 -2 -4 -5 -3 -1 3 units around 2 -4 -5 -3 -2 cycle flows 2 4 3 4 3 1 5 4 6 0 -2 5 2 units in 1 -2 -4 -6 path flows 8

Find a path or cycle using dfs Select a node with deficit if there

Find a path or cycle using dfs Select a node with deficit if there is one. Otherwise, select any node with flow leaving. 2 1 3 4 Carry out a dfs. Determine the capacity of W. 4 5 4 4 6 5 4 6 3 5 The capacity is 1 9

Updates Add the cycle flow to the decomposition 2 1 update the current flow

Updates Add the cycle flow to the decomposition 2 1 update the current flow 2 units around 1 -2 -4 -5 -3 -1 3 units around 2 -4 -5 -3 -2 1 unit around 3 -4 -6 -5 -3 cycle flows 4 3 3 4 3 1 0 5 4 6 5 4 2 units in 1 -2 -4 -6 path flows 10

Find a path or cycle using dfs Select a node with deficit if there

Find a path or cycle using dfs Select a node with deficit if there is one. Otherwise, select any node with flow leaving. 2 4 1 3 3 5 4 6 4 Carry out a dfs. Determine the capacity of W. The capacity of 3 -4 -5 -3 is 3 11

Updates Add the cycle flow to the decomposition 2 1 update the current flow

Updates Add the cycle flow to the decomposition 2 1 update the current flow 2 units around 1 -2 -4 -5 -3 -1 3 units around 2 -4 -5 -3 -2 1 unit around 3 -4 -6 -5 -3 3 units around 3 -4 -5 -3 cycle flows 4 3 3 5 4 6 4 2 units in 1 -2 -4 -6 path flows 12

Find a path or cycle using dfs Select a node with deficit if there

Find a path or cycle using dfs Select a node with deficit if there is one. Otherwise, select any node with flow leaving. 2 4 1 4 3 5 6 4 Carry out a dfs. Determine the capacity of W. 13

Updates and the final flow decomposition Add the cycle flow to the decomposition 2

Updates and the final flow decomposition Add the cycle flow to the decomposition 2 1 update the current flow 2 units around 1 -2 -4 -5 -3 -1 3 units around 2 -4 -5 -3 -2 1 unit around 3 -4 -6 -5 -3 3 units around 3 -4 -5 -3 4 units around 5 -6 -5 cycle flows 4 4 3 5 6 4 2 units in 1 -2 -4 -6 path flows 14

MITOpen. Course. Ware http: //ocw. mit. edu 15. 082 J / 6. 855 J

MITOpen. Course. Ware http: //ocw. mit. edu 15. 082 J / 6. 855 J / ESD. 78 J Network Optimization Fall 2010 For information about citing these materials or our Terms of Use, visit: http: //ocw. mit. edu/terms.