Successive Shortest Path Algorithm 15.082J and 6.855J and ESD.78J Successive Shortest Path Algorithm

The Original Costs and Node Potentials 4 2 4 7 5 2 1 1 6 2 3 5

The Original Capacities and Supplies/Demands 5 -2 10 2 4 30 25 23 20 1 20 20 3 25 5 -7 -19

Select a supply node and find the shortest paths shortest path distance 7 8 4 2 4 7 5 The shortest path tree is marked in bold and blue. 2 1 1 6 2 3 5 8 6

Update the Node Potentials and the Reduced Costs -7 -8 4 3 2 4 7 5 2 6 1 1 6 2 3 5 -6 -8

Send Flow From a Supply Node to a Demand Node Along Shortest Paths (along arcs with reduced costs of 0) 5 -2 Arc numbers are residual capacities. Red arcs have a reduced cost of 0 10 2 4 30 25 23 20 1 20 send 7 units from node 1 to node 3 20 3 25 5 -7 -19

Update the Residual Network 5 -2 If an arc is added to G(x), then it has a reduced cost of 0, and it is red. 10 2 4 30 25 23 20 1 16 20 Arcs in the residual network will always have a non-negative reduced cost 7 13 3 25 5 Arc (3,1) has a reduced cost of 0 -7 -19

A comment At this point, one would choose a source node, and then find the shortest path from the source node to all other nodes, and then update the residual network. However, there are still paths of 0 reduced cost in the residual network, and it makes sense to use them. This heuristic is quite useful in practice.

Send Flow From a Supply Node to a Demand Node Along Shortest Paths 5 -2 10 2 4 Recall that red arcs have a reduced cost of 0 30 25 20 1 16 20 7 Send 2 units of flow from node 1 to node 4 13 3 25 5 -19

Update the Residual Network 5 -2 10 2 4 30 2 units of flow were sent from node 1 to node 4 on 1-3-4 25 18 1 16 20 2 9 14 11 3 25 5 -19

Send Flow From a Supply Node to a Demand Node Along Shortest Paths 5 10 2 4 Send flow from node 1 to node 5 30 25 18 14 1 20 2 How much flow should be sent? 9 11 3 25 5 -19

Update the Residual Network 5 10 2 4 30 11 units of flow were sent from node 1 to node 5 25 18 14 1 20 3 2 20 3 14 5 -8 11 -19

Select a supply node and find the shortest paths -7 -8 3 2 4 The shortest path tree is marked in bold and blue. 6 1 1 The values on the nodes are the current node potentials 3 5 -6 -8

Update the node potentials and the reduced costs -11 -7 -8 3 2 4 To obtain new node potentials, subtract the shortest path distances from the old potentials. 3 1 1 3 3 5 -11 -6 -9 -8

Send Flow From a Supply Node to a Demand Node Along Shortest Paths 5 10 2 4 30 Send flow from node 1 to node 5 25 18 1 3 20 2 20 How much flow will be sent? 3 14 5 -8 11

Update the Residual Network 5 8 2 4 2 28 25 2 units of flow were sent from node 1 to node 5 3 2 20 1 1 20 20 3 12 5 -8 13 -6

Select a supply node and find the shortest paths -7 -11 2 4 The shortest path tree is marked in bold and blue. 3 1 1 3 3 5 -9 -11

Update the Node Potentials and the Reduced Costs -7 -11 2 4 To obtain the new node potential, subtract the shortest path distance from the old potential. 1 2 1 4 3 5 -11 -9 -12 -10

Send Flow From a Supply Node to a Demand Node Along Shortest Paths 5 8 2 4 2 28 Send flow from node 2 to node 5 25 2 20 1 1 20 20 How much flow can be sent? 3 12 5 13 -6

Update the Residual Network 5 3 2 4 7 28 5 units of flow were sent from node 2 to node 6. 25 2 20 1 1 15 5 20 3 12 5 -1 13 -6

Send Flow From a Supply Node to a Demand Node 3 2 4 7 28 Send flow from node 1 to node 5 25 2 20 1 1 15 5 20 3 12 5 -1 13

Update the Residual Network 2 2 4 8 1 unit of flow was sent from node 1 to node 5. 27 25 3 20 1 1 14 6 The resulting flow is feasible, and also optimal. 20 3 12 5 -1 13

The Final Optimal Flow 5 -2 10,8 2 4 30,3 25 23 20 1 20,6 20,20 3 25,13 5 -7 -19

The Final Optimal Node Potentials and the Reduced Costs -7 -11 2 4 Flow is at lower bound. 1 2 1 -4 3 5 Flow is at upper bound -10 -12

15.082J / 6.855J / ESD.78J Network Optimization MITOpenCourseWare http://ocw.mit.edu 15.082J / 6.855J / ESD.78J Network Optimization Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.