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

2. The Original Costs and Node Potentials4 2 4 7 5 2 1 1 6 2 3 5

3. The Original Capacities and Supplies/Demands5 -2 10 2 4 30 25 23 20 1 20 20 3 25 5 -7
-19

4. 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

5. 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

6. 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

7. Update the Residual Network5 -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

8. 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.

9. Send Flow From a Supply Node to a Demand Node Along Shortest Paths5 -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

10. Update the Residual Network5 -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

11. Send Flow From a Supply Node to a Demand Node Along Shortest Paths5 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

12. Update the Residual Network5 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

13. 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

14. 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

15. Send Flow From a Supply Node to a Demand Node Along Shortest Paths5 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

16. Update the Residual Network5 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

17. 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

18. 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

19. Send Flow From a Supply Node to a Demand Node Along Shortest Paths5 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

20. Update the Residual Network5 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

21. Send Flow From a Supply Node to a Demand Node3 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

22. Update the Residual Network2 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

23. 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

24. 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

25. 15.082J / 6.855J / ESD.78J Network Optimization
MITOpenCourseWare
15.082J / 6.855J / ESD.78J Network Optimization
Fall 2010
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