Successive Shortest Path Algorithm 15.082J and 6.855J Successive Shortest Path Algorithm Obtain a network, and use the same network to illustrate the shortest path problem for communication networks, the max flow problem, the minimum cost flow problem, and the multicommodity flow problem. This will be a very efficient way of introducing the four problems. (Perhaps under 10 minutes of class time.)

The Original Costs and Node Potentials 4 2 4 7 5 2 1 1 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 6 2 3 5

The Original Capacities and Supplies/Demands 5 -2 10 2 4 30 25 23 20 1 20 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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? Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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? Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 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 Introduce myself and the Tas. It is critical that students sign up for the virtual campus web site. Hand out instructions on how to use it. Make sure that everyone signs up on the list. Also, create a class list that is separate from the web site. Ask for a show of where students come from. There should be four major contingencies: Grad EECS, ORC, Transportation, Rest 3 5 Flow is at upper bound -10 -12