15.082J and 6.855J and ESD.78J
Successive Shortest Path
Algorithm
The Original Costs and Node Potentials
0
0
4
2
4
7
5
2
0
1
1
6
3
0
2
5
0
2
The Original Capacities and
Supplies/Demands
5
10
2
-2
4
30
25
23
20
20
1
20
3
-7
25
5
-19
3
Select a supply node and find the
shortest paths
7
shortest
path
distance
8
4
2
4
7
0
5
2
1
1
6
3
6
2
5
The
shortest
path tree is
marked in
bold and
blue.
8
4
Update the Node Potentials and the
Reduced Costs
-7
43
2
-8
4
7
5
6
0
0
0
2
1
1
0
6
3
-6
2
0
5
-8
5
Send Flow From a Supply Node to a
Demand Node Along Shortest Paths
(along arcs with reduced costs of 0)
5
10
2
-2
Arc numbers
are residual
capacities.
4
30
25
23
20
20
1
20
3
-7
25
5
Red arcs
have a
reduced cost
of 0
send 7
units from
node 1 to
node 3
-19
6
Update the Residual Network
5
10
2
-2
4
30
25
23
16 1
20
20
7
13
Arc (3,1)
has a
reduced
cost of 0
3
-7
0
25
5
-19
If an arc is
added to G(x),
then it has a
reduced cost
of 0, and it is
red.
Arcs in the
residual
network will
always have a
non-negative
reduced cost
7
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.
8
Send Flow From a Supply Node to a
Demand Node Along Shortest Paths
5
10
2
-2
Recall that red
arcs have a
reduced cost
of 0
4
30
25
16
20
20
1
7
13
3
0
25
5
Send 2 units
of flow from
node 1 to
node 4
-19
9
Update the Residual Network
5
-2 0
10
2
4
30
25
16
18
1
2
9
14
20
11
3
0
25
2 units of flow
were sent from
node 1 to node
4 on 1-3-4
5
-19
10
Send Flow From a Supply Node to a
Demand Node Along Shortest Paths
5
0
10
2
4
Send flow from
node 1 to node 5
30
25
14
18
1
20
2
9
11
3
0
25
5
How much
flow
should be
sent?
-19
11
Update the Residual Network
5
0
10
2
4
30
25
14
3
18
1
20
2
20
3
0
14
11 units of flow
were sent from
node 1 to node 5
5
-8
11
-19
12
Select a supply node and find the
shortest paths
-7
-8
3
2
0
0
0
4
6
1
1
0
0
0
3
-6
0
5
The shortest
path tree is
marked in bold
and blue.
The values on
the nodes are
the current
node potentials
-8
13
Update the node potentials and the
reduced costs
-11
-8
0
3
-7
2
0
0
0
4
3
1
1
0
3
0
3
-6 -9
0
To obtain new
node potentials,
subtract the
shortest path
distances from
the old
potentials.
5
-8 -11
14
Send Flow From a Supply Node to a
Demand Node Along Shortest Paths
5
0
10
2
4
30
Send flow from
node 1 to node 5
25
3
18
1
20
2
20
3
0
14
11
How much
flow will be
sent?
5
-8
15
Update the Residual Network
5
2
4
2
28
3
1
0
8
25
2
20
20
1
20
3
0
12
13
2 units of flow
were sent from
node 1 to node
5
5
-8
-6
16
Select a supply node and find the
shortest paths
-7
0
2
4
0
0
0
0
0
-11
3
1
1
3
The shortest
path tree is
marked in
bold and
blue.
0
3
-9
0
5
-11
17
Update the Node Potentials and the
Reduced Costs
-7
0
2
4
0
0
1
0
0
-11
2
0
1
4
0
3
-9
-10
0
To obtain the
new node
potential,
subtract the
shortest path
distance from
the old potential.
5
-11
-12
18
Send Flow From a Supply Node to a
Demand Node Along Shortest Paths
5
2
4
2
28
25
2
1
0
8
20
20
1
How much
flow can be
sent?
20
3
0
12
13
Send flow from
node 2 to node
5
5
-6
19
Update the Residual Network
5 0
0
3
2
4
7
28
25
2
1
5 units of flow
were sent from
node 2 to node 6.
20
1
15
5
20
3
0
12
13
5
-1
-6
20
Send Flow From a Supply Node to a
Demand Node
0
0
3
2
4
7
28
25
2
1
20
1
15
5
Send flow
from node
1 to node 5
20
3
0
12
13
5
-1
21
Update the Residual Network
0
0
2
2
4
8
27
0
1
1 unit of flow was
sent from node 1
to node 5.
25
3
20
1
14
6
The resulting
flow is feasible,
and also
optimal.
20
3
0
12
13
5
-1
0
22
The Final Optimal Flow
5
10,8
2
-2
4
30,3
25
23
20
20,6
1
20,20
3
-7
25,13
5
-19
23
The Final Optimal Node Potentials and
the Reduced Costs
-7
0
2
4
1
0
0
-11
2
0
1
Flow is
at lower
bound.
-4
Flow is
at upper
bound
3
-10
0
5
-12
24
MITOpenCourseWare
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15.082J / 6.855J / ESD.78J Network Optimization
Fall 2010
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