15.082J and 6.855J and ESD.78J

A minimum cost flow problem
0
2
30, $7
25 1
25, $5
10, $4
20, $2
20, $6
0
4
20, $1
3
0
25, $2
5
-25
2

The Original Capacities and Feasible Flow
0
2
10, 10
0
4
30, 25
25 1
25, 15
20, 10
20, 20
20, 0
The feasible flow can be found by solving a max flow.
3
0
25, 5
5
-25
3

Capacities on the Residual Network
1
10
2
5
20
25
15 10
10
10
3
5
20
4
5
20
4

Costs on the Residual Network
Find a negative cost cycle, if there is one.
1
2
7
-7
-5
5
6
3
-4
2
-2
2
-2
4
5
-1
5

Send flow around the cycle
4 2
Send flow around the negative cost cycle
The capacity of this cycle is 15.
1
25
15
20
3
Form the next residual network.
5
6

Capacities on the residual network
1
20
10
15
5
2
25
10
10
3
5
20
10
4
5
20
7

Costs on the residual network
Find a negative cost cycle, if there is one.
1
7
-7
6
-6
2
5
3
-4
2
-2
2
-2
4
5
-1
8

Send flow around the cycle
2
Send flow around the negative cost cycle
The capacity of this cycle is 10.
1
3
Form the next residual network.
10
20
4
5
20
9

Capacities on the residual network
1
20
10
15
5
2
25
10
20
3
10
15
4
10 10
5
10

Costs in the residual network
Find a negative cost cycle, if there is one.
1
6
7
-7
2
5
-6
3
-4
2
2
-2
4
-1
1
5
11

Send Flow Around the Cycle
2
Send flow around the negative cost cycle
The capacity of this cycle is 5.
1
10
5
3
Form the next residual network.
10
20
4
5
12

Capacities on the residual network
1
25
5
20
5
2
25
5
15
5
4
10 10
3
10
15
5
13

Costs in the residual network
Find a negative cost cycle, if there is one.
1
7
-7
5
2
-6
3
4
-4
2
-2
-1
4
1
2
-2
5
14

Send Flow Around the Cycle
2
Send flow around the negative cost cycle
The capacity of this cycle is 5.
1
3
Form the next residual network.
10
5
10
4
5
15

Capacities on the residual network
1
25
5
20
5
2
25
5
20
4
5 15
3
5
20
5
16

Costs in the residual network
1
7
-7
5
2
Find a negative cost cycle, if there is one.
-6
3
4
-4
2
2
-2
4
-1 1
5
There is no negative cost cycle. But what is the proof?
17

Compute shortest distances in the residual network
7
4
11
2 4
7
-4
-7
0 1 2
5 -1 1
-6
Let d(j) be the shortest path distance from node 1 to node j.
3
10
2
-2
5
12
Next let p
(j) = -d(j)
And compute c p
18

Reduced costs in the residual network
0 1
0
0
7
2
2
0
-0
1
11
4
0
0
4
The reduced costs in G(x*) for the optimal flow x* are all non-negative.
3
10
0
0
5
12
19

MIT OpenCourseWare http://ocw.mit.edu
15.082J / 6.855J / ESD.78J Network Optimization
Fall 2010
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