Bus 140
Professor Denzler
9/28/08
Maximum Flow Problem
Given the network the maximum amount of units we can ship from node 1 to node 8 is 14.
How to solve a Maximum Flow Problem
Given: Each unit between nodes can only be used once.
The maximum amount of units that may travel
along a given path is equal to the lowest value along that path.
You may only go in the same direction the
arrows are pointing.
E.g.: you cannot travel from node 7 to node 5. You may travel as many paths as are
possible as long as there are available units along the path (the smallest value along that path is >0).
E.g.:
if you have used all units from node 1 to 2 you may no longer travel that path.
Steps to solve the given maximum flow problem:
1)
First path to follow is as follows: 1 to 2 to 4 to 6 to 8
a.
The lowest value along this path is 5, so therefore 5 units are shipped.
b.
5 units were shipped so therefore nodes 1 to 2 have no more units available since
they had 5 in the first place.
The 9 units that could be shipped from node 2 to 4 is
now down to 4 since we shipped 5 units along the path.
The value from nodes 4 to 6
is down to 13 since it started at 18 and we shipped 5 units along the path.
The value
from 6 to 8 is down to 1 unit left.
2)
Our next path we will choose is 1 to 3 to either 4 or 5, but in the case we will choose 4,
then 4 to 6 to 8.
a.
The lowest value along the path is 1, so therefore we have 1 unit shipped to bring our
total to 6 units shipped – 5 from the last route and 1 from this route.
b.
Now we must adjust the values along the path.
The 12 from nodes 1 to 3 becomes
11, the 6 from nodes 3 to 4 becomes 5, the 13 we still have from 4 to 6 becomes 12,
and the 1 that we had left from 6 to 8 becomes a 0.
3)
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 '08
 Denzeler,David
 Shortest path problem, Flow network, Maximum flow problem

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