Lecture 25
Network models : Maximal Flow Problem
25.1 Maximal Flow Problem
Algorithm
Step1
Find a path from source to sink that can accommodate a positive flow of material. If no path
exists go to step 5
Step2
Determine the maximum flow that can be shipped from this path and denote by ‘k’ units.
Step3
Decrease the direct capacity (the capacity in the direction of flow of k units) of each branch of
this path ‘k’ and increase the reverse capacity k1. Add ‘k’ units to the amount delivered to sink.
Step4
Goto step1
Step5
The maximal flow is the amount of material delivered to the sink. The optimal shipping schedule
is determined by comparing the original network with the final network. Any reduction in
capacity signifies shipment.
Example 1
Consider the following network and determine the amount of flow between the networks.
1
Solution
Iteration 1: 1 – 3 – 5
Iteration 2: 1 – 2 – 3 – 4 – 5
Iteration 3: 1 – 4 – 5
2
Iteration 4: 1 – 2 – 5
Iteration 5: 1 – 3 – 2 – 5
Maximum flow is 60 units. Therefore the network can be written as
3
Example 2
Solve the maximal flow problem
Solution
Iteration 1: O – A – D – T
Iteration 2: O – B – E – T
4
Iteration 3: O – A – B – D – T
Iteration 4: O – C – E – D – T
Iteration 5: O – C – E – T
5
Iteration 6: O – B – D – T
Therefore there are no more augmenting paths. So the current flow pattern is optimal. The
maximum flow is 13 units.
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