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## Logistics Routing Plans: Max Flow Problem

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### Logistics Routing Plans: Max Flow Problem Objectives and Agenda: 1. Examples for flow of materials over limited capacity channels 2. Finding maximum flows: Ford ... – PowerPoint PPT presentation

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Title: Logistics Routing Plans: Max Flow Problem

1
Logistics Routing Plans Max Flow Problem
Objectives and Agenda 1. Examples for flow of
materials over limited capacity channels 2.
Finding maximum flows Ford-Fulkerson Method
2
Logistics supply problem Example 1
Legend Node Sub-station Edge Power
line Weight Line capacity
What is the maximum power we can supply to Wan
Chai for a Light-n-Sound Show?
3
Logistics supply problem Example 2
Legend nodes train line junctions edges rail
line weights max no. of compartments/day
Maximum number of compartments per day from
Detroit? SF ?
4
Maximum Flow Problem definitions
SOURCE Node with net outflow Production point
SINK Node with net inflow Consumption point
CAPACITY Maximum flow on an edge
Efficient method to solve such problems
Ford-Fulkerson Method
5
Ford-Fulkerson Method..
Three fundamental concepts 1. Flow
cancellation 2. Augmentation flow 3. Residual
network
6
Ford-Fulkerson Method Flow Cancellation
Flow Cancellation Compute the NET FLOW between a
pair of nodes
Network
Flow 5 units, a ? b, 3 units b ? a
Net flows
Additional 7 units from b ? a ?!
7
Ford-Fulkerson Method Augmenting Path
Augmenting Path any path from source ? sink with
positive capacity
Examples
Path Capacity
ltD, M, B, Sgt 8
ltD, M, K, P, Sgt 6

8
Ford-Fulkerson Method Residual network
Given a Network G, with flow, f , on path p
Flow cancellation
residual network
9
Ford-Fulkerson Method..
Initialize the network zero flow
Residual network Carries Max Flow
any augmenting path p in network ?
NO
YES
Apply maximum flow allowed on p
Compute residual network
10
Ford-Fulkerson Method Initialize
edges
11
Ford-Fulkerson Method Find an augmentation path
Step 2. Find a positive flow from Source ? Sink
Flow, f 6 units
12
Ford-Fulkerson Method...
Step 3. Update the residual network due to flow f
Current total flow 6
13
Ford-Fulkerson Method.
Augmentation path ltD, M, B, Sgt Max flow 2
Current total flow 62 Residual network
M
10
0
B
2
8
15
D
0
0
12
12
8
8
2
0
0
10
10
0
0
14
14
7
7
S
6
6
K
6
6
0
0
4
4
P
14
Ford-Fulkerson Method..
Augmentation path ltD, K, M, B, Sgt Max flow 10
Current total flow 6210 Residual network
M
0
0
B
12
8
5
D
0
0
18
2
12
10
10
10
0
0
4
7
7
S
6
K
6
0
4
P
15
Ford-Fulkerson Method
Augmentation path ltD, K, P, B, Sgt Max flow 4
Current total flow 62104 Residual network
M
0
0
B
12
8
1
D
0
0
18
No more Augmentation paths ? DONE
2
16
14
10
10
3
4
0
S
6
10
K
0
0
P
16
Ford-Fulkerson Method Proof
Property 1 We can add augmentation flows
Network G, flow f1 ? residual network Gf1
Network Gf1, flow f2 ? residual network Gf1, f2
Network G, flow (f1 f2) ? residual
network Gf1, f2
gt We can solve the problem in stages!
17
Ford-Fulkerson Method Proof..
Property 2 Every source-containing bag has same
net outflow
Network G, flow f, amount f Each
source-containing bag, net outflow f
Example Compare net flow out of blue bag and red
bag
Why ?
18
Ford-Fulkerson Method Proof...
Definition Outflow capacity of a bag total
capacity of outflows
Examples Outflow capacity of red bag 8610
24 Outflow capacity of blue bag 1210 22
19
Ford-Fulkerson Method Proof.
• Suppose, at termination, total flow in network
f
• Using f, we have no augmentation path from
source ? sink

OUT-OF-BAG Set of nodes with no
augmentation path from source
IN-BAG Set of nodes with augmentation path from
source
Residual network, Gf
gt Existence of f gt f impossible!
20
Concluding remarks
• How to find augmenting paths ?
• -- Need to search all possibilities on the
network
• Classical terminology The Max-flow Min-cut
theorem

(c) Applications (i) Transportation Logistics
(ships, airlines, trains) (ii) Design of supply
networks (water, sewage, chemical plant,