The document discusses network flow problems in operations research and linear programming. It provides examples of the shortest route problem and minimal spanning tree problem. The shortest route problem aims to find the shortest paths from an origin to all destinations in a network. The minimal spanning tree problem seeks to connect all nodes in a network using the minimum total length of branches. Sample problems and step-by-step solutions are presented to illustrate how these network problems can be solved using algorithms that build solutions incrementally by adding nodes to a permanent set.

1. Operations
Research I
Lecture (9)
Network Flow
Linear programming
Fall 2022
IENG 202

2. Overview
A network is an arrangement of paths connected at various points
through which one or more items move from one point to another.
The network is drawn as a diagram providing a picture of the
system thus enabling visual interpretation and enhanced
understanding.
A large number of real-life systems can be modeled as networks
which are relatively easy to construct.
2

3. Network diagrams consist of nodes and branches.
Nodes (circles), represent junction points, or locations.
Branches (lines), connect nodes and represent flow.
Network Components
distance, time, cost, etc.
Origin
3

4.  Shortest-Route Algorithm
 Used for determining the shortest time, distance, or
cost from an origin to a destination through a
network.
 Minimum Spanning Tree Algorithm
 Used in determining the minimum distance (cost,
time) needed to connect a set of locations into a
single system.
 Maximal Flow Algorithm
 Used for determining the greatest amount of flow
that can be transmitted through a system in which
various branches, or connections, have specified
flow capacity limitations.
Network Models
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5. Problem: Determine the shortest routes from the origin to all destinations.
Shipping Routes from Los Angeles
The Shortest Route Problem
Definition and Example Problem Data (1 of 2)
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6. Network of Shipping Routes
The Shortest Route Problem
Definition and Example Problem Data (2 of 2)
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7. Network with Node 1 in the Permanent Set
The Shortest Route Problem
Solution Approach
Determine the initial shortest route from the origin (node 1) to the closest node (3).
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8. Network with Nodes 1 and 3 in the Permanent Set
The Shortest Route Problem
Solution Approach
Determine all nodes directly connected to the permanent set.
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9. Network with Nodes 1, 2, and 3 in the Permanent Set
Redefine the permanent set.
The Shortest Route Problem
Solution Approach
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10. Network with Nodes 1, 2, 3, and 4 in the Permanent Set
The Shortest Route Problem
Solution Approach
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11. The Shortest Route Problem
Solution Approach
Network with Nodes 1, 2, 3, 4, and 6 in the Permanent Set
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12. The Shortest Route Problem
Solution Approach
Network with Nodes 1, 2, 3, 4, 5, and 6 in the Permanent Set
Continue
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13. The Shortest Route Problem
Solution Approach
Network with Optimal Routes from Los Angeles to All Destinations
Optimal Solution
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14. The Shortest Route Problem
Solution Method Summary
Select the node with the shortest direct route from the origin.
Establish a permanent set with the origin node and the node that was
selected in step 1.
Determine all nodes directly connected to the permanent set nodes.
Select the node with the shortest route (branch) from the group of nodes
directly connected to the permanent set nodes.
Repeat steps 3 and 4 until all nodes have joined the permanent set.
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15. The Shortest Route Problem
Example Problem
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16. The Shortest Route Problem
Example Problem
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Permanent Set Path Distance
{ O } O – A 2
O – B 5
O - C 4

17. The Shortest Route Problem
Example Problem
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Permanent Set Path Distance
{ O, A } O – B 5
O – C 4
A – B 4
A - D 9

18. The Shortest Route Problem
Example Problem
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Permanent Set Path Distance
{ O, A, C } O – B 5
A – B 4
A – D 9
C – B 5
C - E 8

19. The Shortest Route Problem
Example Problem
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Permanent Set Path Distance
{ O, A, B, C } A – D 9
C – E 8
B – D 8
B - E 7

20. The Shortest Route Problem
Example Problem
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Permanent Set Path Distance
{ O, A, B, C, E } A – D 9
B – D 8
E – D 8
E – T 14

21. The Shortest Route Problem
Example Problem
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Permanent Set Path Distance
{ O, A, B, C, E } D – T 13
E - T 14

22. The Shortest Route Problem
Example Problem
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Optimal Path Distance
O – A 2
O – C 4
O – A – B 4
O – A – B – E 7
O – A – B – D 8
O – A – B – D – T 13

23. The Shortest Route Problem
Example Problem
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24.  A tree is a set of connected arcs that does not form a
cycle.
 A spanning tree is a tree that connects all nodes of a
network.
 The minimal spanning tree problem seeks to determine
the minimum sum of arc lengths necessary to connect
all nodes in a network.
 The criterion to be minimized in the minimal spanning
tree problem is not limited to distance. Other criteria
include time and cost.
The Minimal Spanning Tree Problem
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25. Required: Connect all nodes at minimum cost.
1
3
2
The Minimal Spanning Tree Problem
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26. Required: Connect all nodes at minimum cost.
Cost is sum of edge weights
1
2
3
2
1
3
The Minimal Spanning Tree Problem
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27. Required: Connect all nodes at minimum cost.
Cost is sum of edge weights
1
2
3
2
1
3
S.T. = 3 S.T. = 5 S.T. = 4
The Minimal Spanning Tree Problem
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28. Required: Connect all nodes at minimum cost.
Cost is sum of edge weights
1
3
2
1
3
2
1
3
2
M.S.T. = 3 S.T. = 5 S.T. = 4
The Minimal Spanning Tree Problem
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29. Network of Possible Cable TV Paths
The Minimal Spanning Tree Problem
Example Problem
Problem: Connect all nodes in a network so that the total branch lengths are
minimized.
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30. The Minimal Spanning Tree Problem
Solution Approach (1 of 6)
Spanning Tree with Nodes 1 and 3
Start with any node in the network and select the closest node to join the spanning
tree.
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31. The Minimal Spanning Tree Problem
Solution Approach (2 of 6)
Spanning Tree with Nodes 1, 3, and 4
Select the closest node not presently in the spanning area (not to create a cycle).
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32. The Minimal Spanning Tree Problem
Solution Approach (3 of 6)
Spanning Tree with Nodes 1, 2, 3, and 4
Continue
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33. The Minimal Spanning Tree Problem
Solution Approach (4 of 6)
Spanning Tree with Nodes 1, 2, 3, 4, and 5
Continue
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34. The Minimal Spanning Tree Problem
Solution Approach (5 of 6)
Spanning Tree with Nodes 1, 2, 3, 4, 5, and 7
Continue
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35. The Minimal Spanning Tree Problem
Solution Approach (6 of 6)
Minimal Spanning Tree for Cable TV Network
Optimal Solution
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36. The Minimal Spanning Tree Problem
Solution Method Summary
Select any starting node (conventionally, node 1).
Select the node closest to the starting node to join the spanning tree (not to
create a cycle).
Select the closest node not presently in the spanning tree.
Repeat step 3 until all nodes have joined the spanning tree.
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The Minimal Spanning Tree Problem-2
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Start
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The Minimal Spanning Tree Problem-2
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The Minimal Spanning Tree Problem-2
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The Minimal Spanning Tree Problem-2
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The Minimal Spanning Tree Problem-2
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The Minimal Spanning Tree Problem-2
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The Minimal Spanning Tree Problem-2
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NO!
The Minimal Spanning Tree Problem-2
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LOOP
The Minimal Spanning Tree Problem-2
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The Minimal Spanning Tree Problem-2
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51. Example Problem Statement and Data
1. Determine the shortest route from Atlanta (node 1) to each of the
other five nodes (branches show travel time between nodes).
2. Assume branches show distance (instead of travel time) between
nodes, develop a minimal spanning tree.
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52. Step 1 (part A): Determine the Shortest Route Solution
1. Permanent Set Branch Time
{1} 1-2 [5]
1-3 5
1-4 7
2. {1,2} 1-3 [5]
1-4 7
2-5 11
3. {1,2,3} 1-4 [7]
2-5 11
3-4 7
4. {1,2,3,4} 4-5 10
4-6 [9]
5. {1,2,3,4,6} 4-5 [10]
6-5 13
6. {1,2,3,4,5,6}
Example Problem, Shortest Route Solution (1 of 2)
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53. Example Problem, Shortest Route Solution (2 of 2)
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54. Example Problem, Minimal Spanning Tree (1 of 2)
1. The closest unconnected node to node 1 is node 2.
2. The closest to 1 and 2 is node 3.
3. The closest to 1, 2, and 3 is node 4.
4. The closest to 1, 2, 3, and 4 is node 6.
5. The closest to 1, 2, 3, 4 and 6 is 5.
6. The shortest total distance is 17 miles.
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55. Example Problem, Minimal Spanning Tree (2 of 2)
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56. Thanks for
Attention