Network Flow Models
MNG221 - Management Science
Network Flow Models
• A Network is an arrangement of paths
connected at various points, through
which one or more items move from
one point to another.
–For example, highway systems,
telephone networks, railroad systems,
and television networks.
Network Flow Models
• Networks are popular because they
provide a picture of a system under
analysis and because a large number of
systems can be easily modeled as
networks.
• Network Flow Models a model that
represents the flow of items through a
system.
Network Flow Models
• Networks consists of two main
components: nodes and branches.
1. Nodes, denoted by circles, represents
junction points connecting branches.
E.g. cities,
2. Branches, represented as lines,
connect nodes and show flow from
one point to another. E.g. roads
connecting cities.
Network Flow Models
• The First node on the Network is
referred to as the origin, and any of the
other remaining nodes could be the
destination.
• The values assigned to branches
typically represent distance, time, or
cost.
Network Flow Models
• Thus, the purpose of the network is to
determine the shortest distance,
shortest length of time, or lowest cost
between points in the network.
Network Flow Models
• A Network Flow of railroad routes
Network Flow Models
• There are three (3) methods for solving
Network Flow Problems:
1. The Shortest Route Problem
2. Minimal Spanning Tree Problem
3. Maximal Flow Problem
Network Flow Models
The Shortest Route Problem
The Shortest Route Problem
• The shortest route problem is to find
the shortest distance between an origin
and various destination points.
• The steps of the shortest route solution
method are as follows:
1. Select the node with the shortest
direct route from the origin.
The Shortest Route Problem
• The steps of the shortest route solution
method are as follows: (Continued)
2. Establish a permanent set with the
origin node and the node that was
selected in step 1.
3. Determine all nodes directly
connected to the permanent set
nodes.
The Shortest Route Problem
• The steps of the shortest route solution method are as
follows: (Continued)
4. Select the node with the shortest
route (branch) from the group of
nodes directly connected to the
permanent set nodes.
5. Repeat steps 3 and 4 until all nodes
have joined the permanent set.
Network Flow Models
The Shortest Route Problem
A Worked Example
The Shortest Route Problem
• For example, the Stagecoach Shipping Company
transports oranges by six trucks from Los Angeles
to six cities in the West and Midwest.
• The shipping company manager wants to
determine the best routes (in terms of the
minimum travel time) for the trucks to take to
reach their destinations.
• We want to determine the shortest routes from
the origin (node 1) to the six destinations (nodes 2
through 7).
The Shortest Route Problem
Shipping routes from Los Angeles
The Shortest Route Problem
Network of shipping routes
The Shortest Route Problem
• Determine the initial shortest route from the origin (node
1) to the closest node directly connected (i.e., adjacent)
node(3).
o The three nodes directly connected to node 1 are 2, 3, and 4.
o 1 and 3 are the permanent set shortest route to these nodes.
Network with node 1 in the permanent set
The Shortest Route Problem
• Determine the shortest route to the three nodes (2, 4,
and 6) directly connected to the permanent set nodes.
 Two branches starting from node 1 (1-2 and 1-4) and two branches
from node 3 (3-4 and 3-6).
 The time to node 6 (branch 3-6) is 31 hours = branch 3-6 (22 hours) +
shortest route to node 3 (9 hours) – Cumulative Time.
Network with nodes 1 and 3 in the permanent set
The Shortest Route Problem
• Redefine the permanent set.
o Five branches lead from the permanent set nodes (1, 2, and 3)
to their directly connected nodes
Network with nodes 1, 2, and 3 in the permanent set
The Shortest Route Problem
• Next, we repeat the process of determining the nodes
directly connected to the permanent set nodes (1, 2, 3, & 4).
– These directly connected nodes are 5, 6, and 7,
– Eliminate the branches from nodes (1 and 2) that is not the shortest
route to permanent set nodes (node 4).
Network with nodes 1, 2, 3,
and 4 in the permanent set
The Shortest Route Problem
• Repeat the process to find the nodes directly connected
(adjacent) to our permanent set (1, 2, 3, 4, and 6) that is
nodes 5 and 7.
Eliminate branches that are not the best route (4-6).
The branch 45 has the shortest cumulative time, of 38 hours.
Network with nodes 1,
2, 3, 4, and 6 in the
permanent set
The Shortest Route Problem
• The only remaining node directly connected to the
permanent set is node 7.
• Of the three branches connecting node 7 to the
permanent set, branch 4-7 has the shortest time, of 43
hours.
Network with nodes
1, 2, 3, 4, 5, and 6 in
the permanent set
The Shortest Route Problem
• The routes with the shortest times from the origin (node
1) to each of the other six nodes and their corresponding
travel times are
Network
with
optimal routes from
Los Angeles to all
destinations
The Shortest Route Problem
Shortest travel time from origin to each destination
From Los Angeles to:
Route
Total Hours
Salt Lake City (node 2)
1-2
16
Phoenix (node 3)
1-3
9
Denver (node 4)
1-3-4
24
1-3-4-5
38
1-3-6
31
1-3-4-7
43
Des Moines (node 5)
Dallas (node 6)
St. Louis (node 7)
Network Flow Models
MNG221 - Management Science
Network Flow Models
Review….
A Network is an arrangement of paths
connected at various points, through
which one or more items move from one
point to another.
Network Flow Models a model that
represents the flow of items through a
system.
Network Flow Models
Review….
• Networks consists of two main
components: nodes and branches.
• There are three (3) methods for solving
Network Flow Problems:
1. The Shortest Route Problem
2. Minimal Spanning Tree Problem
3. Maximal Flow Problem
Network Flow Models
Review….
• The shortest route problem is to find
the shortest distance between an origin
and various destination points.
Network Flow Models
The Minimal Spanning Tree Problem
Minimal Spanning Tree Problem
• The minimal spanning tree problem is
similar to the shortest route problem,
except that the objective is to connect
all the nodes in the network so that the
total branch lengths are minimized.
• The resulting network spans (connects)
all the points in the network at a
minimum total distance (or length).
Minimal Spanning Tree Problem
• The steps of the minimal spanning tree
solution method are as follows:
1. Select
any
starting
node
(conventionally, node 1 is selected).
2. Select the node closest to the starting
node to join the spanning tree.
Minimal Spanning Tree Problem
• The steps of the minimal spanning tree
solution method are as follows: (Continued)
3. Select the closest node not presently
in the spanning tree.
4. Repeat step 3 until all nodes have
joined the spanning tree.
Network Flow Models
The Minimal Spanning Tree Problem
A Worked Example
Minimal Spanning Tree Problem
• EXAMPLE
• The Metro Cable Television Company is to install
a television cable system in a community
consisting of seven suburbs.
• Each of the suburbs must be connected to the
main cable system.
• The cable television company wants to lay out
the main cable network in a way that will
minimize the total length of cable that must be
installed.
Minimal Spanning Tree Problem
• Network of possible cable TV paths
Minimal Spanning Tree Problem
• Start with any node in the network and select the closest
node to join the spanning tree.
• Beginning at node 1, we select the closest node (node 3).
• This branch is indicated with a heavy line.
Spanning tree with
nodes 1 and 3
Minimal Spanning Tree Problem
• The next step is to select the closest node (node
4 -15,000 feet) to any node in the spanning area
(1 or node 3).
Spanning tree with
nodes 1, 3, and 4
Minimal Spanning Tree Problem
• Next, we repeat the process of selecting the closest node
to our present spanning tree (nodes 1, 3, and 4).
• The closest node not now connected to the nodes in our
spanning tree is from node 4 to node 2 is 12,000 feet.
Spanning tree with
nodes 1, 2, 3, and 4
Minimal Spanning Tree Problem
• Our spanning tree now consists of nodes 1, 2, 3, and 4.
• The node closest to this spanning tree is node 5, with a
branch length of 14,000 feet to node 4.
Spanning tree with
nodes 1, 2, 3, 4, and 5
Minimal Spanning Tree Problem
• The spanning tree now contains nodes 1, 2, 3, 4, and 5.
• The closest node not currently connected to the spanning
tree is node 7, with a length of 8,000 feet to node 5.
Spanning tree with
nodes 1, 2, 3, 4, 5, and 7
Minimal Spanning Tree Problem
• The only remaining node not connected to the spanning
tree is node 6.
• The node in the spanning tree closest to node 6 is node 7,
with a branch length of 14,000 feet.
Minimal spanning tree
for cable TV network
Minimal Spanning Tree Problem
• This completes spanning tree, which includes all 7 nodes
and requires the minimum amount of television cable to
connect the seven suburbs72,000 feet.
Route
Total Feet
1,3
9
1,3 &4
24
1,3,4&2,
36
1,3,4,2&5
50
1,3,4,2,5 &7
58
1,3,4,2,5, 7&6
72
Network Flow Models
The Maximal Flow Problem
The Maximal Flow Problem
• The maximal flow problem is to
maximize the amount of flow of items
from an origin to a destination.
• Maximal flow problems can involve the
flow of:
1. Water, gas, or oil through a network of
pipelines;
The Maximal Flow Problem
• Maximal flow problems can involve the
flow of: (Continued)
2. The flow of forms through a paper
processing system (such as a government
agency);
3. The flow of traffic through a road network;
4. The flow of products through a production
line system.
The Maximal Flow Problem
• In each of these examples, the branches
of the network have limited and often
different flow capacities.
• Given these conditions, the decision
maker wants to determine the
maximum flow that can be obtained
through the system.
The Maximal Flow Problem
The steps of the maximal flow solution
method are as follows:
1. Arbitrarily select any path in the
network from origin to destination.
2. Adjust the capacities at each node by
subtracting the maximal flow for the
path selected in step 1.
The Maximal Flow Problem
The steps of the maximal flow solution
method are as follows: (Continued)
3. Add the maximal flow along the
path in the opposite direction at
each node.
4. Repeat steps 1, 2, and 3 until there
are no more paths with available
flow capacity.
Network Analysis
The Maximal Flow Problem
A Worked Example
The Maximal Flow Problem
• Example
• The Scott Tractor Company ships tractor parts from
Omaha to St. Louis by railroad.
• However, a contract limits the number of railroad cars the
company can secure on each branch during a week.
• Given these limiting conditions, the company wants to
know the maximum number of railroad cars containing
tractor parts that can be shipped from Omaha to St. Louis
during a week.
• The number of railroad cars available to the tractor
company on each rail branch is indicated by the number
on the branch to the immediate right of each node (which
represents a rail junction).
The Maximal Flow Problem
• E.g. – Six cars are available from node 1 (Omaha) to
node 2, Eight cars from node 2 to node 5, five cars
from node 4 to node 6 (St. Louis).
The Maximal Flow Problem
• The number on each branch to the immediate left of
each node is the number of cars available for shipping
in the opposite direction.
The Maximal Flow Problem
In Maximal Flow Problems there are two
types of branches
1. Directed Branches – where flow is
possible in only one direction.
 For example -The branch from node 1 to
node 2 is referred to as a directed branch
because flow is possible in only one
direction (from node 1 to node 2, but not
from 2 to 1).
The Maximal Flow Problem
In Maximal Flow Problems there are two
types of branches
1. Undirected Branches – Where flow is
possible in both directions on the
branches
 For example - between nodes 2 and 4
and nodes 3 and 4.
The Maximal Flow Problem
• The 1st step in determining the maximum
possible flow is to choose any path arbitrarily
from origin to destination and ship as much as
possible on that path. We will arbitrarily select
the path 1-2-5-6.
The Maximal Flow Problem
• The maximum number of railroad cars that can
be sent through this route is four (4) cars,
because that is the maximum amount available
on the branch between nodes 5 and 6.
The Maximal Flow Problem
The Maximal Flow Problem
• Next Step is to recompute the branch flow in
both directions.
• Left to Right 1-2-5-6 – Subtract the Minimum
Possible value from each value on right of
Node
•6 - 4 = 2
•8 – 4 = 4
•4 – 4 = 0
The Maximal Flow Problem
• Next Step is to recompute the branch flow in
both directions. (Continued)
• Right to Left 5-6-2-1 - Add the Minimum
Possible value from each value on right of
Node
• 4 + 0= 4
• 4 + 0= 4
• 4 + 0= 4
The Maximal Flow Problem
• Next Select other feasible paths arbitrarily and
determine maximum flow along the paths
until flow is no longer possible.
The Maximal Flow Problem
• We will select path 1-4-6 and the maximum
flow along this path is four cars, which is
subtracted at each of the nodes
The Maximal Flow Problem
• This increases the total flow through the
network to eight cars, because the flow of
four along path 1-4-6 is added to the flow.
The Maximal Flow Problem
• Next Step is to recompute the branch
flow in both directions.
• Left to Right 1-4-6 – Subtract the
Mininum Possible value from each value
on right of Node
•4 – 4 = 0
•5 – 4 = 1
The Maximal Flow Problem
• Next Step is to recompute the branch
flow in both directions.
• Right to Left 6-4-1 - Add the Mininum
Possible value from each value on right
of Node
• 4 + 0= 4
• 4 + 0= 4
The Maximal Flow Problem
• Now we arbitrarily select another path.
• This time we will choose path 1-3-6, with a
maximum possible flow of six cars.
The Maximal Flow Problem
• Next Step is to recompute the branch
flow in both directions.
• Left to Right 1-3-6 – Subtract the
Mininum Possible value from each value
on right of Node
•7 – 6 = 1
•6 – 6 = 0
The Maximal Flow Problem
• Next Step is to recompute the branch
flow in both directions.
• Right to Left 6-3-1 - Add the Mininum
Possible value from each value on right
of Node
• 6 + 0= 6
• 6 + 0= 6
The Maximal Flow Problem
• The available flow capacity along the path 1-34-6 is one car.
• This increases the total flow from 14 cars to
15 cars.
The Maximal Flow Problem
• Next Step is to recompute the branch
flow in both directions.
• Left to Right 1-3-4-6 – Subtract the
Mininum Possible value from each value
on right of Node
•1 – 1 = 0
•2 – 1 = 1
•1 – 1 = 0
The Maximal Flow Problem
• Next Step is to recompute the branch
flow in both directions.
• Right to Left 6-4-3-1 - Add the Mininum
Possible value from each value on right
of Node
• 4 + 1= 5
•2 + 1 = 3
•6 + 1 = 7
The Maximal Flow Problem
• The maximum flow is 15 railroad cars, because there are no more
paths with available flow capacity.
• All paths out of nodes 3, 4, and 5 show zero available capacity,
which prohibits any further paths through the network.