MNG221 - Management Science

MNG221 - Management Science

Network Flow Models

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 Problem2. Minimal Spanning Tree Problem3. Maximal Flow Problem

The Shortest Route ProblemNetwork Flow Models

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.

The Shortest Route ProblemA Worked Example

Network Flow Models

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

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

Des Moines (node 5) 1-3-4-5 38

Dallas (node 6) 1-3-6 31

St. Louis (node 7) 1-3-4-7 43

Shortest travel time from origin to each destination

MNG221 - Management Science

Network Flow Models

Network Flow ModelsReview….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 ModelsReview….• Networks consists of two main

components: nodes and branches.• There are three (3) methods for solving

Network Flow Problems:1. The Shortest Route Problem2. Minimal Spanning Tree Problem3. Maximal Flow Problem

Network Flow ModelsReview….• The shortest route problem is to find

the shortest distance between an origin and various destination points.

The Minimal Spanning Tree ProblemNetwork Flow Models

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.

The Minimal Spanning Tree ProblemA Worked Example

Network Flow Models

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

The Maximal Flow ProblemNetwork Flow Models

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 ProblemThe 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 ProblemThe 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.

The Maximal Flow ProblemA Worked Example

Network Analysis

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 ProblemIn Maximal Flow Problems there are two types of branches1. 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 ProblemIn Maximal Flow Problems there are two types of branches1. 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 1-3-

4-6 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.