Chapter 12
Network Models
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-1
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Learning Objectives
Students will be able to
• Connect all points of a network
while minimizing total distance
using the minimal-spanning tree
technique.
• Determine the maximum flow
through a network using the
maximal-flow technique.
• Find the shortest path through a
network using the shortest-route
technique.
• Understand the important role of
software in solving network
problems.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-2
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Chapter Outline
12.1 Introduction
12.2 Minimal-Spanning Tree
Technique
12.3 Maximal-Flow Technique
12.4 Shortest-Route technique
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-3
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Minimal-Spanning
Tree Technique
• Determines the path through the
network that connects all the
points while minimizing total
distance.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-4
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Minimal-Spanning Tree
Steps
1. Select any node in the network.
2. Connect this node to the nearest
node that minimizes the total
distance.
3. Considering all of the nodes that are
now connected, find and connect the
nearest node that is not connected. If
there is a tie for the nearest node,
select one arbitrarily. A tie suggests
that there may be more than one
optimal solution.
4. Repeat the third step until all nodes
are connected.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-5
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Minimal-Spanning Tree
Lauderdale Construction
3
2
3
1
5
3
2
5
2
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
8
3
2
4
7
7
3
5
4
6
1
6
12-6
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Minimal-Spanning Tree
Iterations 1&2
3
2
3
3
5
2
1
4
5
7
7
2
3
8
3
5
2
4
First Iteration
1
6
3
6
2
3
5
3
Second Iteration
1
5
2
7
7
2
3
2
4
12-7
8
3
5
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4
6
1
6
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Minimal-Spanning Tree
Iterations 3&4
3
2
3
5
3
5
7
2
1
4
7
3
8
3
5
2
4
2
Third Iteration
1
6
6
2
3
1
3
5
3
2
5
7
7
Fourth Iteration
2
3
8
3
5
2
4
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4
12-8
6
1
6
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Minimal-Spanning Tree
Iterations 5 & 6
3
2
3
1
5
5
3
2
2
3
2
5
3
1
4
7
2
7
2
3
2
Sixth iteration
8
3
4
1
6
5
5
6
4
3
8
3
5
3
7
7
Fifth Iteration
2
4
6
1
6
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-9
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
7th Iteration
3
Seventh &
final
iteration
2
3
5
3
1
5
2
7
7
2
3
2
Minimum Distance: 16
4
12-10
8
3
5
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
4
6
1
6
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
The Maximal-Flow
Technique
1. Pick any path from start (source) to
finish (sink) with some flow. If no path
with flow exists, then the optimal
solution has been found.
2. Find the arc on this path with the
smallest capacity available. Call this
capacity C. This represents the
maximum additional capacity that can
be allocated to this route.
3.For each node on this path decrease the
flow capacity in the direction of flow by
the amount C. For each node on this
path, increase the flow capacity in the
reverse direction by C
4. Repeat these steps until an increase in
flow is no longer possible.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-11
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Maximal-Flow Road
Network for Waukesha
2
12
1
2
6
1
3
West 1 2
10
Point
011
0
East
Point
4
1
6
0
5
1
3
3 2
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-12
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Capacity Adjustment
Iteration 1
Add 2
1
2
2
2
Subtract 2
6
3
West
Point 1
3
East
Point
0
2
4
6
1
West
1
Point
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
New path
12-13
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
East
Point
Maximal-Flow Road
Network for Waukesha
Add 2
1
2
2
1
2
Subtract 2
3
West
Point
1
6
1
0
2
0 1 1
10
4
1
6
0
5
3
1
32
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
Path = 1, 2, 6
12-14
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
East
Point
Second Iteration
Add 1
0
3
2
1
4
1
West
Point
1
6
1
0
2
East
Point
0 1 1
10
1
4
Subtract 1
6
0
5
3
1
32
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-15
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Second Iteration
New Path
4
0
2
0
4
0
West
Point
1
6
2
0
2
East
Point
0 2 0
10
1
4
6
0
5
3
1
32
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
Path = 1, 2, 4, 6
12-16
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Third Iteration
0
4
2
0
4
0
West
Point
1
6
2
0
2
0 2 0
10
4
East
Point
1
Subtract 2
6
0
5
3
32
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
1
Add 2
12-17
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Third Iteration
New Path
4
0
2
0
4
0
West
Point
1
6
2
2
2
0 2 0
10
4
East
Point
1
4
0
5
3
30
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
3
Path = 1, 3, 5, 6
12-18
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Road Network for
Waukesha
Third Iteration
4
0
2
0
4
2
0
West
Point 1
6
East
Point
2
2
0 2 0
8
1
4
4
2
Flow
5 Path (Cars Per
3
Hour)
1-2-6
200
1-2-4-6 100
1-3-5-6 200
Total
500
3
30
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-19
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
The Shortest-Route
Technique
1. Find the nearest node to the origin. Put
the distance in a box by the node.
2.Find the next nearest node to the origin,
and put the distance in a box by the node.
In some cases, several paths will have to
be checked to find the nearest node.
3. Repeat this process until you have gone
through the entire network. The last
distance at the ending node will be the
distance of the shortest route. You should
note that the distances placed in the boxes
by each node are the shortest route to this
node. These distances are used as
intermediate results in finding the next
nearest node.
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-20
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Shortest-Route Problem
Ray Design, Inc.
Roads from Ray’s Plant to
the Warehouse
2
200
4
100
100
Plant
50
1
100
150
6 Warehouse
100
200
3
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
40
12-21
5
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Ray Design, Inc.
First Iteration
100
2
4
200
100
100
Plant
50
1
100
150
200
6 Warehouse
100
3
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
40
12-22
5
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Ray Design, Inc.
Second Iteration
100
2
4
200
100
100
Plant
50
1
100
150
200
6 Warehouse
100
3
40
5
150
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-23
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Ray Design, Inc.
Third Iteration
100
2
4
200
100
100
Plant
50
1
100
150
200
6 Warehouse
100
40
3
5
150
190
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-24
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458
Ray Design, Inc.
Fourth Iteration
100
2
4
200
100
100
290
Plant
50
1
100
150
200
6 Warehouse
100
40
3
5
150
190
To accompany Quantitative Analysis
for Management, 8e
by Render/Stair/Hanna
12-25
© 2003 by Prentice Hall, Inc.
Upper Saddle River, NJ 07458