© 2008 pearson addison-wesley. all rights reserved chapter 13 mathematics and business

© 2008 Pearson Addison-Wesley.All rights reserved

Chapter 13

Mathematics and Business

Copyright © 2008 Pearson Education, Inc. Slide 13-2

Chapter 13Mathematics and Business

13A Network Analysis

13B The Traveling Salesman Problem

13C Scheduling Problems


Unit 13A

Network Analysis


Network Representation

13-A

Network A collection of points or objects that are interconnected in some way.

Vertex An object such as a computer, phone, city, island, etc. which makes up a

network.

Edge Represented by a line or curve to be a connection between two vertices.


Bridges of Konigsberg

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A map of the Pregel River flowing through Königsberg and a network representation of the bridges of Königsberg. The vertices represent the land masses (capital letters) and the edges represent the bridges (lowercase letters).

Network Analysis and the War in Iraq and Afghanistan

http://www.npr.org/2010/12/03/131755378/u-s-connects-the-dots-to-catch-roadside-bombers



An Office Intranet

A layout of computers, servers, and cables in a small office intranet.

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An Office Intranet

A network diagram overlaid on the office intranet.

A B E

DC

FG

H

I

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An Office Intranet

A network diagram representing the connections in the office intranet. Vertices represent computers (capital letters) and edges represent cables connecting computers (lowercase letters).

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Euler Circuits

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An Euler circuit is a path through a network that starts and ends at the same point and traverses every edge exactly once. An Euler circuit exists for a network if each vertex has an even number of edges. In the figure below, networks (a) and (b) have Euler circuits, but (c) and (d) do not.


Network Analysis

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Which network has an Euler circuit?

a) b)

c) d)


The Burning Bridges Rule for Finding Euler Circuits

You may begin your circuit from any vertex in the network. However, as you choose edges to follow, never use an edge that is the only connection to a part of the network that you have not already visited.

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Applying the Burning Bridges Rule

Find an Euler circuit for this network.

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Applying the Burning Bridges Rule

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Network Terminology

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Circuit A path within a network that begins and ends at the same vertex without using any edges more than once.

Complete network Every vertex is directly connected to every other vertex.

Tree A network in which all of the vertices are connected and no circuits appear.

Order The number of vertices in a network.

Degree of vertex The number of edges connected to the vertex.


Network Analysis

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What is the order of the network below?

a) 4

b) 5

c) 6

d) 8


Network Analysis

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Which network is a tree?

a) b)

c) d)


Minimum Cost Spanning Networks

A map of seven towns (capital letters) and the routes between them along which telephone lines could be strung, along with the network representation.

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Minimum Cost Spanning Networks

Two spanning networks. The total cost of each spanning network is the sum of the individual costs on its edges. The total cost for spanning network (a) is much higher than the total cost for spanning network (b).

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Kruskal’s Algorithm for Finding Minimum Cost Networks

Step 1: Make a list of the edges from the least expensive to the most expensive.

Step 2: Begin with the least expensive edge. Highlight it to indicate that it is part of the minimum cost spanning network. Continue to select edges in order of increasing cost until every vertex is connected, either directly or indirectly, to every other vertex.

Step 3: If a closed circuit has been created within the spanning network, remove the most expensive edge. The final result is the minimum cost spanning network.

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© 2008 pearson addison-wesley. all rights reserved chapter 13 mathematics and business

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