network models.pdf

8/10/2019 network models.pdf

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To accompanyQuantitative Analysis for Management, Tenth Edition,

by Render, Stair, and HannaPower Point slides created by Jeff HeylEdited by CDCJaurigue

Network Models

2009 Prentice-Hall, Inc.


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Learning Objectives

1. Connect all points of a network whileminimizing total distance using the minimal-spanning tree technique

2. Determine the maximum flow through anetwork using the maximal-flow technique

3. Find the shortest path through a networkusing the shortest-route technique

4. Understand the important role of software insolving network problems

After completing this chapter, students will be able to:After completing this chapter, students will be able to:


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Chapter Outline

1.11.1 Introduction

1.21.2 Minimal-Spanning Tree Technique1.31.3 Maximal-Flow Technique

1.41.4 Shortest-Route Technique


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This chapter covers three network models

that can be used to solve a variety ofproblems

The minimalminimal--spanning tree techniquespanning tree technique

determines a path through a network thatconnects all the points while minimizing thetotal distance

The maximalmaximal--flow techniqueflow techniquefinds themaximum flow of any quantity or substancethrough a network

The shortestshortest--route techniqueroute techniquecan find the

shortest path through a network

Introduction


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Large scale problems may require hundreds

or thousands of iterations making efficientcomputer programs a necessity

All types of networks use a common

terminology The points on a network are called nodesnodesand

may be represented as circles or squares

The lines connecting the nodes are calledarcsarcs

Introduction


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Minimal-Spanning Tree

Terminologies

network consists of a set of nodes linked by arcs

path a sequence of distinct arcs that join twonodes through other nodes

cycle a form of a path where a node is connectedto itself through other nodes

tree a connected network with no cycles allowed

spanning tree a tree that links all the nodes of thenetwork without forming a cycle


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Minimal-Spanning Tree

Objective:

Select a set of arcs in a network that willspan (or connect) ALL the nodes (or points) ofthe network while minimizing total distance/cost.

Remarks:

There may be more than one optimal solution.Total # of connecting segments = # of nodes 1


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Minimal-Spanning Tree Technique Network for Lauderdale Construction

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GulfFigure 1.1


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Minimal-Spanning Tree Technique

Steps for the minimal-spanning tree

technique1. Select any node in the network

2. Connect this node to the nearest node thatminimizes the total distance

3. Considering all the nodes that are nowconnected, find and connect the nearest nodethat is not connected. If there is a tie, select

one arbitrarily. A tie suggests there may bemore than one optimal solution.

4. Repeat the third step until all nodes areconnected


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Start by arbitrarily selecting node 1

The nearest node is node 3 at a distance of 2 (200feet) and we connect those nodes

Considering nodes 1 and 3, we look for the nextnearest node

This is node 4, the closest to node 3 We connect those nodes

We now look for the nearest unconnected node to

nodes 1, 3, and 4 This is either node 2 or node 6

We pick node 2 and connect it to node 3


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Following this same process we connect from

node 2 to node 5 We then connect node 3 to node 6

Node 6 will connect to node 8

The last connection to be made is node 8 to node7

The total distance is found by adding up thedistances in the arcs used in the spanning tree

2 + 2 + 3 + 3 + 3 + 1 + 2 = 16 (or 1,600 feet)


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Minimal-Spanning Tree Technique All iterations for Lauderdale Construction

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Gulf


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Using QM for Windows


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Using QM for Windows


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Exercise

Find the minimum spanning tree for

each of the following networks. Alsofind the minimum total distance.

1.


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Exercise No.2

3.


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The Premiere Bank soon will be hooking up computerterminals at each of its branch offices to thecomputer at its main office using special phonelines with telecommunications devices. Thephone line from a branch office need not beconnected directly to the main office. It can be

connected indirectly by being connected toanother branch office that is connected (directlyor indirectly) to the main office. The onlyrequirement is that every branch office beconnected by some route to the main office. The

charge for the special phone lines is $100 timesthe number of miles involved, where the distance(in miles) between every pair of offices is asfollows:

Assignment


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Assignment No. 1

Management wishes to determine which pairs of offices shouldbe directly connected by special phone lines in order to connect

every branch office (directly or indirectly) to the main office at aminimum total cost.(a) Describe how this problem fits the network description of the

minimum spanning tree problem.(b) How should the connection be done?


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Maximal-Flow

Terminologies

source point of origin sink destination

Objective:

Determine the maximum amount of

material that can flow through anetwork


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Maximal-Flow

Applications:

Finding the maximum number ofautomobiles that can flow through ahighway system

Determining the maximum amount ofchemicals as they flow through a chemicalprocessing plant

Finding the maximum capacity of an oilpipeline network that transport oil fromexploration fields to the refinery and otherlocations


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Maximal-Flow Technique Road network for Waukesha

Capacity in Hundredsof Cars per Hour

WestPoint

East

Point

Figure 12.6

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Maximal-Flow Technique

Four steps of the Maximal-Flow Technique

1. Pick any path from the start (sourcesource) tothe finish (sinksink) with some flow. If no pathwith flow exists, then the optimal solutionhas been found.

2. Find the arc on this path with the smallestflow capacity available. Call this capacity

C. This represents the maximumadditional capacity that can be allocatedto this route.


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Maximal-Flow Technique

Four steps of the Maximal-Flow Technique

3. For each node on this path, decrease theflow capacity in the direction of flow by the

amount C. For each node on the path,increase the flow capacity in the reversedirection by the amount C.

4. Repeat these steps until an increase inflow is no longer possible


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Maximal-Flow Technique Capacity adjustment for path 126 iteration 1

Figure 12.7

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Old Path

New Path

Add 2

Subtract 2


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Maximal-Flow Technique We repeat this process by picking the path 12

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The maximum capacity along this path is 1

The path capacity is adjusted by adding 1 to thewestbound flows and subtracting 1 from the

eastbound flows The result is the new path in Figure 12.8

We repeat this process by picking the path 1356

The maximum capacity along this path is 2 Figure 12.9 shows this adjusted path


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Maximal-Flow Technique Second iteration for Waukesha road system

Figure 12.8

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Old Path

New Network

Add 1

Subtract 1


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Maximal-Flow Technique Third and final iteration for Waukesha road

system

Figure 12.9

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2 0

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ExercisesFor networks (a)

and (b), find theflow patterngiving themaximum flowfrom the sourceto the sink,

given that thearc capacityfrom node i tonodej is thenumber nearest

node i along thearc betweenthese nodes.


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Shortest-Route Technique The shortestshortest--route techniqueroute techniquefinds how a person

or item can travel from one location to anotherwhile minimizing the total distance traveled

It finds the shortest route to a series ofdestinations

Illustration 1: Ray Design, Inc. transports beds, chairs, and

other furniture from the factory to the warehouse They would like to find the route with the shortest

distance The road network is shown in Figure 12.10


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Shortest-Route Technique Roads from Rays plant to warehouse

Plant

Warehouse

50

40

200

1501

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Figure 12.10


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Shortest-Route Technique

Steps of the shortest-route technique

1. Find the nearest node to the origin (plant). Putthe distance in a box by the node.

2. Find the next-nearest node to the origin andput the distance in a box by the node. Several

paths may have to be checked to find thenearest node.

3. Repeat this process until you have gonethrough the entire network. The last distance

at the ending node will be the distance of theshortest route.


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Shortest-Route Technique First iteration for Ray Design

Plant

Warehouse

50

40

200

1501

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Figure 12.11

100


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Shortest-Route Technique Second iteration for Ray Design

Figure 12.12

Plant

Warehouse

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40

200

1501

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100

150


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Shortest-Route Technique Third iteration for Ray Design

Figure 12.13

Plant

Warehouse

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40

200

1501

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100

150 190


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Illustration 2:

Mr. Santos plans to drive from his homein Alabang to one of the resorts inLaguna for a business meeting. Several

paths are available. The number oneach arc is the distance (in kms)between two locations (or nodes). What

path should Mr. Santos take to minimizethe number of kilometers traveled?


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