Quantech Network Models

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2008 Prentice-Hall, Inc.

Ch ap ter 12

Quant i tat ive Analys is for Management , Tenth Edit io n , by Render, Stair, and Hanna

Power Point slides created by Jeff Heyl

Netw o rk Mod els


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2009 Prentice-Hall, Inc. 12 2

L earn ing Ob ject iv es

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 network

using the shortest-route technique4. Understand the important role of software insolving network problems

After completing this chapter, students will be able to:


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Chapter Outl in e

12.1 Introduction12.2 Minimal-Spanning Tree Technique12.3 Maximal-Flow Technique12.4 Shortest-Route Technique


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This chapter covers three network models thatcan be used to solve a variety of problemsThe m inim al-spann ing t ree tech niqu e determinesa path through a network that connects all thepoints while minimizing the total distanceThe m aximal -f low techniqu e finds the maximumflow of any quantity or substance through anetwork

The sh or tes t -route tech niqu e can find theshortest path through a network

In t roduc t ion


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Large scale problems may require hundreds orthousands of iterations making efficient computerprograms a necessityAll types of networks use a common terminologyThe points on a network are called nodes andmay be represented as circles of squaresThe lines connecting the nodes are called arcs

In t roduc t ion


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Minim al-Span n ing Tree Tech niq u e

The minimal-spanning tree technique involvesconnecting all the points of a network togetherwhile minimizing the distance between them


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Steps for the minimal-spanning treetechnique

1. Select any node in the network

2. Connect this node to the nearest node thatminimizes the total distance3. Considering all the nodes that are now

connected, find and connect the nearest nodethat is not connected. If there is a tie, selectone arbitrarily. A tie suggests there may bemore than one optimal solution.

4. Repeat the third step until all nodes areconnected



PROBLEM

Let us consider the Lauderdale ConstructionCompany, which is currently developing aluxurious housing project in Panama City

Beach, Florida. Melvin Lauderdale, ownerand president of Lauderdale Construction,must determine the least expensive way toprovide water and power to each house.




Network for Lauderdale Construction

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GulfFigure 12.1




Start by arbitrarily selecting node 1The nearest node is node 3 at a distance of 2 (200feet) and we connect those nodesConsidering nodes 1 and 3, we look for the nextnearest nodeThis is node 4, the closest to node 3We connect those nodesWe now look for the nearest unconnected node tonodes 1, 3, and 4This is either node 2 or node 6We pick node 2 and connect it to node 3




Following this same process we connect fromnode 2 to node 5We then connect node 3 to node 6Node 6 will connect to node 8The last connection to be made is node 8 to node7The 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)




All iterations for Lauderdale Construction

Figures 12.2 12.5

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Exercise

Roxie LaMothe, owner of a large horsebreeding farm near Orlando, is planning toinstall a complete water system

connecting all of the various stables andbarns. The location of the facilities and thedistances between them is given in thenetwork shown. Roxie must determine theleast expensive way to provide water toeach facility. What do you recommend?



Exercise


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Maxim al-Flow Tech n iqu e

The maximal-flow technique allows us todetermine the maximum amount of a material thatcan flow through a networkWaukesha Wisconsin is in the process ofdeveloping a road system for the downtown areaThey want to determine the maximum number ofcars that can flow through the town from west toeast

The road network is shown in Figure 12.7The numbers by the nodes indicate the number ofcars that can flow f rom the node


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Four steps of the Maximal-Flow Technique1. Pick any path from the start ( s o u r c e ) to the

finish ( s ink ) with some flow. If no path withflow exists, then the optimal solution hasbeen found.

2. Find the arc on this path with the smallestflow capacity available. Call this capacity C.This represents the maximum additional

capacity that can be allocated to this route.


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Four steps of the Maximal-Flow Technique3. For each node on this path, decrease the flow

capacity in the direction of flow by theamount 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 in flow isno longer possible


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Road network for Waukesha

Capacity in Hundredsof Cars per Hour

WestPoint

EastPoint

Figure 12.6

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We start by arbitrarily picking the path 1 2 6which is at the top of the networkThe maximum flow is 2 units from node 2 tonode 6The path capacity is adjusted by adding 2 to thewestbound flows and subtracting 2 from theeastbound flowsThe result is the new path in Figure 12.7 which

shows the new relative capacity of the path atthis stage


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Capacity adjustment for path 1 2 6 iteration 1

Figure 12.7

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Old Path

New Path

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We repeat this process by picking the path 1 2 4 6The maximum capacity along this path is 1The path capacity is adjusted by adding 1 to thewestbound flows and subtracting 1 from theeastbound flowsThe result is the new path in Figure 12.8We repeat this process by picking the path 1 3

5 6The maximum capacity along this path is 2Figure 12.9 shows this adjusted path


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Second iteration for Waukesha road system

Figure 12.8

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New Network

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Third and final iteration for Waukesha roadsystem

Figure 12.9

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There are no more paths from nodes 1 to 6 withunused capacity so this represents a finaliterationThe maximum flow through this network is 500cars

PATH FLOW (CARS PER HOUR)

1 2 6 200

1 2 4 6 100

1 3 5 6 200

Total 500


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Sho rtes t -Rou te Tech n iqu e

The sh or tes t -route tech niqu e finds how a personor item can travel from one location to anotherwhile minimizing the total distance traveledIt finds the shortest route to a series ofdestinationsRay Design, Inc. transports beds, chairs, andother furniture from the factory to the warehouseThey would like to find the route with the shortest

distanceThe road network is shown in Figure 12.10


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Roads from Rays plant to warehouse

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Warehouse

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Figure 12.10


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Steps of the shortest-route technique1. Find the nearest node to the origin (plant). Put

the 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. Severalpaths may have to be checked to find thenearest node.

3. Repeat this process until you have gone

through the entire network. The last distanceat the ending node will be the distance of theshortest route.


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We can see that the nearest node to the plant isnode 2We connect these two nodesAfter investigation, we find node 3 is the next

nearest node but there are two possible pathsThe shortest path is 1 2 3 with a distance of 150We repeat the process and find the next node isnode 5 by going through node 3

The next nearest node is either 4 or 6 and 6 turnsout to be closerThe shortest path is 1 2 3 5 6 with a distance of290 miles


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Second iteration for Ray Design

Figure 12.12

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Warehouse

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Third iteration for Ray Design

Figure 12.13

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Fourth and final iteration for Ray Design

Figure 12.14

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Warehouse

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