network models (1)

8/2/2019 Network Models (1)

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NETWORK MODELS

By:

Ankur Yadav - 06Kanika Sachdeva - 16Poorva Mishra - 26Shivam Awasthi - 36Shruti Sanklecha - 46Vinayak Naik - 56


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Introduction

Minimum spanning tree problem

Maximum flow problem

Shortest route problem

All node pairs shortest path

Practical Applications


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Minimal Spanning Tree

Definition:

The minimal spanning tree techniquedetermines the path through which networkthat connects all points while minimizingtotal distance

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Algorithm:

o Select any node in the network

o Connect this node to the nearest node

minimizing the total distanceo Select the node out of unconnected nodes

which can be connected with minimumdistance by adding one edge only

o If there is a tie, select arbitrarilyo A tie suggests more than one optimal solution

o Repeat till all nodes are connected


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Example:


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Example:

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Total distance=200+200+300+300+300+100+200=1500 units


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Maximal Flow Problem

Maximal Flow Technique

Linear Programming


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The maximal-flow technique allows themaximum amount of a material that can flowthrough a network to be determined.

For example:

It has been used to find the maximumnumber of automobiles that can flow through a

state highway system.


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Algorithm Select any path Find the arc on this path with the smallest flow

capacity (C)available 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 theflow capacity in the reverse direction by

the amount C. Repeat these steps until an increase in flow is no

longer possible.


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Road Network


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Capacity Adjustment

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3EastPointWest

Point

Add 2

Subtract 2

Iteration 1

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1EastPointWest

Point

New path


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New Arrangement

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EastPoint

WestPoint

New path


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Final Iteration

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EastPoint

WestPoint


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Linear Programming

Variable Xij= flow from node i to jMaximize flow = X61

X12


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Shortest PathAlgorithm

Shortest distance from one location to

another.

Used to minimize total distance from anystarting node to a final node


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Steps of the shortest routetechnique

Find the nearest node to theorigin(plant).Put the distance in a box bythe node.

Find the next nearest node to theorigin(plant) and put the distance in a boxby the node.Repeat this process till the entire network is

scanned.The last distance at the ending node will bethe distance of the shortest route.


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Warehouse

Plant


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ALL-NODE-PAIRS SHORTEST PATH

Floyd-Warshall algorithm is useful for finding theshortest path between all pairs of nodes in anetwork

Check if d(i,j)>d(i,k) + d(k,j) ,then

the shortest route from i to j is through k

Algorithm:1. Initialize distance and node adjacency matrices.

2. Check distance matrix for shorter paths between

nodes,using node 1 as an intermediate node.Replacecorresponding nodes in adjacency matrix with node 1.

3. Repeat the second step using the other nodes insequence,as the intermediate node.


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First Iteration


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Second Iteration


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Fourth Iteration


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Eighth Iteration


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Company: Digital equipment corporation

Problem:

Connect computer systems to LAN usingethernet

Ensure effective transport of packets ofinformation

Solution:

A network model was developed

Least cost paths were found using the spanningtree algorithm


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Traffic control system on Hanshin Expressway Objective:

Maximize flow of traffic through the network

Reduce congestion and bottlenecks caused by

accidents Solution:

Direct & indirect systems developed to controltraffic

System was developed using maximal flowtechnique


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Other Applications

Network design in molecular biology

Transportation problem

Minimize transportation costs

Used for deciding warehouse or factorylocations

Project Management techniques (CPM/PERT)

Completion time for a project

Determine critical and non critical activities


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QUERIES???

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