1_network flow models

8/2/2019 1_network Flow Models

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NETWORK FLOW MODELS


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Chapter Topics

The Minimal Spanning Tree ProblemThe Shortest Route Problem

The Maximal Flow Problem

Critical Path


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Overview

A network is an arrangement of paths connected at variouspoints through which one or more items move from one point toanother.

The network is drawn as a diagram providing a picture of thesystem thus enabling visual interpretation and enhanced

understanding.A large number of real-life systems can be modeled as networkswhich are relatively easy to conceive and construct.

Network diagrams consist of nodes and branches.

Nodes(circles), represent junction points, or locations.

Branches(lines), connect nodes and represent flow.


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Characteristics of Network Models

A node is a specific location

An arc connects 2 nodes

Arcs can be 1-way or 2-way


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Application

Nodes

(vertices)

Arcs (edges) Flow

Intersection

Airports

Switching points

Pumping Station

Work Centers

Roads

Air lanes

Wires, Channels

Pipes

Material-

handling routes

Vehicles

Aircraft

Messages,

Fluids

Jobs


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Figure 1

Network of Railroad Routes

Four nodes, four branches in figure.

Atlanta, node 1, termed origin, any of others destination.

Branches identified by beginning and ending node numbers.

Value assigned to each branch (distance, time, cost, etc.).

Example


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Minimum Spanning Trees


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Minimum Spanning Tree (MST)

it is a tree (i.e., it is acyclic)

it covers all the vertices V

contains |V| - 1 edges

the total cost associated with tree edges is the

minimum among all possible spanning trees not necessarily unique

A minimum spanning tree is a subgraph of anundirected weighted graph G, such that


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Minimum-Spanning Trees

Imagine: You wish to connect telephone system in a town orall the computers in an office building usingthe least amount of cable

a weighted graph problem !!

- Each vertex in a graph Grepresents a home (computer)

- Each edge represents the amount of cable needed toconnect all computers

Concrete example


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Problem: Laying Telephone Wire

Central office


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Wiring: Nave Approach

Central office

Expensive!


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Wiring: Better Approach

Central office

Minimize the total length of wire connecting the customers


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How Can We Generate a MST?

a

ce

d

b2

45

9

6

4

5

5

a

ce

d

b2

45

9

6

4

5

5


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Network of Possible Cable TV Paths

The Minimal Spanning Tree ProblemDefinition and Example Problem Data

Problem: Connect all nodes in a network so that the total branch lengths areminimized.


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The Minimal Spanning Tree ProblemSolution Approach (1 of 6)

Spanning Tree with Nodes 1 and 3

Start with any node in the network and select the closest node to join thespanning tree.


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Spanning Tree with Nodes 1, 3, and 4

Select the closest node not presently in the spanning area.


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Spanning Tree with Nodes 1, 2, 3, and 4

Continue


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Spanning Tree with Nodes 1, 2, 3, 4, and 5

Continue


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Spanning Tree with Nodes 1, 2, 3, 4, 5, and 7

Continue


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Minimal Spanning Tree for Cable TV Network

Optimal Solution


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The Minimal Spanning Tree ProblemSolution Method Summary

Select any starting node (conventionally, node 1).Select the node closest to the starting node to join thespanning tree.

Select the closest node not presently in the spanning tree

(if there is a tie, select one arbitrarily).

Repeat step 3 until all nodes have joined the spanning tree.


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Lauderdale Construction Example

Building a network of water pipes to supply water to 8

houses (distance in hundreds of feet)


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Steps 1 and 2

Starting arbitrarily with node (house) 1, the closest

node is node 3


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Second and Third Iterations


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Fourth and Fifth Iterations


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Sixth and Seventh Iterations

After all nodes (homes) are connected the total distance

is 16 or 1,600 feet of water pipe


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Problem: Determine the shortest routes from the origin to all destinations.

Shipping Routes from Los Angeles

The Shortest Route ProblemDefinition and Example Problem Data


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The Shortest Route ProblemSolution Method

Select the node with the shortest direct route from theorigin.

Establish a permanent setwith the origin node and thenode that was selected in step 1.

Determine all nodes directly connected to the permanentset nodes.

Select the node with the shortest route (branch) from thegroup of nodes directly connected to the permanent set

nodes.Repeat steps 3 and 4 until all nodes have joined thepermanent set.


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Network of Shipping Routes

The Shortest Route ProblemDefinition and Example Problem Data (2 of 2)


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Network with Node 1 in the Permanent Set

The Shortest Route ProblemSolution Approach (1 of 8)

Determine the initial shortest route from the origin (node 1) to the closest node(3).


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Network with Nodes 1 and 3 in the Permanent Set


Determine all nodes directly connected to the permanent set.


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Network with Nodes 1, 2, and 3 in the Permanent Set

Redefine the permanent set.



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Network with Nodes 1, 2, 3, and 4 in the Permanent Set


Continue


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Network with Nodes 1, 2, 3, 4, and 6 in the Permanent Set

Continue


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Network with Nodes 1, 2, 3, 4, 5, and 6 in the Permanent Set

Continue


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Network with Optimal Routes from Los Angeles to All Destinations

Optimal Solution


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Shortest Travel Time from Origin to Each Destination


Solution Summary


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Using LP for SPP

Want to find the shortest path from the factory to

the warehouse

Supply of 1 at factory

Demand of 1 at warehouse


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Decision Variables

Xij = flow from node i to node j

Note: flow on arc ij will be 1 if arc ij is used,

and 0 if not used

Roads are bi-directional, so the 9 roads require

18 decision variables

Sh P h P bl


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Shortest Path Problem

LP Formulation

)andallforbinary,is(

0

1

,10

11

1

i

1

2 2i

1 1

1 1

jix

x

njforxx

njiforxx

iforxx

xcZ

ij

ij

j

ai a

jiij

n

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ij ij

ijji

m

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ijij

Minimize

s.t.

Total Cost


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Example


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Objective Function (in distance)

Min 100X12 + 200X13 + 100X21 + 50X23 +

200X24 + 100X25 + 200X31 + 50X32 +

40X35 + 200X42 + 150X45 + 100X46 +

40X53 + 100X52 + 150X54 + 100X56 +

100X64 + 100X65

Subject to the constraints:

(see next slide)


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Constraints: Flow Balance For Each Node

Node

(X21 + X31) (X12 + X13) = -1 1

(X12+X32+X42+X52)(X21+X23+X24+X25)=0 2

(X13 + X23 + X53) (X31 + X32 + X35) = 0 3

(X24 + X54 + X64) (X42 + X45 + X46) = 0 4

(X25+X35+X45+X65)(X52+X53+X54+X56)=0 5

(X46 + X56) (X64 + X65) = 1 6

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