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15.053
Thursday, March 14
z
Introduction to Network Flows
z
Handouts:
Lecture Notes
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Network Models
z
Linear Programming models that exhibit a
very special structure
z
Can use this structure to dramatically
reduce computational complexity
z
First widespread application of LP to
problems of industrial logistics
z
Addresses huge number of diverse
applications
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Notation and Terminology
Note: Network terminology is not (and never will be)
standardized.
The same concept may be denoted in
many different ways.
Called:
•
NETWORK
•
directed graph
•
digraph
•
graph
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1
4
3
Class Handouts (Ahuja,
Magnanti, Orlin)
Node set
N = {1,2,3,4}
Arc Set
Network
G = (N,A)
{(1,2),(1,3),(3,2),(3,4),(2,4)}
A={12,13,32,34,24}
Graph
G = (V,E)
Edge set:
Also Seen
Vertex set
V = {1,2,3,4}
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Directed and Undirected Networks
2
3
4
1
a
b
c
d
e
An Undirected Graph
2
3
4
1
a
b
c
d
e
A Directed Graph
•
The field of Network Optimization concerns
optimization problems on networks
•
Networks are used to transport commodities
•
physical goods
(products, liquids)
•
communication
•
electricity, etc.
Applications
Physical analog
Physical analog
Flow
Communication
systems
phone exchanges,
computers,
transmission
facilities, satellites
Cables, fiber optic
links, microwave
relay links
Voice messages,
Data,
Video transmissions
Hydraulic systems
Pumping stations
Reservoirs, Lakes
Pipelines
Water, Gas, Oil,
Hydraulic fluids
Integrated
computer circuits
Gates, registers,
processors
Wires
Electrical current
Mechanical systems
Joints
Rods, Beams,
Springs
Heat, Energy
Transportation
systems
Intersections,
Airports,
Rail yards
Highways,
Airline routes
Railbeds
Passengers,
freight,
vehicles,
operators
An Overview of Some Applications of
Network Optimization
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Examples of
terms.
2
3
4
Directed Path
.
Example:
1, 2, 3, 4
(or 1, a, 2, b, 3, e)
Directions are important.
Two paths abe (or 1234)
and acde (or 12534)
a
b
c
1
Cycle or circuit (or loop)
1, 2, 3, 1.
(or 1, a, 2, b, 3, e)
Note that directions are ignored.
Directed Cycle:
(1, 2, 3, 4, 1) or
1, a, 2, b, 3, c, 4, d, 1
Directions are important.
Cycles (loops):
abcd (or 12341)
badc (or 32143)
eba (or 1321)
cde (or 3413)
5
d
e
2
3
4
a
b
c
d
1
e
Path:
Example:
5, 2, 3, 4.
(or 5, c, 2, b, 3, e, 4)
Note that directions are ignored.
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More Definitions
2
1
4
3
A network is
connected
if every node
can be reached from every other
node by following a sequence of
arcs in which direction is ignored.
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This note was uploaded on 12/20/2011 for the course BUS 15.053 taught by Professor Prof.jamesorlin during the Spring '05 term at MIT.
 Spring '05
 Prof.JamesOrlin

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