SA305 – Linear Programming
Asst. Prof. Nelson Uhan
Spring 2013
Lesson 1. Introduction
What is operations research?
• “The most influential academic discipline field you’ve never heard of ”
[Boston Globe, 2004]
• Operations Research (OR) is the discipline of applying advanced mathematical methods to help make
better decisions
• “The Science of Better”
[INFORMS slogan]
• “A liberal education in a technological world”
[Thomas Magnanti, former Dean of Engineering at MIT]
• Numerous applications
– manufacturing
– operating systems
– logistics
– airline pricing
– communications
– finance
– marketing
Historical connections between OR and the military
• 1930s: Britain – deployment of radar, aircraft maintenance and inspection, antisubmarine operations
• 1940s: US Navy – Antisubmarine Warfare Operations Research Group, Operations Evaluation Group
established
• 1950s: Naval Postgraduate School OR program, Naval Research Logistics Quarterly established
• 1962: Center for Naval Analyses established
The traveling salesperson problem
• A saleswoman located in Annapolis wants to visit all 48 state capitals of the continental US sell her
wares
• What is shortest way of visiting all the capitals and then returning to Annapolis?
• Entire books have been written on the TSP
• 1962: contest by Proctor and Gamble - best TSP tour through 33 US cities
• 1998: The Florida Sun-Sentinel’s Science page ponders Santa Claus’s traveling problem
1
• One of the most popular problems in operations research
• Numerous applications in expected and unexpected places
– Circuit board manufacturing
– Genome sequencing
• Your turn! Try to find the shortest way of visiting all the capitals and then returning to Annapolis
• What about 13,509 cities in the US?
• Sophisticated mathematical techniques are our best bet
The OR approach
Interpreting
Problem statement and data
Solution
C A N A D
A
Olympia WASH
INGTO
H
M ONTANA
Bismarck
M
M INNESOTA
ON
C
I F I
P A C
CALI
FORN
Cheyenne
City
UTAH
Springfield
KANSAS
RU
OC
EA
SS
OCE
Oklahoma City
Frankfort
WEST
VIRGINIA
DC
Boston
YORKAlbany
Atlanta
M ARYLAND
AN
Austin
M
C
A
N
A
D
G
A
RI N
SE A
GU L F
0
AN
A
Richmond
CAROLI
PSHIRE
SETTS
ISLAND
ICUT
NA
Columbia
SOUTH
CAROLINA
GEORGIA
Montgomery
J ackson
Tallahassee
Baton Rouge
IA
ALASKA
OCE
U.S. Departmentof the Interior
U.S. Geological Survey
ALABAM
HAM
RHODE
J ERSEY
NEW
IA
ONT
M ASSACHU
ce
Providen
CONNECT
Trenton
Dover
DELAWARE
Annapolis
on
Washingt
VIRGIN
NORTH
TENNESSEE
Little Rock
NEW
Concord
io
NEW
Hartford
Charleston
ARKANSAS
LOUISIANA
N
BE
IC
ta r
Raleigh
N
E A
O C
IC
Columbus
Indianapolis
J efferson City
OKLAHOM A
NEW M EXICO
M ISSISSIPPI
HAWAII
IF
100 mi
CI F
INDIANA
Nashville
Santa Fe
A
Phoenix
ARC T I C
100 km
PA
ke O n
A
YLVANI
PENNS Harrisburg
KENTUCKY
ARIZON
Honolulu
PAC
0
Lake Michigan
ILLINOIS
M ISSOURI
TEXAS
0
VERM
ke
OHIO
Lincoln
Topeka
M AINE
a
August
Montpeli
La
ie
Er
Des Moines
COLORADO
KA
La
Lansing
IOWA
NEBRASKA
Denver
IA
N
Madison
Salt Lake
NEVAD
A
Carson
City
Sacrame
nto
A
WISCONSIN
IG
St Paul
Pierre
AS
er
H
SOUTH DAKOTA
W YOM ING
IC O
C E AN
La
ron
Hu
ke
IDAHO
A
P AI I
CI F
I
C
Boise
AW
Lake S
uper
i or
NORTH DAKOTA
Helena
OREG
AL
N
Salem
O C E
A N
What is the shortest
way of visiting all
the capitals and
then returning
to Annapolis?
AT
O L AN
CE TI C
AN
STATES AND CAPITALS
TM
A T
L A
N T
I C
nationalatlas.gov
Where We Are
0
OF
AL A
J uneau
SK
A
200 mi
200 km
E
X
IC
GU
O
XI C
F ME
LF O
0
0
100
100
200
200
O
FLORIDA
T
300 mi
300 km
H
E
B
A
H
A
M
AS
CUBA
The National Atlas of the United States of AmericaO
R
states_capitals2.pdf INTERIOR-GEOLOGICAL SURVEY, RESTON, VIRGINIA-2003
Optimization (this course),
Stochastic Processes
Modeling
min
s.t.
∑{i , j}∈E c{i , j} x{i , j}
∑{i ,k}∈E x{i ,k} = 2
∀i ∈ N
x{i , j} ∈ {0, 1}
∀{i, j} ∈ E
Solving
∑{i , j}∈δ(S) x{i , j} ≥ 2 ∀S ⊆ N
Mathematical model
Goals for this course
• Modeling
– Recognize opportunities for mathematical optimization
– Formulate optimization models – linear programs – that capture the essence of the problem
– Illustrate applications of real-world problems
• Solving
– Algorithms to solve these mathematical models
• Detailed topic list and schedule is on the syllabus
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Optimization is everywhere
• “Minimize” time it takes to get from class to class
• “Maximize” the company’s profits
• (Moneyball) “Best” lineup for the Oakland A’s
• We are always trying to make decisions in a way that meets some objective subject to some constraints
• Some success stories of optimization helping solve complex real-world decision-making problems ...
Package delivery
• UPS has an air network consisting of 7 hubs, nearly 100 additional airports in the US, 160 aircraft of
nine different types
• Decision:
• Objective:
• Constraints:
• UPS credits optimization-based planning tools with identifying operational changes that have saved
over $87 million to date, reduced planning times, peak and non-peak costs, fleet requirements
Sports scheduling
• ACC Basketball earns over $30 million in revenue annually, almost all from TV and radio
• TV networks need a steady stream of “high quality” games, NCAA rules, school preferences and traditions
• Decision:
• Objective:
3
• Constraints:
• Optimization approaches yields reasonable schedules very quickly
Radiation therapy
• High doses of radiation can kill cancer cells and/or prevent them from growing and dividing
• Can also kill healthy cells!
• Radiation can be delivered at different angles and intensities
• Decision:
• Objective:
• Constraints:
• Many successes reported using different types of optimization models
Next time...
• A small example
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