SA305 – Linear Programming
Asst. Prof. Nelson Uhan
Spring 2016
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, e.g.
– logistics
– manufacturing
– workforce scheduling
– finance
– marketing
OR and the military
• The military uses OR to improve decision making in a variety of ways, e.g.
– force composition
– weapon selection
– search and detection
– flight operations scheduling
– training and personnel assignment
• Assessment Division (OPNAV N81) at the Pentagon
• The Naval Postgraduate School has one of the oldest and most well-respected OR departments in the US
• Naval Research Logistics is a prominent academic journal featuring research in OR
The traveling salesperson problem
• A salesperson located in Annapolis wants to visit clients in each of the 48 state capitals of the continental
US and Washington DC
• What is shortest way of visiting all the capitals and then returning to Annapolis?
• Entire books have been written on the TSP
1
• 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
• The TSP has even been used to make art
• 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
INGTO
AL
N
H
Helena
OREG
MONTANA
La k e S
upe
NORTH DAKOTA
Bismarck
ON
UTAH
100 mi
OC
EA
SS
ri
Boston
YORKAlbany
Oklahoma City
WEST
VIRGINIA
DC
Little Rock
Austin
Atlanta
M
N
C
A
N
A
D
G
A
RIN
SE A
GULF
0
OCE
AN
U.S. Department of the Interior
U.S. Geological Survey
ALABAMA
ce
RHODE
CONNECT
Trenton
JERSEY
NEW
Dover
DELAWARE
Annapolis
on
Washingt
VIRGIN
MARYLAND
IA
Richmond
NORTH
TENNESSEE
ARKANSAS
CAROLI
ISLAND
ICUT
NA
Columbia
SOUTH
CAROLINA
GEORGIA
Montgomery
Jackson
Tallahassee
LOUISIANA
AN
T
VERMON
RE
HAMPSHI
NEW
USETTS
MASSACH
Providen
Hartford
Charleston
Frankfort
OKLAHOMA
NEW MEXICO
OCE
August
ier
Concord
o
NEW
IA
YLVAN
PENNS Harrisburg
Baton Rouge
IA
ALASKA
IC
Columbus
Indianapolis
Jefferson City
MISSISSIPPI
BE
CIF
ta
ke O n
ie
Raleigh
Santa Fe
A
Phoenix
ARCTIC
RU
100 km
PA
INDIANA
MISSOURI
Nashville
ARIZON
N
E A
O C
0
0
La k e Michigan
ILLINOIS
Springfield
Topeka
TEXAS
IC
Er
KENTUCKY
HAWAII
IF
ke
OHIO
Lincoln
KANSAS
Honolulu
PA C
La
Des Moines
COLORADO
E
MAIN
Montpel
La
Lansing
IOWA
NEBRASKA
Denver
IA
N
FORN
A
C
I F I
P A C
nto
CALI
Cheyenne
City
KA
ron
Madison
Salt Lake
NEVAD
Carson
A
City
Sacrame
IG
WISCONSIN
Hu
St Paul
Pierre
AS
a
H
SOUTH DAKOTA
WYOMING
IC O
CEAN
La
ke
IDAHO
A
PA I I
CIF
I
C
Boise
AW
r i or
M
MINNESOTA
AT
O LAN
CE TIC
AN
STATES AND CAPITALS
C A N A D
A
Olympia WASH
Salem
O C E
A N
What is the shortest
way of visiting all
the capitals and
then returning
to Annapolis?
TM
A T
L A
N T
I C
nationalatlas.gov
Where We Are
0
OF
ALA
SK
A
200 mi
200 km
E
X
I C
GU
O
XIC
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
Juneau
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
2
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...
• Formulating a small optimization model
4
STATES AND CAPITALS
TM
C A N A D
A
ON
H
ORE
NA
NO RTH DAK OTA
Bismarck
Madison
Carso
n
C
I F I
P A C
Sacra
mento
CAL
IFOR
N E VA
City
Salt Lake
DA
Cheyenne
City
U TA H
La
Des Moines
Lincoln
ILL INO IS
Springfield
CO LO RA DO
Topeka
KANSA S
N
E A
O C
A R IZ O
NA
N EW M EX IC
O
Phoenix
Jackson
ARCTIC
HAWAII
Honolulu
0
0
IF
IC
100 mi
RU
OC
EA
SS
OCE
AN
Austin
M
ALASKA
C
A
N
A
CIF
IC
RIN
A
PA
D
BE
G
SE A
GULF
0
OCE
AN
U.S. Department of the Interior
U.S. Geological Survey
IA
P S H IR
E
EY
Richmo
IC UT
nd
IN A
Columbia
S O UT H
IN A
C A RO L
Montgomery
Tallahassee
LOUI SIAN A
Baton Rouge
IA
N
100 km
G E O RG
JE R S
MAR
IA
V IR G IN
Raleigh
Atlanta
MISSISSIPPI
NEW
NECT
over
E
lis D D E LAWA R
Annapo
n
to
g
D
N
Washin
A
DC
L
Y
C A RO L
N O RT H
ARKA NSAS
A
A LA B A M
n
Trento
Harris
W E S TIA
V IR G IN
TE NN ES SE E
Little Rock
CON
LVA N IA
N N SY
burg
KE NT UC KY
OKLAH OMA
Oklahoma City
ke
ie
PE
Nashville
Santa Fe
Er
Charleston
Frankfort
Jefferson City
ke
Columbus
Indianapolis
T
NE
rd
TS
Conco
U S ET
o
S AC H
ri
M AS
O nta
n
to
s
o
ence
YO R KAlbany B
Provid
NEW
D
LAN
rd
fo
D E IS
rt
Ha
R HO
O H IO
IN D IA N A
MIS SOU RI
TE XA S
PA C
La
Lansing
IOW A
NEB RAS KA
Denver
NIA
ron
G
MON
AM
W H
Hu
WIS CO NS IN
Au
VER
ke
W YO M IN
St Paul
Pierre
N
SOUTH DAK OTA
A
O
NE
gusta
Montp
La
IG
I DA H
MAI
IC O
CEAN
elier
I
H
Boise
r i or
M
MINNESOTA
A
PA I I
CIF
C
GON
KA
O C E
A N
Helena
M O N TA
AW
AS
A T
L A
N T
I C
Salem
La k e S
upe
La k e Michigan
Olym W A S H
pia
INGT
AL
AT
O LAN
CE TIC
AN
nationalatlas.gov
Where We Are
0
OF
ALA
SK
A
200 mi
200 km
E
X
I C
GU
O
XICO
F ME
LF O
0
0
100
100
200
200
F L O R ID
A
T
300 mi
300 km
H
E
B
A
H
A
M
AS
Juneau
CUBA
The National Atlas of the United States of AmericaO
R
states_capitals2.pdf INTERIOR-GEOLOGICAL SURVEY, RESTON, VIRGINIA-2003