Keys What is OR?
SA305: Linear models and optimization
Max Wakefield
United States Naval Academy
January 7, 2014
Spring 2014
SA305
1
Keys What is OR?
SA305: Linear Models and Optimization
Today’s agenda:
Course policies
Spring 2014
SA305
2
Keys What is OR?
SA305: Linear Models and Optimization
Today’s agenda:
Course policies
Syllabus
Spring 2014
SA305
2
Keys What is OR?
SA305: Linear Models and Optimization
Today’s agenda:
Course policies
Syllabus
Keys to success
Spring 2014
SA305
2
Keys What is OR?
SA305: Linear Models and Optimization
Today’s agenda:
Course policies
Syllabus
Keys to success
What is operations research?
Spring 2014
SA305
2
Keys What is OR?
SUCCESS!
For success in this course:
This course is very difficult!!!!!!
Spring 2014
SA305
3
Keys What is OR?
SUCCESS!
For success in this course:
This course is very difficult!!!!!!
Work on your homework every day
Spring 2014
SA305
3
Keys What is OR?
SUCCESS!
For success in this course:
This course is very difficult!!!!!!
Work on your homework every day
Read your text book
Spring 2014
SA305
3
Keys What is OR?
SUCCESS!
For success in this course:
This course is very difficult!!!!!!
Work on your homework every day
Read your text book
No really, read your textbook
Spring 2014
SA305
3
Keys What is OR?
SUCCESS!
For success in this course:
This course is very difficult!!!!!!
Work on your homework every day
Read your text book
No really, read your textbook
...please, please, please, read your textbook
Spring 2014
SA305
3
Keys What is OR?
SUCCESS!
For success in this course:
This course is very difficult!!!!!!
Work on your homework every day
Read your text book
No really, read your textbook
...please, please, please, read your textbook
Talk to me: Ask questions, go to EI, etc.
Spring 2014
SA305
3
Keys What is OR?
SUCCESS!
For success in this course:
This course is very difficult!!!!!!
Work on your homework every day
Read your text book
No really, read your textbook
...please, please, please, read your textbook
Talk to me: Ask questions, go to EI, etc.
Forming a study group is encouraged, but....
You need to make sure you can actually do the problems.
Spring 2014
SA305
3
Keys What is OR?
What is operations research?
Spring 2014
SA305
4
Keys What is OR?
What is operations research?
“The most influential academic discipline field you’ve never
heard of”
Boston Globe, 2004
Spring 2014
SA305
4
Keys What is OR?
What is operations research?
“The most influential academic discipline field you’ve never
heard of”
Boston Globe, 2004
“The Science of Better”
Spring 2014
INFORMS slogan
SA305
4
Keys What is OR?
What is operations research?
“The most influential academic discipline field you’ve never
heard of”
Boston Globe, 2004
“The Science of Better”
INFORMS slogan
“Useful applied math”
Spring 2014
SA305
4
Keys What is OR?
What is operations research?
“The most influential academic discipline field you’ve never
heard of”
Boston Globe, 2004
“The Science of Better”
INFORMS slogan
“Useful applied math”
Operations Research (OR) is the discipline of applying
advanced mathematical methods to help make better
decisions.
Spring 2014
SA305
4
Keys What is OR?
doomed to repeat
A little history
17th and 18th century: Some mathematical underpinnings
Expected value, B. Pascal (1654)
Newton’s Method, Newton (1665)
Bridges of Königsberg, Euler (1736)
Bayes Rule, Bayes (1763)
Lagrangian multipliers, Lagrange (1788)
Least Squares, Gauss (1795)
Spring 2014
SA305
5
Keys What is OR?
doomed to repeat
Father of OR (perhaps): Charles Babbage (1791–1871)
Spring 2014
SA305
5
Keys What is OR?
doomed to repeat
Father of OR (perhaps): Charles Babbage (1791–1871)
Uses OR to analyze mail delivery, leads to Sir Rowland Hill introducing the
Penny Post to Britain
Key quote from the Babbage Room of the Totnes Museum: He broke down
the service into various elements – the operations involved, the manpower
required, the expense of each process – and discovered that the cost of
handling mail was greater than the cost of transportation.
Spring 2014
SA305
5
Keys What is OR?
doomed to repeat
19th and early 20th century results
Solution of inequalities, Fourier (1826)
Solution of linear inequalities, Gauss (1826)
Gantt charts, Gantt, Taylor (1900)
Farkas lemma, Farkas (1902)
Pareto optimality, Pareto (1906)
Markov chains, Markov (1907)
“The Theory of Probabilities and Telephone
Conversations”, Erlang (1909)
Spring 2014
SA305
5
Keys What is OR?
doomed to repeat
Formal beginnings:
1936 The term “operational research” is first used in Great Britain to
describe experiments studying the most effective use of radar.
1939,1947 Kantorovich (1939) and Dantzig (1947) discover linear
programming. Dantzig (1947) describes the simplex method.
1941 Operational Research Section (ORS) was established in Britain
directed by P. Blackett.
Organizes flying maintainance and inspection
Improvement of antisubmarine operations
1949 Monte Carlo simulation, S. M. Ulam, J. von Neumann
1952 First graduate programs (M.A. and Ph.D.) established at Case
Western.
Spring 2014
SA305
5
Keys What is OR?
doomed to repeat
Historical connections to the Navy:
1942,1945 U.S. Navy begins to use OR formally via the Antisubmarine
Warfare Operations Research Group (1942) and the Operations
Evaluation Group (OEG, 1945).
1951 Naval Post Graduate School OR program established.
1954 Naval Research Logistics Quarterly established.
1962 Center for Naval Analyses established.
Spring 2014
SA305
5
Keys What is OR?
doomed to repeat
The traveling salesperson problem
Spring 2014
SA305
6
Keys What is OR?
doomed to repeat
The traveling salesperson problem
STATES AND CAPITALS
TM
C A N A D
A
H
MON TAN
A
La k e S
upe
NORTH DAKOTA
Bismarck
ON
M
MINNESOTA
Salt Lake
DA
Cheyenne
City
UTA H
IA
ILLINOIS
Springfield
Topeka
KANSAS
100 mi
AR IZO
Santa Fe
NA
Oklahoma City
ARCTIC
RU
OC
EA
SS
OCE
Atlanta
Austin
M
C
A
N
A
D
A
G
SE A
GULF
0
OCE
AN
A
HAM
Trenton
Harr
DC
NEW
JERS
NEC TICU
PSH IRE
EY
er
E
is Dov DELAWAR
Annapol
ton
D
Washing
YLAN
CAR OLI
T
MAR
NA
Columbia
SOUTH
A
CAR OLIN
Tallahassee
Baton Rouge
IA
N
RIN
ALAB AMA
GEO RGI
MO NT
Montgomery
Jackson
LOUISIANA
AN
ALASKA
IC
NIA
SYLVAisburg
Raleigh
ARKANSAS
Little Rock
CON
IA
VIR GIN mond
Rich
NO RTH
TENNE SSEE
MISSISSIPPI
BE
CIF
PE NN
WE STIA
VIR GIN
Charleston
Frankfort
Jefferson City
OKLAHOMA
NEW MEXI
CO
Phoenix
100 km
PA
Columbus
Indianapolis
Nashville
N
E A
O C
0
0
ie
KENT UCKY
Honolulu
IC
Er
ke
NEW
d
ETT S
Concor
o
SAC HUS
ri
MAS
O nta
ton
YO RKAlbany Bos Providence
NE W
ND
d
DE ISLA
Hartfor
RHO
OHIO
INDIA NA
MISSOURI
TEXAS
IF
La k e Michigan
L
e
ak
Des Moines
Lincoln
COLOR ADO
HAWAII
PA C
La
Lansing
IOWA
NEBRASKA
Denver
N
C
I F I
P A C
FO RN
A
NE VA
Carson
City
Sacram
ento
CA LI
VER
ron
Madison
IG
WISCONS IN
NE
Augus
ier
Montpel
La
Hu
St Paul
Pierre
G
M AI
ta
I
H
SOUTH DAKOTA
WYO MIN
KA
IC O
CEAN
ke
IDA HO
AS
A
PA I I
CIF
C
Boise
AW
r i or
O C E
A N
Helena
OR EG
AL
GTON
A T
L A
N T
I C
Olymp WA SH
ia
IN
Salem
AT
O LAN
CE TIC
AN
nationalatlas.gov
Where We Are
0
E
X
I C
GU
O
0
OF
ALA
SK
A
200 mi
XIC
F ME
LF O
0
100
100
200
200
O
FLO RIDA
T
300 mi
300 km
H
E
B
A
H
A
M
AS
Juneau
CUBA
200 km
U.S. Department of the Interior
U.S. Geological Survey
The National Atlas of the United States of AmericaO
R
A saleswoman in Annapolis wants to visit the 47 other state
capitals of the continental United States to sell her wares
states_capitals2.pdf INTERIOR-GEOLOGICAL SURVEY, RESTON, VIRGINIA-2003
Spring 2014
SA305
6
Keys What is OR?
doomed to repeat
The traveling salesperson problem
STATES AND CAPITALS
TM
C A N A D
A
H
MON TAN
A
La k e S
upe
NORTH DAKOTA
Bismarck
ON
M
MINNESOTA
Salt Lake
DA
Cheyenne
City
UTA H
IA
ILLINOIS
Springfield
Topeka
KANSAS
100 mi
AR IZO
Santa Fe
NA
Oklahoma City
ARCTIC
RU
OC
EA
SS
OCE
Atlanta
Austin
M
C
A
N
A
D
A
G
SE A
GULF
0
OCE
AN
A
HAM
Trenton
Harr
DC
NEW
JERS
NEC TICU
PSH IRE
EY
er
E
is Dov DELAWAR
Annapol
ton
D
Washing
YLAN
CAR OLI
T
MAR
NA
Columbia
SOUTH
A
CAR OLIN
Tallahassee
Baton Rouge
IA
N
RIN
ALAB AMA
GEO RGI
MO NT
Montgomery
Jackson
LOUISIANA
AN
ALASKA
IC
NIA
SYLVAisburg
Raleigh
ARKANSAS
Little Rock
CON
IA
VIR GIN mond
Rich
NO RTH
TENNE SSEE
MISSISSIPPI
BE
CIF
PE NN
WE STIA
VIR GIN
Charleston
Frankfort
Jefferson City
OKLAHOMA
NEW MEXI
CO
Phoenix
100 km
PA
Columbus
Indianapolis
Nashville
N
E A
O C
0
0
ie
KENT UCKY
Honolulu
IC
Er
ke
NEW
d
ETT S
Concor
o
SAC HUS
ri
MAS
O nta
ton
YO RKAlbany Bos Providence
NE W
ND
d
DE ISLA
Hartfor
RHO
OHIO
INDIA NA
MISSOURI
TEXAS
IF
La k e Michigan
L
e
ak
Des Moines
Lincoln
COLOR ADO
HAWAII
PA C
La
Lansing
IOWA
NEBRASKA
Denver
N
C
I F I
P A C
FO RN
A
NE VA
Carson
City
Sacram
ento
CA LI
VER
ron
Madison
IG
WISCONS IN
NE
Augus
ier
Montpel
La
Hu
St Paul
Pierre
G
M AI
ta
I
H
SOUTH DAKOTA
WYO MIN
KA
IC O
CEAN
ke
IDA HO
AS
A
PA I I
CIF
C
Boise
AW
r i or
O C E
A N
Helena
OR EG
AL
GTON
A T
L A
N T
I C
Olymp WA SH
ia
IN
Salem
AT
O LAN
CE TIC
AN
nationalatlas.gov
Where We Are
0
E
X
I C
GU
O
0
OF
ALA
SK
A
200 mi
XIC
F ME
LF O
0
100
100
200
200
O
FLO RIDA
T
300 mi
300 km
H
E
B
A
H
A
M
AS
Juneau
CUBA
200 km
U.S. Department of the Interior
U.S. Geological Survey
The National Atlas of the United States of AmericaO
R
A saleswoman in Annapolis wants to visit the 47 other state
capitals of the continental United States to sell her wares
states_capitals2.pdf INTERIOR-GEOLOGICAL SURVEY, RESTON, VIRGINIA-2003
What is shortest way to visit the other capitals and return
to Annapolis?
Spring 2014
SA305
6
Keys What is OR?
doomed to repeat
The traveling salesperson problem
Entire books have been written on the TSP
Spring 2014
SA305
7
Keys What is OR?
doomed to repeat
The traveling salesperson problem
1962: contest by Proctor and Gamble - best TSP tour
through 33 US cities
Spring 2014
SA305
8
Keys What is OR?
doomed to repeat
The traveling salesperson problem
1998: The Florida Sun-Sentinel’s Science page ponders
Santa Claus’s traveling problem
http://www.tsp.gatech.edu
Spring 2014
SA305
9
Keys What is OR?
doomed to repeat
The traveling salesperson problem
One of the most popular problems in operations research
Spring 2014
SA305
10
Keys What is OR?
doomed to repeat
The traveling salesperson problem
One of the most popular problems in operations research
Numerous applications in expected and unexpected places
Spring 2014
SA305
10
Keys What is OR?
doomed to repeat
The traveling salesperson problem
One of the most popular problems in operations research
Numerous applications in expected and unexpected places
Circuit board manufacturing
Spring 2014
SA305
10
Keys What is OR?
doomed to repeat
The traveling salesperson problem
One of the most popular problems in operations research
Numerous applications in expected and unexpected places
Circuit board manufacturing
Genome sequencing
Spring 2014
SA305
10
Keys What is OR?
doomed to repeat
The traveling salesperson problem
Your turn! Try to find the shortest way of visiting the
other capitals and then returning to Annapolis
Spring 2014
SA305
11
Keys What is OR?
doomed to repeat
The traveling salesperson problem
STATES AND CAPITALS
TM
Where We Are
C A N A D
A
H
Helena
A
Bismarck
M
M INNESOTA
C
I F I
P A C
Carso
n City
Sacra
mento
FO R
DA
Cheyenne
City
UTAH
Lincoln
ILLINO IS
Springfield
COLO RADO
Topeka
KANSAS
100 mi
Santa Fe
Oklahoma City
NEW M EXIC
O
ARC T I C
RU
OC
EA
SS
OCE
WE STIA
VIRGIN
YO RKAlbany
rd
Hartfo
IA
SY LVANurg
Harrisb
DC
ARKANSAS
Atlanta
NE W
M
A
N
A
D
A
SE A
GU L F
0
OCE
AN
U.S. Departmentof the Interior
U.S. Geological Survey
GE ORGIA
PSH IRE
AN
E ISL
ECTICU
EY
M ARY
D
T
nd
Richmo
A
Columbia
SO UTH
A
CA RO LIN
J ackson
Tallahassee
Baton Rouge
IA
C
G
A
J ERS
ver
lis Do DEL AW ARE
Annapo
on
Washingt
LAN D
CA RO LIN
HA M
Montgomery
LOUISIANA
Austin
N
RI N
ALA BAM
RHOD
CO NN
n
Trento
IA
VI RGIN
NO RTH
TENN ESSE E
Little Rock
AN
ALASKA
IC
PE NN
ON T
NE W
SETTS
Concord
SAC HU
M AS
n
Bosto
ence
Provid
io
NE W
ie
n
Charlesto
Frankfort
M ISSISSIPPI
BE
CI F
Columbus
Indianapolis
J efferson City
OKLAHOM A
NA
Phoenix
100 km
PA
Er
ta r
Raleigh
N
E A
O C
0
0
ke
ke O n
OH IO
INDI ANA
Nashville
AR IZO
Honolulu
IC
La
M ISSOURI
TEXA S
IF
La
KEN TUC KY
HAWAII
PAC
VE RM
Lansing
Des Moines
E
sta
Augu
elier
Montp
IOWA
NEBRASK A
Denver
N IA
N
Madison
Salt Lake
A
W YOM ING
NE VA
IG
WISCON SIN
Lake Michigan
H
St Paul
M A IN
IC O
C E AN
ron
Hu
ke
SOUTH DAKOTA
KA
La
C
IDAH
O
Pierre
C A LI
I
AS
A
P AI I
CI F
Lake S
uper
i or
NORTH DAKOTA
ON
Boise
AW
O C E
A N
M ON TAN
Salem
O R EG
AL
TO N
A T
L A
N T
I C
Olym W AS HI
pia
NG
AT
O L AN
CE TI C
AN
The nationalatlas.gov
answer:
0
E
X
IC
GU
O
XI C
F ME
LF O
0
0
100
100
200
200
O
A
FLO RID
T
300 mi
300 km
H
E
B
A
H
A
M
AS
OF
AL A
J uneau
SK
A
200 mi
CUBA
200 km
Spring 2014
SA305
The National Atlas of12
the United States of AmericaO
R
Keys What is OR?
doomed to repeat
The traveling salesperson problem
What about 13,509 cities in the US?
Spring 2014
SA305
13
Keys What is OR?
doomed to repeat
The traveling salesperson problem
What about 13,509 cities in the US?
Spring 2014
SA305
13
Keys What is OR?
doomed to repeat
The traveling salesperson problem
What about 13,509 cities in the US?
Sophisticated mathematical techniques are our best bet
Spring 2014
SA305
13
Keys What is OR?
doomed to repeat
The OR approach
Problem definition
What is the shortest
way to visit all capitals? What are the
distances between
all pairs of capitals?
Spring 2014
SA305
14
Keys What is OR?
doomed to repeat
The OR approach
Problem definition
What is the shortest
way to visit all capitals? What are the
distances between
all pairs of capitals?
Modeling
min
P
s.t.
P
P
{i,j}∈E
c{i,j} x{i,j}
{i,k}∈E
x{i,k} = 2
{i,j}∈δ(S)
∀i ∈ N
x{i,j} ≥ 2 ∀S ⊆ N
x{i,j} ∈ {0, 1}
∀{i, j} ∈ E
Mathematical model
Spring 2014
SA305
14
Keys What is OR?
doomed to repeat
The OR approach
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
FORN
Cheyenne
City
UTAH
IA
Topeka
IC
100 mi
OC
EA
SS
OCE
Oklahoma City
ARKANSAS
Little Rock
Boston
YORKAlbany
Austin
Atlanta
M
N
C
A
N
A
D
G
A
RI N
SE A
GU L F
0
OCE
AN
ALABAM
A
J ERSEY
NEW
Dover
DELAWARE
Annapolis
on
Washingt
VIRGIN
M ARYLAND
IA
Richmond
CAROLI
HAM
RHODE
CONNECT
Trenton
ONT
PSHIRE
SETTS
M ASSACHU
ce
Providen
Hartford
DC
ISLAND
ICUT
NA
Columbia
SOUTH
CAROLINA
GEORGIA
Montgomery
J ackson
Tallahassee
LOUISIANA
AN
NEW
Concord
io
NEW
Baton Rouge
IA
ALASKA
IC
ta r
A
YLVANI
PENNS Harrisburg
NORTH
TENNESSEE
OKLAHOM A
NEW M EXICO
M ISSISSIPPI
BE
CI F
ke O n
ie
Raleigh
Santa Fe
A
Phoenix
ARC T I C
RU
100 km
PA
Er
WEST
VIRGINIA
Charleston
Frankfort
J efferson City
Nashville
ARIZON
N
E A
O C
0
Columbus
Indianapolis
M ISSOURI
TEXAS
0
INDIANA
KENTUCKY
HAWAII
IF
M AINE
a
August
VERM
ke
OHIO
ILLINOIS
Springfield
COLORADO
KA
Montpeli
La
Des Moines
Lincoln
KANSAS
Honolulu
PAC
AS
La
Lansing
IOWA
NEBRASKA
Denver
Lake Michigan
Madison
Salt Lake
NEVAD
A
Carson
City
Sacrame
nto
CALI
N
WISCONSIN
A
St Paul
Pierre
IG
SOUTH DAKOTA
W YOM ING
IC O
C E AN
er
H
IDAHO
A
La
ron
Hu
ke
Boise
AW
P AI I
CI F
I
C
What is the shortest
way to visit all capitals? What are the
distances between
all pairs of capitals?
Lake S
uper
i or
NORTH DAKOTA
Helena
OREG
AL
N
Salem
AT
O L AN
CE TI C
AN
STATES AND CAPITALS
TM
O C E
A N
nationalatlas.gov
Where We Are
A T
L A
N T
I C
Problem definition
0
OF
AL A
J uneau
SK
A
200 mi
200 km
U.S. Departmentof the Interior
U.S. Geological Survey
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
Modeling
min
P
s.t.
P
P
{i,j}∈E
c{i,j} x{i,j}
{i,k}∈E
x{i,k} = 2
{i,j}∈δ(S)
Algorithms
∀i ∈ N
x{i,j} ≥ 2 ∀S ⊆ N
x{i,j} ∈ {0, 1}
∀{i, j} ∈ E
Mathematical model
Spring 2014
SA305
14
Keys What is OR?
doomed to repeat
The OR approach
Reality: What is actual route? More cities? roads/trains/airplanes? Costs?
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
FORN
Cheyenne
City
UTAH
IA
Topeka
IC
100 mi
OC
EA
SS
OCE
Oklahoma City
ARKANSAS
Little Rock
Boston
YORKAlbany
Austin
Atlanta
M
N
C
A
N
A
D
G
A
RI N
SE A
GU L F
0
OCE
AN
ALABAM
A
J ERSEY
NEW
Dover
DELAWARE
Annapolis
on
Washingt
VIRGIN
M ARYLAND
IA
Richmond
CAROLI
HAM
RHODE
CONNECT
Trenton
ONT
PSHIRE
SETTS
M ASSACHU
ce
Providen
Hartford
DC
ISLAND
ICUT
NA
Columbia
SOUTH
CAROLINA
GEORGIA
Montgomery
J ackson
Tallahassee
LOUISIANA
AN
NEW
Concord
io
NEW
Baton Rouge
IA
ALASKA
IC
ta r
A
YLVANI
PENNS Harrisburg
NORTH
TENNESSEE
OKLAHOM A
NEW M EXICO
M ISSISSIPPI
BE
CI F
ke O n
ie
Raleigh
Santa Fe
A
Phoenix
ARC T I C
RU
100 km
PA
Er
WEST
VIRGINIA
Charleston
Frankfort
J efferson City
Nashville
ARIZON
N
E A
O C
0
Columbus
Indianapolis
M ISSOURI
TEXAS
0
INDIANA
KENTUCKY
HAWAII
IF
M AINE
a
August
VERM
ke
OHIO
ILLINOIS
Springfield
COLORADO
KA
Montpeli
La
Des Moines
Lincoln
KANSAS
Honolulu
PAC
AS
La
Lansing
IOWA
NEBRASKA
Denver
Lake Michigan
Madison
Salt Lake
NEVAD
A
Carson
City
Sacrame
nto
CALI
N
WISCONSIN
A
St Paul
Pierre
IG
SOUTH DAKOTA
W YOM ING
IC O
C E AN
er
H
IDAHO
A
La
ron
Hu
ke
Boise
AW
P AI I
CI F
I
C
What is the shortest
way to visit all capitals? What are the
distances between
all pairs of capitals?
Lake S
uper
i or
NORTH DAKOTA
Helena
OREG
AL
N
Salem
AT
O L AN
CE TI C
AN
STATES AND CAPITALS
TM
O C E
A N
nationalatlas.gov
Where We Are
A T
L A
N T
I C
Problem definition
0
OF
AL A
J uneau
SK
A
200 mi
200 km
U.S. Departmentof the Interior
U.S. Geological Survey
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
Modeling
min
P
s.t.
P
P
{i,j}∈E
c{i,j} x{i,j}
{i,k}∈E
x{i,k} = 2
{i,j}∈δ(S)
Algorithms
∀i ∈ N
x{i,j} ≥ 2 ∀S ⊆ N
x{i,j} ∈ {0, 1}
∀{i, j} ∈ E
Mathematical model
Spring 2014
SA305
14
Keys What is OR?
doomed to repeat
The OR approach
Reality: What is actual route? More cities? roads/trains/airplanes? Costs?
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
Cheyenne
City
UTAH
Topeka
0
IC
100 mi
ta r
OC
EA
OCE
Oklahoma City
ARKANSAS
Little Rock
Boston
YORKAlbany
Austin
Atlanta
M ARYLAND
IA
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
Tallahassee
Baton Rouge
IA
N
OCE
ALABAM
J ERSEY
NEW
Dover
DELAWARE
Annapolis
on
Washingt
VIRGIN
HAM
RHODE
CONNECT
Trenton
ONT
M ASSACHU
ce
Providen
Hartford
DC
Montgomery
J ackson
LOUISIANA
AN
NEW
Concord
io
NEW
A
YLVANI
PENNS Harrisburg
NORTH
TENNESSEE
OKLAHOM A
NEW M EXICO
M ISSISSIPPI
SS
ALASKA
IC
ke O n
ie
Raleigh
Santa Fe
A
Phoenix
BE
CI F
Er
WEST
VIRGINIA
Charleston
Frankfort
J efferson City
Nashville
ARIZON
ARC T I C
RU
100 km
PA
Columbus
Indianapolis
M ISSOURI
TEXAS
0
INDIANA
KENTUCKY
HAWAII
IF
M AINE
a
August
VERM
ke
OHIO
ILLINOIS
Springfield
COLORADO
KA
Montpeli
La
Des Moines
Lincoln
KANSAS
Honolulu
PAC
AS
La
Lansing
IOWA
NEBRASKA
Denver
IA
N
E A
O C
Optimization (this course),
Stochastic Processes
FORN
Lake Michigan
Madison
Salt Lake
NEVAD
A
Carson
City
Sacrame
nto
CALI
N
WISCONSIN
A
St Paul
Pierre
IG
SOUTH DAKOTA
W YOM ING
IC O
C E AN
er
H
IDAHO
A
La
ron
Hu
ke
Boise
AW
P AI I
CI F
I
C
What is the shortest
way to visit all capitals? What are the
distances between
all pairs of capitals?
Lake S
uper
i or
NORTH DAKOTA
Helena
OREG
AL
N
Salem
AT
O L AN
CE TI C
AN
STATES AND CAPITALS
TM
O C E
A N
nationalatlas.gov
Where We Are
A T
L A
N T
I C
Problem definition
0
OF
AL A
J uneau
SK
A
200 mi
200 km
U.S. Departmentof the Interior
U.S. Geological Survey
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
Modeling
min
P
s.t.
P
P
{i,j}∈E
c{i,j} x{i,j}
{i,k}∈E
x{i,k} = 2
{i,j}∈δ(S)
Algorithms
∀i ∈ N
x{i,j} ≥ 2 ∀S ⊆ N
x{i,j} ∈ {0, 1}
∀{i, j} ∈ E
Mathematical model
Spring 2014
SA305
14
Keys What is OR?
doomed to repeat
OR applications
Airline: scheduling planes and crews, pricing tickets, taking
reservations, and planning the size of the fleet,
Pharmaceutical: R & D management,
Delivery companies: routing and planning,
Financial services: credit scoring, marketing, and internal
operations,
Government: deployment of emergency services, scheduling
police and fire department, regulation of environmental
pollution, air traffic safety, AIDS intervention
Healthcare: kidney transplant donor matching,
departmental staffing and scheduling, radiation therapy
Spring 2014
SA305
15
Keys What is OR?
doomed to repeat
A general OR approach
1
Problem definition and data gathering
1 Discussing problem with decision makers
2 Gathering data
3 Assumptions
Spring 2014
SA305
16
Keys What is OR?
doomed to repeat
A general OR approach
1
Problem definition and data gathering
1 Discussing problem with decision makers
2 Gathering data
3 Assumptions
2
Creating a mathematical model
1 What are the decision variables? Constraints?
2 How is uncertainty being taken into account?
Spring 2014
SA305
16
Keys What is OR?
doomed to repeat
A general OR approach
1
Problem definition and data gathering
1 Discussing problem with decision makers
2 Gathering data
3 Assumptions
2
Creating a mathematical model
1 What are the decision variables? Constraints?
2 How is uncertainty being taken into account?
3
Solving the mathematical model
1 This course–optimization algorithms: can we find the
optimal solution efficiently? If not, how close can we come?
Spring 2014
SA305
16
Keys What is OR?
doomed to repeat
A general OR approach
1
Problem definition and data gathering
1 Discussing problem with decision makers
2 Gathering data
3 Assumptions
2
Creating a mathematical model
1 What are the decision variables? Constraints?
2 How is uncertainty being taken into account?
3
Solving the mathematical model
1 This course–optimization algorithms: can we find the
optimal solution efficiently? If not, how close can we come?
4
Interpreting and implementing the model
1 Do the results make sense?
2 How do we interpret the results?
3 How can we relax our assumptions?
Spring 2014
SA305
16
Keys What is OR?
doomed to repeat
A general OR approach
1
Problem definition and data gathering
1 Discussing problem with decision makers
2 Gathering data
3 Assumptions
2
Creating a mathematical model
1 What are the decision variables? Constraints?
2 How is uncertainty being taken into account?
3
Solving the mathematical model
1 This course–optimization algorithms: can we find the
optimal solution efficiently? If not, how close can we come?
4
Interpreting and implementing the model
1 Do the results make sense?
2 How do we interpret the results?
3 How can we relax our assumptions?
Repeat!
Spring 2014
SA305
16