SM223 – Calculus III with Optimization
Asst. Prof. Nelson Uhan
Fall 2012
Lesson 30. Introduction to Operations Research and Optimization
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
–
–
–
–
–
–
–
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military logistics in World War II
manufacturing
operating systems
logistics
airline pricing
communications
finance
marketing
• We’ll talk about some applications later
The traveling salesperson problem
• A saleswoman located in Annapolis wants to visit all 48 state capitals of the continental United States
to 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
• 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 Indianapolis
• What about 13,509 cities in the US?
• Sophisticated mathematical techniques are our best bet
1
The OR approach
Problem statement
Solution
C A N A D
A
Olympia WASH
INGTO
H
M ONTANA
Bismarck
M
M INNESOTA
ON
Salt Lake
NEVAD
A
C
I F I
P A C
Carson
City
Sacrame
nto
CALI
FORN
Cheyenne
City
UTAH
IC
RU
OC
EA
SS
OCE
Oklahoma City
Boston
YORKAlbany
WEST
VIRGINIA
DC
Little Rock
Austin
M ARYLAND
Richmond
M
ALASKA
C
A
N
A
D
G
A
RI N
SE A
GU L F
0
AN
A
CAROLI
PSHIRE
SETTS
ISLAND
ICUT
NA
Columbia
SOUTH
CAROLINA
GEORGIA
Montgomery
J ackson
Tallahassee
Baton Rouge
IA
N
OCE
U.S. Departmentof the Interior
U.S. Geological Survey
NORTH
Atlanta
ALABAM
J ERSEY
NEW
DELAWARE
IA
HAM
RHODE
CONNECT
Annapolis
on
Washingt
VIRGIN
ONT
M ASSACHU
ce
Providen
Hartford
Trenton
Dover
ARKANSAS
LOUISIANA
AN
NEW
Concord
io
NEW
Raleigh
BE
IC
ta r
A
YLVANI
PENNS Harrisburg
Charleston
Nashville
N
E A
O C
IF
100 mi
Columbus
Indianapolis
TENNESSEE
OKLAHOM A
NEW M EXICO
M ISSISSIPPI
100 km
CI F
ke O n
ie
KENTUCKY
Santa Fe
A
Phoenix
Honolulu
PAC
PA
INDIANA
Frankfort
TEXAS
0
Lake Michigan
ILLINOIS
M ISSOURI
J efferson City
KANSAS
ARIZON
ARC T I C
HAWAII
0
VERM
Er
OHIO
Springfield
Topeka
M AINE
a
August
Montpeli
L
e
ak
Des Moines
Lincoln
COLORADO
KA
La
Lansing
IOWA
NEBRASKA
Denver
IA
N
Madison
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 Indianapolis?
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
Calculus,
Algorithms
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
∑{i , j}∈δ(S) x{i , j} ≥ 2 ∀S ⊆ N
Mathematical model
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 ...
Sports scheduling
• ACC Basketball earns over $30 million in revenue annually, almost all from TV and radio
• TV networks need a steady stream of “high quality” games, NCAA rules, school preferences and traditions
• Decision:
• Objective:
2
• Constraints:
• Optimization approaches yields reasonable schedules very quickly
Radiation therapy
• High doses of radiation can kill cancer cells and/or prevent them from growing and dividing
• Can also kill healthy cells!
• Radiation can be delivered at different angles and intensities
• Decision:
• Objective:
• Constraints:
• Many successes reported using different types of optimization models
My first optimization model
• Let’s try to build a simple optimization model
The Anteater-Bugs Corporation
[Adapted from MIT 15.053, Spring 2007]
• The Anteater-Bugs Corporation has two main brands of beer: Bugwheezer and Bug-Lite
• Each of these products contains two main ingredients: malt and hops
• Beer production also requires some labor
• Ann Anteater is in charge of beer production for October
3
• For October:
Ingredient / Beer
Malt (grams/bottle)
Hops (grams/bottle)
Profit ($/bottle)
Bugwheezer
10
20
1
Bug-Lite
3
12
2
Total available
1000
2000
What goes into an optimization model?
1. Determine the decision variables
2. Write the objective function (and goal) in terms of the decision variables
3. Write the constraints in terms of the decision variables
• Get something that looks like this:
minimize or maximize
subject to
(objective function)
(constraints)
Step 1: Determine the decision variables
• The decision variables in an optimization model represent the decisions to be taken, for example
– “How much of product A should be produced?”
• Decision variables should completely describe the decisions to be made
• Ann needs to determine the Anteater-Bugs’s October production quantities of Bugwheezer and BugLite
•
•
Step 2: Write the objective function
• The objective function of an optimization model quantifies the quality of the decisions described by
the decision variables, for example
– total cost of producing products A, B and C
• The objective function can be maximized or minimized (the goal)
• The Anteater-Bugs Corporation wants to maximize profits
4
• Assume all bottles produced are sold
• If W bottles of Bugwheezer and L bottles of Bug-Lite are produced, what is the profit?
• Objective function + goal:
Step 3: Write the constraints
• Constraints specify the the values that decision variables can take through equalities and inequalities
• If W bottles of Bugwheezer and L bottles of Bug-Lite are produced, how much malt is used?
• This quantity needs to be less than 1000
• This constraint can be written as
• Similar constraint for hops:
• Can we produce a negative number of beer bottles?
• Can we produce a fractional number of beer bottles?
• Nonnegativity:
• Integrality:
5
Anteater-Bugs’s Optimization Model
• Putting this all together...
maximize
subject to
1W + 2L
10W + 3L ≤ 1000
20W + 12L ≤ 2000
W ≥0
L≥0
W integer
L integer
Goals for this course
• Formulate very simple models
• Use calculus to solve even simpler models
Next time...
• The calculus of optimization
6
(total profit)
(malt capacity)
(hops capacity)
(nonnegativity)
(integrality)