Mathematics of Operations Research
John E. Mitchell
Department of Mathematical Sciences
RPI, Troy, NY 12180 USA
February 8, 2012
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Operations Research
Art&Sci of Math
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Outline
1
Courses
2
What is OR?
3
Optimization
4
Probability and Statistics
Mitchell ()
Operations Research
Art&Sci of Math
2 / 17
Courses
Outline
1
Courses
2
What is OR?
3
Optimization
4
Probability and Statistics
Mitchell ()
Operations Research
Art&Sci of Math
3 / 17
Courses
Courses in Operations Research
Required:
MATP 4700 Math Models of Operations Research
OR Option: (take at least one)
MATP 4600 Probability Theory and Applications
MATP 4820 Computational Optimization
Also available:
MATP 4620 Mathematical Statistics
Mitchell ()
Operations Research
Art&Sci of Math
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Courses
Prerequisites
MATP 4600 Probability Theory and Applications:
Calculus II.
The level of rigor in the course depends on the instructor.
MATP 4700 Math Models of Operations Research:
Need the matrix algebra in MATH 2010.
Emphasis is on algorithms for solving optimization problems.
MATP 4620 Mathematical Statistics:
Requires MATP 4600.
MATP 4820 Computational Optimization:
Requires MATP 4700. Uses MATLAB.
Mitchell ()
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What is OR?
Outline
1
Courses
2
What is OR?
3
Optimization
4
Probability and Statistics
Mitchell ()
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What is OR?
Popular culture
Mitchell ()
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What is OR?
Overview
“Operations research is the application of advanced analytical
methods to help make better decisions.
O.R. has contributed billions of dollars in benefits and savings to
corporations, government, and the nonprofit sector.” (informs.org)
Enable consideration of all available options.
Careful predictions of outcomes and estimates of risk.
One current challenge: making sense of huge data sets.
Mitchell ()
Operations Research
Art&Sci of Math
8 / 17
Optimization
Outline
1
Courses
2
What is OR?
3
Optimization
4
Probability and Statistics
Mitchell ()
Operations Research
Art&Sci of Math
9 / 17
Optimization
The Traveling Salesman Problem
TSP Webpage
Bill Cook, Georgia Tech
Robert Bosch
Oberlin College
Find shortest route that visits each city exactly once.
Largest problem solved to date has more than 85000 cities.
Eg: VLSI, vehicle routing (UPS, school buses,...)
Mitchell ()
Operations Research
Art&Sci of Math
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Optimization
The Diet Problem
Find the cheapest combination of foods that will satisfy all your
nutritional requirements.
Results from the
NEOS Server
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Optimization
Portfolio Optimization
expected
return
Minimize the risk in an
investment portfolio,
subject to achieving a
certain return.
optimal
risk
Many investment companies are now using optimization extensively to
decide how to allocate assets.
Improved hardware and software has enabled the solution of far larger
problems, taking some of the guesswork out of the allocation of assets.
Mitchell ()
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Optimization
Crew scheduling
Assign crews to flights.
ORD
- s JFK
:s
]
J
SFO
Ensure each flight covered.
J
s J
OC
Meet regulations.
J sATL
CsX
LAX
*
B
i
XXX
Xz
Minimize costs.
XIAH
B
s NB s
MIA
Would like a robust schedule.
The airlines run on small profit margins, so saving a few percent
through good scheduling can make an enormous difference in terms of
profitability.
They also use linear programming for yield management.
Mitchell ()
Operations Research
Art&Sci of Math
13 / 17
Optimization
Manufacturing and transportation
For example, an oil company has oil fields in Saudi Arabia and
Borneo, refineries in Japan and Australia, and customers in the US
and New Zealand.
The fields produce different qualities of oil, which is refined and
combined into different grades of gasoline. What raw oil should be
shipped to which refinery? How much of each type of gasoline should
each refinery produce?
Mitchell ()
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Optimization
Telecommunications
Call routing: Many telephone calls from New York to Los Angeles,
from Houston to Atlanta, etc. How should these calls be routed
through the telephone network?
Network design: If we need to build extra capacity, which links should
we concentrate on? Should we build new switching stations?
Mitchell ()
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Probability and Statistics
Outline
1
Courses
2
What is OR?
3
Optimization
4
Probability and Statistics
Mitchell ()
Operations Research
Art&Sci of Math
16 / 17
Probability and Statistics
Probability and Statistics
benefit
How do we handle
uncertainty?
Eg: investment returns, travel
times, efficacy of medicine,
...
dose
Mitchell ()
Operations Research
Art&Sci of Math
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Probability and Statistics
Probability and Statistics
benefit
How do we handle
uncertainty?
Eg: investment returns, travel
times, efficacy of medicine,
...
dose
Regression: find best model for the data. Eg: how does a medical
outcome depend on drugs and other aspects of care?
Can then use this to design improved medicines;
what aspects of current medicines contribute most to efficacy?
Mitchell ()
Operations Research
Art&Sci of Math
17 / 17