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Mathematics of Operations Research
John E. Mitchell
Department of Mathematical SciencesRPI, Troy, NY 12180 USA
February 8, 2012
Mitchell () Operations Research Art&Sci of Math 1 / 17
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 ApplicationsMATP 4820 Computational Optimization
Also available:MATP 4620 Mathematical Statistics
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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.
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What is OR?
Outline
1 Courses
2 What is OR?
3 Optimization
4 Probability and Statistics
Mitchell () Operations Research Art&Sci of Math 6 / 17
What is OR?
Popular culture
Mitchell () Operations Research Art&Sci of Math 7 / 17
What is OR?
Overview
“Operations research is the application of advanced analyticalmethods to help make better decisions.O.R. has contributed billions of dollars in benefits and savings tocorporations, 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 WebpageBill Cook, Georgia Tech
Robert BoschOberlin 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 10 / 17
Optimization
The Diet Problem
Find the cheapest combination of foods that will satisfy all yournutritional requirements.
Results from theNEOS Server
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Optimization
Portfolio Optimization
Minimize the risk in aninvestment portfolio,subject to achieving acertain return.
risk
expectedreturn
optimal
Many investment companies are now using optimization extensively todecide how to allocate assets.Improved hardware and software has enabled the solution of far largerproblems, taking some of the guesswork out of the allocation of assets.
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Optimization
Crew scheduling
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sORD -Assign crews to flights.
Ensure each flight covered.
Meet regulations.
Minimize costs.
Would like a robust schedule.
The airlines run on small profit margins, so saving a few percentthrough good scheduling can make an enormous difference in terms ofprofitability.
They also use linear programming for yield management.
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Optimization
Manufacturing and transportation
For example, an oil company has oil fields in Saudi Arabia andBorneo, refineries in Japan and Australia, and customers in the USand New Zealand.
The fields produce different qualities of oil, which is refined andcombined into different grades of gasoline. What raw oil should beshipped to which refinery? How much of each type of gasoline shouldeach refinery produce?
Mitchell () Operations Research Art&Sci of Math 14 / 17
Optimization
Telecommunications
Call routing: Many telephone calls from New York to Los Angeles,from Houston to Atlanta, etc. How should these calls be routedthrough the telephone network?
Network design: If we need to build extra capacity, which links shouldwe concentrate on? Should we build new switching stations?
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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
How do we handleuncertainty?
Eg: investment returns, traveltimes, efficacy of medicine,. . .
dose
benefit
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Probability and Statistics
Probability and Statistics
How do we handleuncertainty?
Eg: investment returns, traveltimes, efficacy of medicine,. . .
dose
benefit
Regression: find best model for the data. Eg: how does a medicaloutcome 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?
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