Math 319: Math Modeling
Prof. Andrew Ross
Note: this syllabus is temporary, and may change up to the first day of class.
This version posted on: 2017-09-05
Math Modeling is the art of taking a real-world problem and stating it
in mathematical terms. It often involves making simplifying assumptions.
In our class, we get in the habit of doing all the parts of the
math modeling cycle: modeling, solving, checking, and guessing.
Often, a large part of the problem is even deciding which problem to solve.
For example, should you find the best schedule for your staff at one
location, or consider opening new locations? Should you
start with a theoretical model then match it to data, or just model
the data directly? We will also consider a lot of common
mathematical models, and explore their properties.
Course Catalog Entry
The modeling process; model building and evaluation, techniques of
modeling; model fitting and models requiring optimization; empirical
model construction---experimental models, dimensional analysis, simulation
models, dynamic models; use of derivatives in the modeling process,
single and multivariable dynamic models.
Math 120 and Math 122, but really we don't use them very heavily
Some experience using Excel, Python, R, VBA, Mathematica, Maple, or Matlab will also
be VERY helpful, but it is not strictly a prerequisite.
Follow-up courses: Math 325 Differential Equations, Math 418 Modeling with Linear Algebra, Math 419W Introduction to Stochastic Mathematical Modeling, Math 425 Math for Scientists, Math 436 Numerical Analysis
The U of M has two related courses: Math 462, "Mathematical
Modeling", and Math 463, "Mathematical Modeling in Biology". However,
focus on differential equation models, while this class focuses on
regression, operations research, and dynamical systems.
Section 0, CRN 11093: Tue/Thu 12:30-1:45 in Pray-Harrold 324
Brief schedule overview:
3 credit hours.
- Wed Sep 6 : First day of our class
- Thu Nov 2: Proposal 1 due
- Thu Nov 16: Project 1 due, presentations start
- Wed Nov 22: Thanksgiving break, no classes
- Tue Dec 5: Proposal 2 due
- Tue Dec 19: Final Presentations (instead of exam), 11:30 (AN HOUR EARLY!); Project 2 due
Class meetings will be mostly interactive lectures, with some time to work on
problems in class, and some time to go over problems from the homework.
You should bring a laptop to most class sessions, since I couldn't get a computer lab for us. If you don't have a laptop, buddy up with someone who does.
I expect that you will work on Math 319 for 6 to 9 hours per week outside of class
during a regular (Fall or Winter) semester, and 2 times that during a Summer semester (7.5-week session)
Professor Andrew Ross
(734) 487-1658, but I strongly prefer e-mail instead of phone contact.
Math department main office: Pray-Harrold 515, (734) 487-1444
Office Hours and other help
Here is my complete schedule.
9:30-10:45 Stat 360-0, Pray-H 405
10:45-12:00 office hours
1:30-3 grant work
3-4 grant meeting (Wed)
busy all morning.
11:30-12:30 Office hours and lunch
12:30-1:45 Math 319, Pray-H 324
2:00-3:15 Stat 319, Pray-H 324
3:15-4:15 Office Hour
4:15-5:00 Student research meeting (K)
10-11 Student research meeting (P)
once per month, busy 11am-2:30pm with department meetings
4-5 Student research meeting (N)
I am also happy to make appointments if you cannot come to the general
office hours. Please send me e-mail to arrange an appointment.
The Mathematics Student Services Center (or "Math Lab") is also here to
help you, in Pray-Harrold 411
Their hours are posted here.
Please give them a call at 734-487-0983 or just drop by.
A good place to study, if the Math Lab doesn't suit you, is the Math Den, Pray-Harrold room 501.
Another resource on campus is the
Holman Success Center, formerly the Holman Learning Center.
Many assignments in this course will be in the form of papers, which I
want to be well written. Please consult with
The Writing Center
for help in tuning up your writing.
Teaching philosophy, interests
I am a very applied mathematician. Applied, applied, applied. Not pure. Impure.
I try to focus on real-world problems, rather than artificial drill problems (though
I do recognize the need for some drill). My classes spend much more time on formulating
problems (going from the real world to math notation and back) than on proving theorems.
If you want the theoretical basis for anything we are discussing, please ask!
My general math interests are in Industrial Engineering and Operations Research (IEOR).
In particular, I do research in applied probability and queueing theory,
the mathematics of predicting how long
it takes to wait in line for service. You can learn more about this in Math 319 and 419
when I teach them. I also enjoy teaching about cost-minimizing/profit-maximizing methods
called Non-Linear Programming (NLP) in Math 560.
Most students do well in this course without a textbook. For those who
feel the need to have one just in case, I suggest finding
"A First Course in Mathematical Modeling", any edition, by
Giordano, Weir, and Fox, in a library or the Math Den (PH 501).
A lot of our work will be done on computers, specifically in Excel or other spreadsheet software (except Apple Numbers). If you
had been waiting for a good reason to buy a laptop, this is it.
Spreadsheets other than Excel (such as OpenOffice/LibreOffice, Google Docs, etc.) work reasonably well for most things in the class, but some things really don't work well without name-brand Excel. Fortunately, it's available free to EMU students (as of Fall 2016). Email me to ask for details.
Course Web Pages
I will post data files, homework assignment files, etc. on
my home page.
We will use an on-line gradebook (via
EMU Canvas) to keep track of grades.
You are expected to keep an eye
on your scores using the system, and get extra help if your scores
indicate the need.
Nearly everything will be submitted via the various dropboxes inside EMU Canvas. The rule is: if it's not in a dropbox, it doesn't exist (for grading purposes).
Here is a list of books that I have found interesting and related to
math modeling. Perhaps some of them will strike your fancy, too.
I own the ones that are starred (*) and can lend them to you.
Others you will have to find at the library or on the usual Internet
booksellers. Links are given to Amazon, but I do not specifically endorse
them or any particular bookseller. Of course, if you like a book you can
see what similar books the online bookseller recommends.
Here are some journals that you might be interested in:
- Mathematical Modeling: A Comprehensive Introduction, Gerhard Dangelmayr and Michael Kirby, a free e-text!
- Optimization Modeling with Spreadsheets, Second Edition
by Kenneth R. Baker, 2011 (EMU library has electronic access)
- An Introduction to Linear Programming, Stephen J. Miller, 2007 (free on the web)
- An Introduction to Statistical Learning, with Applications in R by James, Witten, Hastie and Tibshirani (Springer, 2013) (free on the web)
- Practical Optimization by John Chinneck (free on the web)
- Invitation to Dynamical Systems, Edward R. Scheinerman (free on the web)
- Elementary Mathematical Modeling: A Dynamic Approach/ Sandefur, James
- *Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics by Robert B. Banks, and its sequel:
- *Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics by Robert B. Banks
- Race car vehicle dynamics by William F. Milliken, Douglas L. Milliken
- Going Faster: Mastering the Art of Race Driving by Carl Lopez
- Logistics Systems Analysis by Carlos Daganzo
- The nature of mathematical modeling by Neil Gershenfeld
- Mathematical and Experimental Modeling of Physical and Biological Processes by Banks and Tran
- *Small is Profitable by Amory B. Lovins, et al.
- Statistical Orbit Determination by Tapley and Schultz
- Street-Fighting Mathematics (free online book!)
- Applied Mathematical Programming by Bradley, Hax, and Magnanti (Addison-Wesley, 1977) (free online book! A bit old, though).
- Beasley's O.R. Notes (free!)
- Quantitative Methods in Health Care Management: Techniques and Applications, by Yasar A. Ozcan (EMU library)
- Patient Flow: Reducing Delay in Healthcare Delivery, edited by Randolph W. Hall (EMU Library, incl. electronic copy)
- *Queueing Methods: for Services and Manufacturing by Randolph W. Hall
- Mathematical Modeling for the Life Sciences, by Jacques Istas
- Turning Numbers into Knowledge: Mastering the Art of Problem Solving
by Jonathan G. Koomey (EMU library, incl. electronic copy)
- *The Active Modeler: Mathematical Modeling with Microsoft Excel by Erich Neuwirth
- * Mathematical Modeling With Excel by Brian Albright
- Dynamic Modeling and Control of Engineering Systems, 3rd edition, Kulakowski, Gardner, and Shearer
- Guerilla data analysis using Microsoft Excel
- *Modeling for Insight: A Master Class for Business Analysts by Powell and Batt
- Management Science: The Art of Modeling with Spreadsheets by Powell and Baker
- Methods of Operations Research (1951) by Morse and Kimball--especially for military applications
- Military operations research (1997) By N. K. Jaiswal
- Applied operations research (1988) By Ronald William Shephard--especially for military applications
- Mathematics and Sports edited by Joseph A. Gallian
- Excel for scientists and engineers : numerical methods / E. Joseph Billo.
- Guerilla data analysis using Microsoft Excel / Jelen, Bill
- Handbook of nonlinear regression models / David A. Ratkowsky
- Nonlinear regression modeling : a unified practical approach / David A. Ratkowsky.
- Nonlinear regression / G.A.F. Seber and C.J. Wild.
- Applied logistic regression By David W. Hosmer, Stanley Lemeshow
- Microsoft Excel, or other spreadsheet software like Gnumeric or OpenOffice or Google Docs
- Python or R is used for only a few people's projects--most people don't use them.
- Mathematica, Maple, or Matlab/Octave/Scilab are even more rarely used in this class.
Our primary goal is to teach you to be a good (or great!) math modeler.
To be a good modeler, you need:
We have a few secondary goals, which may be more or less applicable to your
- Good habits and procedures, just like a scientist, and
- Knowledge of common math models.
- Get enough people together to form a few teams for the
Math Contest in Modeling (MCM), Feb 8-12, 2018.
I participated in this 3 times as an undergrad and had a lot of fun. Recent EMU teams have done well!
- Give future teachers some great ideas to
show your kids how high-power math is used in the real world. You may enjoy reading Meaningful Math.
- Give computer-science students lots of interesting things to program. You may like reading this blog entry about math for programmers.
- Teach you how to communicate your math models by writing math papers and giving math presentations.
By the end of the course, students will be able to:
(optional topics that we might not get to are marked with a ?)
Also compare this list of outcomes to the
CUPM 2015 course guide for math modeling.
- (General modeling skills):
- categorize problems into operational/tactical/strategic categories,
- identify nearby problems in the oper./tact./strat. hierarchy,
- evaluate models by constructing simple test cases,
- conduct cross-validation when needed,
- select the most important variables to start modeling with,
- (Empirical modeling skills):
- use ordinary, semilog, and loglog plots to evaluate relationships in data sets,
- perform linear regression in software,
- interpret the correlation coefficient,
- perform transformations before regression as appropriate,
- perform multiple variable linear regression in software,
- fit sine/cosine functions to data using multiple linear regression (simplified Fourier),
- fit a function to data using nonlinear regression,
- decide when to use logistic regression (logit), and interpret the results
- (Communication skills):
- write a technical report,
- differentiate between literature of varying quality, e.g. peer-reviewed vs. working paper vs. white paper vs. web site,
- design appropriate figures to communicate models and results,
- (Optimization skills):
- Formulate non-linear programs (NLP) as appropriate,
- solve NLP using software,
- describe the (im)possibility of multiple optimal solutions (convexity/concavity)
- formulate linear programs (LP) as appropriate,
- solve LP using software,
- describe the nature of LP solutions,
- identify common LP models: network flow, diet/blending, inventory, assignment, (minimax ?)
- formulate Integer programs (IP) as appropriate,
- identify common IP models: knapsack, scheduling, (fixed charge ?)
- (Dynamical Systems skills):
- use Dynamical Systems to model:
- single-variable, incl. financial models (credit cards, mortgages),
- single-variable limited population growth/ carrying capacity
- two-population: predator/prey, competition, cooperation
- age-structured populations (Leslie models)
- Markov-chain models
- Fit model parameters to data,
- Describe equilibrium behavior,
- Implement and interpret x=time plots, phase-plane plots, and delta-a_n versus a_n plots
- (Other models):
- (?) describe basic Queueing models,
- describe the Traveling Salesperson problem (TSP)
- (?) describe project-scheduling models (PERT)
- (?) describe dynamic-programming models (DP)
See the nicely printable 1-page schedule.
This course was originally organized around the Giordano modeling textbook, though it is not required for the course.
Here we show which chapters from that book we cover,
in roughly the order we will cover them. A star (*)
denotes full coverage, a plus (+) denotes partial coverage, and no
symbol denotes no coverage. For example, DTMCs (as cool as they are)
will be covered in Math 419 rather than 319.
Ch 2:+ proportionality, similarity
Ch 3:* model fitting, least-squares
Ch 4:+ experimental modeling, high-order polynom, low-order polynom, splines
Ch 5:+ simulation
Ch 6: Discrete Time Markov Chains (DTMCs)
Ch 8:+ modeling using graph theory
Ch 7:+ Linear Programming (LP), one-dim. line search
(and add Integer Programming?)
Ch 13:* Non-Linear Programming (NLP), inventory
Ch 9:+ dimensional analysis and similitude
Ch 10: graphs of functions as models
Ch 1:* difference equations, dynamical systems
Ch 11:+ one-dim ODEs
Ch 12:+ systems of ODEs
Some variations in this outline are to be expected.
Regular attendance is strongly recommended. Since there is no formal textbook, missing class means you will miss a lot!
If you must miss a class,
arrange to get a copy of the notes from someone, and arrange for
someone to ask your questions for you.
My lectures and discussions mostly use the document camera, along with
demonstrations in Excel and other mathematical software. I do not
usually have PowerPoint-like presentations, and thus cannot hand
out copies of slides.
Homework will be assigned about once per class meeting, though some assignments are short. It will
sometimes be a small problem set designed to help you
understand the behavior of math models. Other times, it will involve
writing up a little paper on an assigned topic. All homework should
be typed unless noted.
Homework papers should be submitted on-line, where they might be
checked by TurnItIn or a similar
service. This is partly to help keep you honest, and partly to help you
learn acceptable ways to cite the work of others. A side benefit is that
sometimes TurnItIn finds papers relevant to your work that you would not
have found otherwise!
There will be no exams, unless the class demonstrates an unwillingness
to be motivated any other way.
Instead of a mid-term and a final exam, you will do a mid-term and a final
project. Your results will be reported in a paper and a presentation to
the class. You may work by yourself or in a team of 2 people, but no
groups larger than 2 will be allowed. You may switch project partners
at your will. Your project grades will each be split something like this:
The final presentations will be made during the
time slot reserved for the final exam.
- 10 pct: proposal
- 80 pct: work and written report
- 10 pct: presentation
On average, students should spend a total of about 30 minutes in office hours
discussing the project. Plan for this in advance!
No scores will be dropped, unless a valid excuse with
evidence (as needed) is given.
In the unfortunate event of a medical need, the appropriate grade or
grades might be dropped entirely, rather than giving a make-up, at the instructor's discretion.
You are highly encouraged to still complete the relevant assignments
and consult with me during office hours to ensure you know the
Your final score will be computed as follows:
Final percentage scores will be given letter grades as follows:
- 50 percent for all the homework together,
- 20 percent for the mid-term project, and
- 30 percent for the final project.
- 92.0 and above : A
- 88.0 to 92.0: A-
- 84.0 to 88.0: B+
- 80.0 to 84.0: B
- 76.0 to 80.0: B-
- 72.0 to 76.0: C+, etc.
The instructor reserves the right to make changes to this syllabus
throughout the semester. Notification will be given in class or
by e-mail or both. If you miss class, it is your responsibility
to find out about syllabus and schedule changes, especially
the due dates and times of projects, assignments, or presentations.
Advice from Other Math Modeling Students
In the last two semesters, I've asked my math modeling students to give
advice to you, future math modeling students, based on their experiences
in my course. Here are some of the highlights:
- * work in groups * start the first day assignment is given *
don't take too many credits w/ this class * ask a lot of questions *
utilize Dr. Ross
Do go to his office hours more than you normally would; if you have a question ask don't wait.
See Prof. Ross in office hours and don't be afraid to email him. He is usually very helpful and approachable.
Plan on visiting Prof. Ross during office hours in order to do well in
the class. You will learn a lot in the end, but be ready to work.
- [prof ross:] add a note to the syllabus stating something
to the effect of, "This class will not be like other math classes.
Instead of straight-up problems or proofs, the biggest amount of work
will be setting up the models, exercises, etc. and in analysing what
your results mean. It will not be the mathematical work done to obtain
the results that is the tricky part." But word the note better.
- attend the office hours Prof Ross is really good at explaining & helping out with the homework
- WORK TOGETHER!
- Take notes during the computer lab days and send yourself the excel sheets.
- Go to class. The computer lab days help even if you know excel well.
Go to class. Go to office hours and pick project that you're energized
about and interested in even if they're harder. It will make this math
class the best one you've ever taken.
- Don't drop the class! It sounds impossible in the beginning, but stick with it.
- Don't procrastinate.
- Take differential equations close to this class, it will make more sense!
- Start projects ASAP.
- Ask questions!!! The professor will guide you along the way like Yoda.
- Talking to anyone about your projects or the homework, be it Prof. Ross or other students, is a really, really good idea.
- Never be afraid to ask for help.
- If project falls through, have backups.
University Writing Center
The University Writing Center (115 Halle Library; 487-0694) offers one-to-one writing consulting for both undergraduate and graduate students. Students can make appointments or drop in between the hours of 10 a.m. and 6 p.m. Mondays through Thursdays and from 11 a.m. to 4 p.m. on Fridays. The UWC opens for the Fall 2017 semester on Monday, September 11, and will close on Thursday, December 14. Students are encouraged to come to the UWC at any stage of the writing process.
The UWC also has several satellite locations across campus (in Owen, Sill, Marshall, Pray-Harrold, and Mark Jefferson). These satellites provide drop-in writing support to students in various colleges and programs. The Pray-Harrold UWC satellite (rm. 211) is open Mondays through Thursdays from 11 a.m. to 4 p.m. The locations and hours for the other satellites can be found on the UWC web site: http://www.emich.edu/uwc
UWC writing consultants also work in the Academic Projects Center (116 Halle Library), which offers drop-in consulting for students on writing, research, and technology-related issues. The APC is open
10 a.m. to 5 p.m. Mondays through Thursdays. Additional information about the APC can be found at http://www.emich.edu/apc.
Students seeking writing support at any location of the University Writing Center should bring a draft of their writing (along with any relevant instructions or rubrics) to work on during the consultation.
Standard University Policies
In addition to the articulated course specific policies and expectations, students are responsible for understanding all applicable University guidelines, policies, and procedures. The EMU Student Handbook is the primary resource provided to students to ensure that they have access to all university policies, support resources, and student's rights and responsibilities. Changes may be made to the EMU Student Handbook whenever necessary, and shall be effective immediately, and/or as of the date on which a policy is formally adopted, and/or on the date specified in the amendment. Please note: Electing not to access the link provided below does not absolve a student of responsibility. For questions about any university policy, procedure, practice, or resource, please contact the Office of the Ombuds: 248 Student Center, 734.487.0074, email@example.com, or visit the website: www.emich.edu/ombuds
CLICK HERE to access the University Course Policies
Student Handbook Link: https://www.emich.edu/studenthandbook/index.php
Graduate School Policies: http://www.emich.edu/graduate/policies/index.php
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