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Should a football team run or pass? A game theory and linear programming approach

Last week I visited Oberlin College to deliver the Fuzzy Vance Lecture in Mathematics (see post here). In addition, I gave two lectures to Bob Bosch’s undergraduate optimization course. My post about my lecture on ambulance location models is here.

My second lecture was about how to solve two player zero-sum games using linear programming. The application was a sports analytics application of whether a football team should run or pass. The purpose of the lecture was to learn about zero-sum games (it was a new topic to most students) and learn how to solve zero-sum games with two decision-makers using linear programming.

This lecture tied into my Badger Bracketology work, but since I do not use optimization in my college football playoff forecasting model, I selected another football application.


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integer programming for locating ambulances

Last week I visited Oberlin College to deliver the Fuzzy Vance Lecture in Mathematics (see post here). In addition, I gave two lectures to Bob Bosch’s undergraduate optimization course. I will post my materials for both of my lectures on my blog. The first lecture was related to my evening talk and focused on ambulance location models and modeling integer programs.

The purpose of the lecture was to work on modeling in integer programming. We focused on coverage models and worked through two of the three models that successively lift simplifying assumptions (in a 75 minute lecture). The “Integer Programming Bag of Tricks” on slide 18 contains a series of constraints for modeling conditional constraints (courtesy of Jeff Linderoth and Jim Luedtke). We use these tricks to assign at least L calls for service (demand) to stations–but only stations that are “open”–in the modeling exercise. Slides are below.

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Punk Rock OR goes to Oberlin College

This week I visited Oberlin College to deliver the Fuzzy Vance Lecture in Mathematics. I was honored to be the 20th Fuzzy Vance lecturer. Each year, Oberlin invites one mathematician (or an operations researcher/fake mathematician in my case!) to visit campus, participate in classes, and give a lecture (the “Fuzzy Vance Lecture”) to the general public.

My evening talk to the public was about my research in emergency medical services and emergency response. My slides and some pictures from my visit are below. I will post my teaching materials on my blog next week.

Oberlin is a small liberal arts college that attracts intelligent students who have eclectic interests. Many students are interested in music, creative writing, and computer science in addition to math. I enjoyed meeting with students when I taught Bob Bosch’s undergraduate course in optimization, which mostly has students from math and computer science.

Bob Bosch and his colleagues in the Oberlin math department were fantastic hosts. They filled me in on the history of the Fuzzy Vance Lecture series, but there was some disagreement about whether Fuzzy Vance was actually fuzzy (nicknamed for fuzzy hair or for another mysterious reason). I was also pleasantly surprised to learn that Oberlin is known for its unusual albino squirrel population. I am a fan of campus squirrels: the squirrels at my alma mater have had an interesting history.

Here are some memories from my visit.


The poster for the Fuzzy Vance Lecture Series in Mathematics

The poster for the Fuzzy Vance Lecture Series in mathematics



Rock and Roll Hall of Fame advertisements were everywhere in Cleveland. I was thrilled to be able to visit the museum during my visit.


Bob Bosch and I found The Clash exhibit at the Rock and Roll Hall of Fame.


My favorite crosswalk in Oberlin, which boasts one of the best music conservancies in the US.


Pablo Picasso, Chair and Owl (1947) from the Oberlin art museum


Claude Monet, The Red Kerchief from the Oberlin art museum

what punk rock #orms is reading

  1. When radiation isn’t the real risk. Hospital patients were evacuated after the nuclear accident at Fukushima to protect them from radiation. The evacuation led to 1600 deaths, but the radiation would have led to none. An interesting article on decision-making under uncertainty.
  2. The dad who wrote the check using “Common Core” math doesn’t know what he is talking about. Different isn’t always worse.
  3. Tallys Yunes has a model for building the best fantasy football team using optimization.
  4. Selfies are killing more people than shark attacks. And yet I’m not replacing “being eaten by a shark” with “taking a selfie” on my list of irrational fears.
  5. The unwritten rules of college. One educator argues that giving “transparent” assignments with clear rationales and grading criteria can make a significant impact in student performance, confidence, and retention. I can’t disagree.
  6. K12 schools are continuing to overlook introverts
  7. A PNAS study finds that women are just as ambitious as men but weigh the potential downsides of promotion more. This may result in a gender gap in promotions. Why is this the case that women focus on the negative? Maybe because men in general tend to take more risks and exhibit overconfidence and underestimate the negative. Interesting stuff.

improving diagnosis in health care

Five percent of adults seeking healthcare (12 million adults) have an incorrect or delayed medical diagnosis. These mistakes are costly. They account for 6 to 17 percent of adverse events at hospitals and result in death more than other types of mistakes. Most people will experience at least one of these diagnostic errors in their lifetime, sometimes with fatal consequences.

The Institute of Medicine, of the National Academy of Sciences, issued a report entitled Improving Diagnosis in Health Care that addresses diagnostic errors. The report contains many suggestions for how diagnostic healthcare errors can be reduced. Check it out. The Washington Post also has a nice article on this report.

A previous Institute of Medicine 2000 report on patient safety (“To Err is Human“) addressed other types of safety issues that involve human factors issues after a diagnosis has been made. This was a landmark report that led to many safety and quality improvements in healthcare (and great research at the ISYE department at UW-Madison!). However, diagnostic errors received little attention since the publication of this report despite being a problem.

The committee for the Improving Diagnosis in Health Care report is largely composed of medical personnel with at least one notable exception: my UW-Madison ISYE colleague Pascale Carayon at UW-Madison. To a large extent, diagnosis is a medical problem: there are thousands of conditions, many of which are rare, and it’s hard to match the correct single diagnosis with a set of ambiguous outcomes and test results. I appreciate how hard this problem is, and I’m impressed that so many doctors get it right the first time. Medical expertise is a necessary first ingredient.

But medical diagnosis is also a systems problem. Earlier I blogged about the report “Operations Research – A Catalyst for Engineering Grand Challenges” that summarized ways OR can address engineering grand challenges from the National Academy of Engineering. One of the four challenge areas in this report was “OR for human health,” and treatment and diagnostic issues fell under this area. Diagnosis is increasingly a systems issue, since diagnosis is often a function of medical tests and medical imaging. OR is good at weighing the costs and benefits of diagnosis and treatment since Type II errors are often really costly.

I’ve just made a plug for OR and medical diagnosis, but to be honest, I mainly read articles for planning treatment once a positive diagnosis has been made. One important paper in the literature develops linear programming-based machine learning techniques to improve breast cancer diagnosis (more UW-Madison research!):

Mangasarian, O. L., Street, W. N., & Wolberg, W. H. (1995). Breast cancer diagnosis and prognosis via linear programming. Operations Research, 43(4), 570-577.

Let me know of any OR work in the area of medical diagnosis in the comments. Kudos to the committee who produced the  Improving Diagnosis in Health Care report – I hope it leads to new important research in OR and industrial engineering.

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dividing up a large class into discussion sections using integer programming

I am team teaching a freshman course on engineering grand challenges with five other instructors. Each of the six instructors teaches a “theme” (mine is about systems, critical infrastructure, and logistics) to a subgroup of students. We must partition the students into six themes twice, since we repeat our themes in the second half of the semester with another group of students (we call these modules). To facilitate the assignments, students can submit a rank ordering of their top four choices. I was asked to use optimization to make the pairings. I thought about manually making the matches, but I could see that it would be a lot of work to redo the matchings if someone wanted to make some changes later on. So I put together a quick optimization model to do the work for me.

Here are the constraints:

  • Theme class sizes must be between 21 and 24 students
  • Each student should get one of their top two choices (this is not guaranteed)
  • Every student needs to be assigned to a theme in each module
  • Every student needs to be assigned to different themes across the two modules
  • Two groups of students are in first year interest groups (FIGs) and should all be assigned to the same first theme, regardless of preferences. This is a hard-but-flexible constraint (is that a thing?). These students needed to be assigned to the same theme, but I could choose any theme for them.

I had about two days to make the assignments and possibly field some requests to change the assignments. Integer programming was appealing because it would allow me to quickly make changes to the assignments and do a “what if” analysis. I could quickly see that it was going to be impossible to fill some of the themes based on student preferences. Luckily, some students did not submit preferences, so I could use these students to “pad” the assignments while maintaining a good solution. The objective function captures the quality of the assignments based on student preferences, so I would always have a feasible solution. I assigned weights for each student-theme pair as the objective function coefficients. I assigned the first choice theme a weight of 8, the second choice theme a weight of 4, a third choice theme a weight of 2, and a fourth choice theme a weight of 1. I assigned all other student-theme pairs a weight of zero. The goal was to mostly assign students to first and second preferred choices, so I chose weights that would “encourage” this. After solving the model I could easily check to see how many students were assigned to one of their top two matches (we promise to do this, if its possible).

The FIGs was a bit tricky. This constraint made it hard to eyeball a good solution and motivated the use of integer programming. I wanted to be able to manually test out different theme assignments for the FIGs and possibly have the flexibility to change the assignments really quickly if one of the instructors did or did not want a FIG. I manually selected themes for the FIG students by fixing variables and then optimized the assignment of everyone else. I could have treated the FIG assignments as a decision instead of an input.

Here are the parameters:

  • N = 138 students
  • T = 6 themes
  • M = 2 modules
  • w_{nt}: preference weight for student n and theme t (either 0, 1, 2, 4, or 8).

Here are the binary decision variables:

  • x_{ntm}: 1 if student n is assigned to theme t in module m.

The integer programming model is as follows.

maximize \sum_{n,t,m} w_{nt} x_{ntm}

subject to

21 \le \sum_n x_{ntm} \le 24, t=1,...,T,\ m=1,2

x_{nt1} + x_{nt2} \le 1, n=1,...,N, t=1,...,T

\sum_t x_{ntm} = 1,\ i=1,...,N,\ m=1,2

x_{ntm} \in \{0,1\}, n=1,...,N,\ t=1,...,T,\ m=1,2

The objective function maximizes the value of the assignments. The first constraint sets class sizes between 21 and 24. The second constraint ensures that a student is not assigned to the same theme across both modules. The third constraint ensures that each student is assigned to one of the themes in both modules. I do not require the model to assign students to one of their top two choices, and therefore, a feasible solution is always easy to find. However, I checked afterward to see if students got what they wanted (according to my results, they did!). What isn’t in this model is fixed variables associated with FIG student assignments, but that is straightforward to change.

The model was easy to set up except for setting the weights w_{nt} based on student preferences. This information came from a survey we conducted on the course management system. Downloading the data in a spreadsheet did not give us a flat file, so it took some extra parsing to get the right data. We would have needed to parse the survey data even if we made the assignments manually, so this was unavoidable. I would have liked to ensure diversity in teams somehow, but I did not have the data for this. Next time.

All in all, this was fun, and I would recommend the use of integer programming. It took me longer to write this post than to set up and solve the integer program to assign students to themes :)



healthcare in the age of analytics

INFORMS has a new volume of its Editor’s Cut that is a collection of resources for healthcare in the age of analytics [Link]. Healthcare is starting to adopt advanced analytical methods to improve health and healthcare delivery, and this volume is a starting point for learning more about analytics for healthcare. Volume resources include research articles, trade journal articles, videos, and podcasts. Here is the introductory video starring volume editor M. Eric Johnson of Vanderbilt University.

Here is the list of recent research articles about healthcare analytics: Sadly, the articles are paywalled, but you can access the articles if your institution has a subscription.

The Vital Role of Operations Analysis in Improving Healthcare Delivery
Linda V. Green

Predictive Analytics for Readmission of Patients with Congestive Heart Failure
Indranil Bardhan, Jeong-ha (Cath) Oh, Zhiqiang (Eric) Zheng, Kirk Kirksey

Feeling Blue? Go Online: An Empirical Study of Social Support Among Patients
Lu Yan, Yong Tan

Electronic Medical Records and Physician Productivity: Evidence from Panel Data Analysis
Hemant K. Bhargava, Abhay Nath Mishra

Business Analytics Assists Transitioning Traditional Medicine to Telemedicine at Virtual Radiologic
Ersin Körpeoğlu, Zachary Kurtz, Fatma Kılınç-Karzan, Sunder Kekre, Pat A. Basu

Offering Pharmaceutical Samples: The Role of Physician Learning and Patient Payment Ability
Ram Bala, Pradeep Bhardwaj, Yuxin Chen

The Weighted Set Covering Game: A Vaccine Pricing Model for Pediatric Immunization
Matthew J. Robbins, Sheldon H. Jacobson, Uday V. Shanbhag, Banafsheh Behzad

Multiregional Dynamic Vaccine Allocation During an Influenza Epidemic
Anna Teytelman, Richard C. Larson

The Digitization of Healthcare: Boundary Risks, Emotion, and Consumer Willingness to Disclose Personal Health Information
Catherine L. Anderson, Ritu Agarwal

Process Management Impact on Clinical and Experiential Quality: Managing Tensions Between Safe and Patient-Centered Healthcare
Aravind Chandrasekaran, Claire Senot, Kenneth K. Boyer

Waiting Patiently: An Empirical Study of Queue Abandonment in an Emergency Department
Robert J. Batt, Christian Terwiesch

Commentaries to “The Vital Role of Operations Analysis in Improving Healthcare Delivery”

Multilevel Simulations of Health Delivery Systems: A Prospective Tool for Policy, Strategy, Planning, and Management
Hyunwoo Park, Trustin Clear, William B. Rouse, Rahul C. Basole, Mark L. Braunstein, Kenneth L. Brigham, Lynn Cunningham

Value-in-Context of Healthcare: What Human Factors Differentiate Quality of Nursing Services?
Hironobu Matsushita, Kyoichi Kijima

Information Hang-overs in Healthcare Service Systems
Atanu Lahiri, Abraham Seidmann

Active Social Media Management: The Case of Health Care
Amalia R. Miller, Catherine Tucker

Privacy Protection and Technology Diffusion: The Case of Electronic Medical Records
Amalia R. Miller, Catherine Tucker


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