Optimization Games for the Young (Part 2)

 A couple of weeks ago, I wrote a blog post in which I tell a story that motivated me to start creating games for children and young adults to get them excited about Operations Research/Prescriptive Analytics/Optimization and STEM in general. You can download my first two games here: Lego Furniture and Pack That Bag!

Since then, I can’t stop thinking about other games to create. I have four game ideas in my head right now and today I’m back to share my third one with you: Stop the Fire! This is actually based on a research problem on which I am working right now.

I hope you have a chance to play this and my other games with your young ones. Please let me know your experience with these games and any feedback you may have toward improving them in the comments below.

Keep playing!

Optimization Games for the Young

I’ve recently volunteered to be a part of the INFORMS K-12 Outreach Sub-Committee and the story behind this post has everything to do with my goals in helping the committee in its mission.

The 8-year-old daughter of one of my student’s significant other saw her doing homework and asked what it was. My student responded “optimization,” to which the 8yo replied “what’s optimization?” My student said she wasn’t able to provide a succinct enough explanation but the girl was interested and likes math in general. Therefore, I decided to take the opportunity and try to encourage this little girl. There are too few women in STEM fields, not because of lack of ability, but because insufficient encouragement and motivation. Maybe this gesture will make no difference in this girl’s life, but maybe it will. I can’t tell for sure, but I sure am going to try each and every time one of these opportunities comes up. I put together an envelope with 3 optimization games (with a little message on the outside) for my student to take home with her:

It’s often the case that little gestures can change a person’s life. I remember very clearly the tiny things my teachers / mentors / advisors / friends did that inspired me tremendously. I’m sure these people won’t even remember what they did because to them it might have been nothing. To the person on the receiving end, however, it meant a lot. This gesture is my attempt at paying it forward. To all of those who inspired and encouraged me throughout my life, thank you.

Several friends, upon hearing about this story, asked me for the games so they could play with their kids. One could argue that these are more like puzzles than games. I see them as games because you can have several people playing together, each trying it their own way, and teaching each other, or challenging each other, which creates a back-and-forth discussion where everybody learns something.

Here are the ones I created:

Lego Furniture game: http://moya.bus.miami.edu/~tallys/games/lego-furniture.pdf

Pack That Bag! game: http://moya.bus.miami.edu/~tallys/games/pack-that-bag.pdf

The third one, I got from the puzzlor.com website: Good Burger.

I hope these games become a source of fun for you and your kids as well. Enjoy!

By the way, the Lego pieces cut out of paper work pretty well:

(Unexpected) Lessons Learned From a Prescriptive Analytics Course Refresh

I’ve been teaching the core prescriptive analytics course in our MBA program since 2005, and it was once again time for a major refresh/improvement. Therefore, this past December I went for it, and then proceeded to ask myself the following question over and over during the first half of the spring semester: “Why did you do this to yourself?” It was grueling. It was hard. It was overwhelming. However, when I look at the end result, I am really proud of it. It’s far from perfect, but I collected tons of feedback from my students (THANK YOU!) and I’m already polishing it for the next iteration (that one will be close to spotless). The goal of this post is to share a portion of the feedback I received, which turned out to contain some unexpected surprises.

To give you some context, I improved the existing homework problems, Excel files, and lecture notes (everything now is in PDFs created with LaTeX; no Powerpoint anymore!). I also created a lot of new content (including a hands-on Lego exercise and a case study I developed from scratch), especially when it comes to decision-making under uncertainty. All in all, I’d say 50% of the course is new material, and the remaining 50% is a much better version of the previous material.

Here are the lecture themes (unless otherwise noted, each lecture is a 2-hour session; there are 2 sessions per week). An asterisk * indicates a brand new lecture:

• Lecture 1: Introduction to Optimization: hands-on Lego exercise, investment, scheduling.
• Lecture 2: Mixing and Carryovers: blending, cash flow, production and inventory planning.
• Lecture 3: Networks: assignment, transshipment, shortest path.
• Lecture 4: Integer and Binary Variables: budget allocation, logical conditions, covering, fixed costs.
• Lecture 5: Sensitivity Analysis: shadow prices and reduced costs.
• MIDTERM EXAM
• Lecture 6*: Optimization Under Uncertainty: two-stage stochastic linear programming.
• Lecture 7*: Decision Analysis and Decision Trees (incl. EVPI, EVSI, risk profiles, Bayes’ Rule).
• Lecture 8*: Sequential Decision Processes: stochastic dynamic programming.
• Lecture 9* (two sessions): Monte Carlo Simulation.
• Lecture 10*: Case Discussion.
• FINAL EXAM

On my final exam, I had a question asking students to name their favorite and least favorite lecture. I wanted to see where the home run was and where the big fail was. I had a total of 76 students, and here is the final tally:

You’ll notice that the columns don’t add up to 76 as they should. Some people provided two favorites, and some people did not indicate a least favorite lecture. Here’s what I infer from the table above. (Let me know if you see something I didn’t!)

• Networks as the clear “home run” lecture: I would never have guessed that in a million years! This was so surprising to me. Some of the comments mentioned the visual aspect of it as an attractiveness. I can see that, but still…
• Stochastic DP as the clear “big fail” lecture: I can actually see that because I wasn’t very fond of the lecture myself after I taught it. It looks OK on paper, but the classroom feeling I got from it was lukewarm. I’m thinking of replacing it with a second case study to take place after the first four lectures.
• Two-stage stochastic LP as pretty much another big fail: This one I attribute to a fail on my part. The material in there is pretty good. It’s just a matter of my doing a better job motivating and explaining it (the written feedback points to its being hard to follow). I’ll keep it for sure and refine the delivery. (Two people actually loved it! :-)
• Simulation as a bit of a polarizing topic: I honestly thought the simulation class was going to be a big home run. The 11 votes in the right column are likely my fault again (based on feedback). I fully believe in the quality of the content, but I need to do a better job connecting it with the statistics course that precedes mine (I assumed too much of the students).
• Beloved decision trees: I had a lot of fun putting together and delivering the decision tree lecture, but I didn’t expect it would get so many “fave” votes. Being a new lecture, there were some hiccups that I can easily fix (the same being true for lectures 6, 8, and 9), and seeing how successful it already was gives me even more motivation.
• Case study: I had never written a full-length case study before, but got excited about doing so after an instructional session organized for the faculty at our school. Despite liking the case on paper (it combines marketing and optimization aspects), I was a bit anxious before the actual discussion took place. My students did an amazing job and performed way beyond my expectations. After seeing where they went and what issues they brought up, I feel like I can polish the case to make it even better.
• Lego exercise: Another pleasant surprise. I was worried the Lego exercise might be seen as too simple or silly. I was wrong! The feedback indicated it worked well as a motivational tool, a visual aid, and a good ice breaker (it’s the very first thing I do in Lecture 1).

I’m pretty happy overall and really enjoyed what I learned from this fave/least fave question. It can be pretty informative. Do you have any other creative ways to elicit student feedback that you find particularly helpful? If so, please share them in the comments below.

We’re Hiring: Open-Rank, Tenure-Track

Open-Rank Tenure-Track Faculty Position in Management Science

The Management Science Department at the Miami Business School invites applications for an open-rank, tenure-track faculty position to begin in the fall of 2019. Applicants with research interests in all areas of Operations Research and Analytics will be considered.

Applicants should possess, or be close to completing, a Ph.D. in Operations Research, Operations Management, Industrial Engineering, or a related discipline by the start date of employment. The Management Science Department consists of a diverse group of faculty with expertise in Statistics and Operations Research. We seek candidates whose research and teaching interests complement and strengthen our existing departmental strengths. Responsibilities include research, which is expected to lead to top-tier publications, teaching at both the undergraduate and graduate levels, and service.

Applications should be submitted by e-mail to MASrecruiting@bus.miami.edu, and should include the following: a curriculum vitae, up to three representative publications, brief research and teaching statements, an official graduate transcript, information about teaching experience and evaluations (if available), and three letters of recommendation. All applications completed by December 1, 2018 will receive full consideration, but candidates are urged to submit all required material as soon as possible. Applications will be accepted until the position is filled. Candidates who are presenting at the INFORMS Annual Meeting in Phoenix are encouraged to include the details of their session in their cover letter. Our faculty attending the meeting will try to attend these presentations and may arrange for interviews whenever possible.

Salary is competitive and commensurate with qualifications and experience. This is a nine-month appointment and summer research support is anticipated from the Business School.

The University of Miami offers a comprehensive benefits package including medical and dental benefits, tuition remission, and much more. For additional information, visit https://www.hr.miami.edu/working-at-the-u/new-employee-total-rewards/index.html. The University is an equal opportunity employer and encourages candidates regardless of gender, race, color, ethnicity, age, disability status or sexual orientation to apply.

How to Optimally Allocate “Bribe” Money

 As I sat through a 4-hour meeting the other day, I had an idea for a very useful and generic prescriptive analytics model to be used in conjunction with the results of a statistical learning study (e.g. a regression model). I can immediately see a large number of applications for this model, some of which I’ll illustrate in this post. I also believe this idea could be turned into an interesting case study (combining predictive and prescriptive analytics) for class discussion in a master’s level course (MBA or specialized MS), and I’d love to partner with someone to turn that into reality. Let me know if you’d like to pursue this! (Disclaimer: I don’t have much experience writing cases, but I want to get better at it.)

Without further ado…

You’re interested in gaining a better understanding of the likelihood, or probability , that certain agents will perform a given action that benefits you. So you hire a group of business analytics consultants who create a statistical learning model M that predicts the value of with high accuracy given an agent (and their attributes/characteristics) as input.

Those agents with high-enough probability of performing the action, say (a subjective choice), don’t concern you too much because they’re already on your side. Your focus is on those whose value of  is less than 50%. One of the outcomes of model M is a number for each agent (or group of similar agents) indicating how much of an effect giving that agent a financial incentive would have on the value of their action probability . For example, this number could be 0.1 for a given agent, indicating that they’d be 10% more likely to perform the action for each $1000 of incentive they receive. So, assuming a linear model, if their originally predicted were 35%, giving them $2000 would be enough to push them beyond your target 50% threshold. (Considering incentives are given in multiples of $1000.)

Before we go any further, let me make this more concrete with a few examples.

Example 1: You are the seller of a product. Agents are customers. The action is purchasing your product. The financial incentive is a discount. You have a budget for the overall discount you can give and you want to optimally allocate different discount amounts to different customers to bring as many of them as possible to the brink of buying your product (say, a 50% chance).

Example 2: You are a politician. Agents are voters. The action is voting for you. The financial incentive is a bribe. This happens in Brazil and likely in other countries as well. (Disclaimer: I’m not advocating that this is an ethical or moral thing to do. Like any superpower, however, mathematics can be used by the dark side as well.)

Example 3: You are a university. Agents are admitted students. The action is picking you to attend. The financial incentive is a scholarship.

I’m sure you can come up with other examples. Let me know in the comments!

So the big question is: How do we optimally allocate the limited pool of financial incentives? Optimization to the rescue! I’ll provide a link to an Excel spreadsheet below, but first let’s understand the math behind it.

Given  agents, for each agent , let  be how much the agent’s action probability (before the incentive) is below my target threshold, and let  be how much agent ‘s action probability increases for each dollars of financial incentive. In my example above,  (50% minus 35%), , and .

Let’s create two variables for each agent . Variable  is an integer number indicating how many -dollar incentives we decide to give to that agent. And variable  is a binary (yes/no) variable indicating whether or not we manage to bring agent  to our side (i.e. whether we raised his/her action probability beyond the threshold).

To indicate that our goal is to push as many agents beyond the action threshold as possible, we write

If our total budget for financial incentives is , we respect the budget with the following constraint (note that the sum of all  equals the total number of-dollar financial incentive packages given away):

Then, if we want  to be equal to 1 (meaning “yes”), we need  to be high enough for the increase in the agent’s action probability to exceed the threshold value. This can be accomplished with these constraints

In my earlier example, the above constraint would read

Therefore, unless  is at least 2, the value of  cannot be equal to 1.

That’s it! We are done with the math. Wasn’t that beautiful?

Here’s an Excel spreadsheet that implements this model for a random instance of this problem with 20 agents. It’s already set up with all you need to run the Solver add-in. Feel free to play with it and let me know if you have any questions.

Theory versus Practice, Intangibles, Intuition, and the Usefulness of Imperfection

I felt like putting in writing a few thoughts I often find myself telling my students, and hence this post. You can download a (nicer) PDF version of it here.

Theory versus Practice: A Challenge

It was a rainy Saturday afternoon…

“Ready? Go!”

As she reached for her pencil and began scribbling, I opened an empty Excel spreadsheet and started typing in the data and thinking out loud.

“Eight integer variables, three constraints, plus lower and upper bounds. How is it going over there?”

“I’m doing OK. Let me think.”

A minute later, I click “Solve” and declare victory.

“There you go: $2.80. Can’t beat that!”

“Five buns? Do you expect people to eat a burger with five buns?”

“Hmm…you’re right. One more constraint… voilà $2.62. The best burger on the market! What did you get?”

“Mine costs $2.61.”

“I win! Ha, ha!”

“But look at your burger! Nobody would ever want to eat this crap. Mine is much more appetizing.”

As I reluctantly examined the two solutions, there was no denying the obvious.

That was me and my wife solving the Good Burger puzzle. The challenge is to create the most expensive burger out of a given set of ingredients such as beef patties, cheese, lettuce, tomato, etc., while keeping sodium, fat, and calorie counts in check. Her hand-calculated solution was suboptimal by one cent. It was technically worse than mine, but what does optimality really mean in practice?

Optimality or Bust?

Once upon a time, at the board meeting of a for-profit company…

“Today, I’m excited to report the results from the cost-cutting initiative that my team of analytics experts developed over the last six months. We managed to produce a solution to our problem that improves profits by 6%.”

“But is this solution optimal?”

Said no one ever after hearing “improves profits by 6%.”

Intangibles and Intuition

The two anecdotes above illustrate an important idea that I emphatically stress to my students every spring semester: models aren’t perfect, and that’s perfectly OK. There’s a reason why business analytics is known as “the science of better” rather than “the science of provably optimal.” More often than not, it is impossible to capture all nuances of a real-life problem into a mathematical model. Therefore, solutions produced by such a model are to be taken with a grain of salt and cautious optimism. Do these numbers still make sense after the omitted intangibles are brought back into the picture? If so, great. If not, can we slightly modify the solution? Do we need to revise the model? When building my burger, I ignored flavor considerations and overall gastronomic appeal, whereas my wife didn’t.

Another matter with which non-experts struggle is keeping their intuition from negatively affecting their modeling choices. Or, as I like to say in class, “don’t try to solve the problem; focus on representing the problem.” A mathematical model is a translation, say, from English to formulas, of the story that warrants investigation. Letting your understanding of the story bias you into thinking the solution should/shouldn’t look a certain way, and adding that assumption into your model, can be detrimental. As humans, we tend to think intuitively, which in turn limits the universe of possibilities we look at. Good solutions to complex problems can be, at least at first sight, counterintuitive. Make sure your model has the freedom to consider “weird” courses of action as well.

The Usefulness of Imperfection

Imperfect models create imperfect solutions that can still be useful. Having a solution in the ballpark of good answers is better than looking at a blank slate and not knowing where to begin. Solving a model many times with a range of input values improves your understanding of how certain numbers relate to others. As your understanding of the problem improves, that (initially) counterintuitive solution starts to make sense. Improved understanding leads to revising your initial assumptions and even realizing that what you thought mattered is secondary. What really matters is this other number to which you didn’t pay much attention to begin with. In extreme cases, this first model may lead you to conclude that you need to create a completely different model to solve a completely different problem, and that’s OK too! You are learning about your problem in the process of trying to solve your problem. Finally, it may very well be that your problem, as originally stated, has no solution. This would have been difficult to detect by hand because complex problems have several moving parts with intricate relationships of cause and effect. Having a model whose output says “infeasible” can be a valuable tool in a meeting. It is concrete proof that the original question and/or assumptions are inconsistent in some way. (Assuming there’s no mistake in the model, of course.) From there, it will be a much shorter path to convincing people to revise the question/assumptions than it would have been otherwise. Who would have thought? Modeling also makes your meetings more efficient!

Buying Metrorail Tickets in Miami

(Credits: Thanks to my MBA student Mike Plytynski for the idea for this post.)

Instead of a subway system, Miami has surface trains like the one depicted below. This is known as our Metrorail system. The Metrorail connects to the Tri-Rail system (which takes you north as far as West Palm Beach) and also to the free-to-ride, fully-automated, electrically-powered Metromover.

If you are a (reasonably) frequent Metrorail rider, you may consider buying a rechargeable card at one of the ticket vending machines. There is a 1-day pass ($5.65), a 7-day pass ($29.25), and a monthly pass ($112.50) that offer unlimited rides, but let’s focus on the case when you don’t ride often enough to justify the cost of such passes.

The current cost of a one-way trip is $2.25, regardless of distance traveled. The vending machines at the entrance of each Metrorail station allow you to load the following pre-specified amounts onto your card: $1, $2.25, $3, $4.50, $5, $10, $20, and $40.

No one likes to waste time having to stop by the vending machine to reload the Metrorail card. Besides, by Murphy’s Law, the next train will arrive and depart exactly when you’re stuck in line at the machine trying to reload your card. Therefore, we’ll consider that our objective is to minimize the number of visits to the vending machine to load our card with enough money for one-way trips. If we were free to load any amount on the card, we could simply load  times $2.25 and be done. The absence of this flexibility is what makes this situation both interesting and a reason for us to hate the Metrorail. So let’s write an optimization model to solve this problem.

Let variable be the number of times we go to the vending machine to load $1 on the card. Let  be the number of times we load $2.25 on the card, and so on all the way to  (how many times we load $40). To minimize the number of visits to the vending machine, we write the objective function

To make sure we load enough money on the card for  trips, we write the constraint

If we are not careful, we may end up with unnecessarily too much money left on the card. So it’s wise to limit that excess with the constraint

where is a limit on what we’d be willing to have left on the card after the  trips are completed. Conceivably, this leftover amount could be applied toward your next batch of  trips. Hold that thought; we’ll return to it later.

I put together an Excel Solver spreadsheet model for this problem, which you can download from here (sheet named “standard” in the Excel file).

If you change the second constraint above to an equality and solve this problem for different values of  and , you can create the following heat map. The numbers in the colored squares indicate the minimum number of required visits to the vending machine.

Interestingly, some values of , such as 10, 18, and 20, are more friendly in the sense that they allow you to visit the machine at most twice while incurring a small leftover amount (no more than $0.50). On the other hand, to avoid visiting the machine more than twice when paying for 12, 14, or 16 trips, you must be willing to accept a much higher ending balance: $3, $8.50, and $4, respectively. In addition, some of the reload amounts that might at first seem useless, such as $5 and $10, actually do get used in the optimal solutions for some values of  and .

Ending Balances As Multiples of $2.25

As we saw above, allowing for larger ending balances on the card can help reduce the number of visits to the vending machine. One way to make these balances useful is to apply them toward future trips. If the balance is a multiple of the cost of a one-way trip, even better. To do that, we can replace the second constraint above with

where  is a new, non-negative integer variable. The sheet called “leftover is multiple” in this Excel file implements this version of the optimization model. The conclusion is that you pretty much only need 2 visits to the vending machine for , and the value of  seems to decrease as increases. (See Excel file for detailed results.)

Now, if only I could get someone from Miami-Dade Transit to read this post…

We Are Hiring! Tenure-Track Position in Business Analytics, University of Miami

I’m excited to announce that my department at the University of Miami’s School of Business is hiring this year. Check out the job description below and please help me spread the word!

Tenure-Track Position in Business Analytics

The Department of Management Science in the School of Business at the University of Miami invites applications for a tenure-track Assistant Professor position in Business Analytics to start in the fall of 2018.

Applicants with research and teaching interests in all areas of business analytics or data science will be considered. The Management Science Department is home to a diverse group of faculty with expertise in data analytics and operations research, and offers a Master of Science program in Business Analytics, in addition to participating in the undergraduate, MBA, and Ph.D. programs of the School. The position affords the successful candidate the opportunity to have an immediate impact in a dynamic department that is rapidly growing in the area of business analytics. Duties will include research and teaching at both the graduate and undergraduate levels. Salary is competitive and commensurate with background and experience.

Applicants should possess, or be close to completing, a Ph.D. in a discipline related to business analytics or data science by the start date of employment. Applications should be submitted by e-mail to MASrecruiting@bus.miami.edu, and should include the following: a curriculum vitae, up to three representative publications, brief research and teaching statements, an official graduate transcript, information about teaching experience and performance evaluations (if available), and three letters of recommendation. All applications completed by December 1, 2017 will receive full consideration, but candidates are urged to submit all required material as soon as possible. Applications will be accepted until the position is filled.

The University of Miami offers a comprehensive benefits package including medical and dental benefits, tuition remission, and much more.

The University is an equal opportunity employer and encourages candidates regardless of gender, race, color, ethnicity, age, disability status or sexual orientation to apply.

Bringing Research into the Classroom: Can Relevant and Impactful be Easy to Explain?

 O.R. researchers and practitioners are constantly churning out papers that tackle a wide variety of important and hard-to-solve practical problems. On one hand, as a researcher, I understand how difficult these problems can be and how it’s often the case that fancy math and complex algorithms need to be used. On the other hand, as someone who teaches optimization to MBA students who aren’t easily excited by mathematics, I’m always looking for motivational examples that are both interesting and not too complex to be understood in 5 minutes. (That’s the little slot of time I reserve at the beginning of my lectures to go over an application before the lecture itself starts.)

Every now and then, I come across a paper that fits the bill perfectly: it addresses an important problem, produces impactful results, and (here comes the rare part), accomplishes the previous two goals by using math that my MBA students can follow 100%, while being confident that they themselves could replicate it given what they learned in my course (the optimization models).

The paper to which I’m referring has recently appeared in Operations Research (Articles in Advance, January 2017): The Impact of Linear Optimization on Promotion Planning, by Maxime C. Cohen, Ngai-Hang Zachary Leung, Kiran Panchamgam, Georgia Perakis, and Anthony Smith (http://dx.doi.org/10.1287/opre.2016.1573).

If I had to pick one word to describe this paper, it would be BEAUTIFUL.

I immediately proceeded to put together a 5-minute summary presentation (8 slides) to cover the problem, approach, and results. I’ll be showing this to 100 of my MBA students on this coming Tuesday (Valentine’s Day!). I hope they love it as much as I did. Feel free to show this presentation to your own students if you wish, and let me know how it went down in the comments.

A recent Poets & Quants article explains how business schools with the highest quality teaching strive to bring their faculty’s research into the classroom so that students get to learn the latest and greatest ideas. The O.R. paper above is a perfect example of when this can be done effectively.

Portuguese Pronunciation and Language Tips

I’ll be taking a group of 34 MBA students on an international business immersion trip to my native Brazil this Spring. We’ll be visiting about a dozen companies in the cities of São Paulo and Rio de Janeiro. This is an initiative created by the awesome Center for International Business Education and Research (CIBER) at the University of Miami.

I’d like my students to be able to pronounce some of the main sounds in Portuguese correctly because I know Brazilians pay attention and really enjoy when foreigners make an effort to say things properly. Therefore, I created a video in which I go over what I consider to be some of the most important things to know when speaking Portuguese (there are others, but I didn’t want the video to be too long).

You can access it on my YouTube channel here: https://www.youtube.com/watch?v=LzgoYFokBPk

Moreover, the 2016 Olympic Games are coming, so I figured these tips could be useful for a larger audience as well. I wish American sports casters would watch this video because they murdered the pronunciation of everything during the World Cup in 2014.

Bonus material: My daughter, Lavinia Lilith, a.k.a. #LLCoolBaby, makes a short appearance at around the halfway mark.

Enjoy!