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.

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.

A Successful Optimization Conference

Three days, 116 attendees, 105 talks, 8 posters, 3 plenary speakers, 2 featured speakers, and a small ceremony in honor of Robert Fourer. That’s a one-sentence summary of the Fourth INFORMS Optimization Society Conference that took place from February 24th to the 26th on the University of Miami campus. Mother nature cooperated with good weather and the participants did not hesitate to tell us (the organizers) how much they enjoyed their time here in Coral Gables. It was a lot of work, but it was all worth it. I’m exhausted but happy. Thank you all for your kind words, handshakes, and feedback.

The conference was much more than simply giving and attending talks (to paraphrase Michael Trick). There were old friends getting together, there were new friendships (and papers) getting started, there were old/long-forgotten, unfinished papers coming back to life (my case), there were laughs, jokes, and camaraderie. There were inspiring plenary talks that made some of us think (and tweet) about new research directions, and wonder whether we chose the right path. A true networking event.

In Brazil, in the context of a soccer match, we say that a good referee is one whose presence we don’t notice. The conference wasn’t glitch-free (on my end of things), but it was pretty much glitch-free to everyone else not involved in the organization (at least this was the feedback I received). That’s as good as it gets, in my opinion. We wanted to be the soccer referee who wears an invisibility cloak, and it seems to have worked. We (the conference chairs and organizing committee) couldn’t have pulled it off without the tremendous help of many people, and I want to take this opportunity to thank them once more:

From the Management Science Department: our awesome office manager Vanessa Ferguson, and dedicated students Jannelle Chaviano, Jen Verdon, Meiyin Cheng, and William Barnard. Thank you for taking care of the catering, logistics, registration desk, signage, receipts, printing and binding of programs, bags & badges, table decorations, and the 1000 other things that seem to be small but amount to a whole lot when put together.

From the IT and Budget departments at UM: Emil Diego, May Peralta, and Richard Mencke. Thank you for your help with the web site and payment processing.

From the INFORMS offices: Terry Cryan, Ellen Tralongo and Paulette Bronis. Thank you for all your help with the abstract submission system, special requests of all sorts, and formatting of the final program.

From my family: Madeline Keller (a.k.a. my darling wife). Thank you for helping out with all sorts of little things, and for lending us your computer expertise and attention to detail (and for helping me get a beer early at the receptions :-) there must be some advantage to being an organizer, right?). And most importantly, thank you for your patience during my stressful days.

Finally, here are a few of photos taken during the conference. If you have more photos, please send them to me and I’ll be happy to add them to this page.

My vegetarian boxed lunch. Eaten during Manoj Saxena’s plenary:

Last slide of Dimitris Bertsimas’s plenary (click on it to enlarge it, and note the “stochastic analysis *without* probability”):

Me standing on stage minutes before officially closing the conference and introducing our last featured speaker, David Alderson (taken by Mike Trick):

Curious photo of an art piece at the Lowe Art Museum (where the Friday reception was held). According to a number of participants, the man in this photo looks like one of the conference organizers. Can you guess who it is? (Copyright Carlos Betancourt)

Operations Research Memes

In the spirit of bringing awareness about O.R. to the masses, I created the memes below. Perhaps they’ll gain some traction or at least get a few people to wonder about what O.R. is. Who knows, they may even motivate someone to Google the term! If you end up making your own O.R.-inspired meme, please send me a link to it via the comments section. To create mine, I used the quickmeme.com web site.

UPDATE: A few other OR bloggers and tweeps joined the meme crusade! Here are their creations (in chronological order of my becoming aware of them):

Laura McLay created the memes below:

Michael Trick created these:

Paul Rubin suggested the creation of this one:

Guido Diepen created this one:

Bill Cook made this cool TSP meme:

Paul Rubin made this one, western style:

My MBA student William Bucciero got inspired by these O.R. memes and made some of his own. He was kind enough to share them with me. I think he did a great job! Here they are:

Another one of my MBA students, Jason Siem, also joined the O.R. meme bandwagon. Here’s one of his (pretty funny and true):

INFORMS Optimization Society Conference: Submission Deadline in Four Days!

The abstract/poster submission deadline for the Fourth INFORMS Optimization Society Conference is this Friday, January 6! There’s still time to get your abstract in. For information about the conference and submission instructions, check the conference web site at http://bus.miami.edu/ios, as well as this previous post.

If you’ve already submitted an abstract, please log into the system and make sure your submission is complete. We now have a tentative schedule online.

I’m looking forward to seeing you in Miami in February!

INFORMS Optimization Society Conference 2nd Call for Abstracts, Posters, and Participation

Back in September I wrote about the Fourth INFORMS Optimization Society Conference that’s taking place on the University of Miami campus, February 24-26, 2012. Now I’m writing again to remind all of you that the early registration deadline is approaching fast: it’s December 15. Make sure to take advantage of the discount! Moreover, keep in mind that the abstract/poster submission deadline is also close by: January 6, 2012.

If you haven’t done so yet, make sure to check out the conference web site: http://bus.miami.edu/ios. The conference is shaping up to be a great event and an amazing opportunity to network with — and present your work to — some of the best minds in the optimization community from all over the world. Even if your work is still in progress, consider submitting an abstract to our poster session; it’s an opportunity to get additional feedback on what you’re doing. In addition, we have an exceptional line-up of plenary and semi-plenary speakers.

Finally, since I’ve been building a reputation as someone who likes to talk about food (e.g. here, here, here, here, and here), keep this in mind: your registration fee includes 3 breakfasts, 3 lunches, 2 receptions, and 6 coffee breaks! It doesn’t get any better than this.

So, recapitulating, make sure to register before December 15, and send your abstract/poster in by January 6!

I hope to see you all here in Miami!

Winter Blues Have You Down? Miami in February is Your Town!

Want the perfect reason to come to Miami in February? What about the 2012 INFORMS Optimization Society Conference? The conference, whose theme is “Optimization and Analytics: New Frontiers in Theory and Practice”, will be hosted by the University of Miami School of Business Administration from Friday, February 24 to Sunday, February 26 on its beautiful campus in Coral Gables, Florida. We are very fortunate to have many of the top researchers in Optimization and Analytics as members of our advisory and program committees. I expect the final conference program to be full of high-quality talks.

This is my first time as a member of an organizing committee and I’m happy to say that, despite all the work, it’s been a lot of fun!

Here’s a link to the call for abstracts and posters (we’re already accepting submissions). For more information, including important dates, registration rates, plenary speakers, and hotel reservations, visit the conference web site at http://bus.miami.edu/ios, or send me an e-mail (tallys at miami dot edu). I hope to see you all here!

Happy Holidays O.R. Style

The previous owners of our house were kind enough to leave their piano behind. It’s a beautiful old-school player piano. It uses paper rolls to encode the songs like punch cards (we inherited a number of rolls as well). I’ve never taken piano classes, but every now and then I enjoy poking around the keys until I manage to play something simple. Here’s a picture:

After managing to play “We Wish You a Merry Christmas”, I decided to create a happy-holidays video card as a thank you to all my readers (yes, all nine of you). However, having a tendency to think about alternative lyrics to well-known songs, I could not help doing it again. The result is an O.R.-ish version of the song, which I hope you’ll enjoy.