Those who know me are aware of my strong passion for basketball, so I had to find a way to relate this post to my favorite sport. Fans of basketball in general, and of the Miami Heat in particular, might be familiar with this video clip in which LeBron James makes a bold prediction. Back in 2010, when asked how many titles the Heat’s big three would win together, he replies “Not one, not two, not three, not four, not five, not six, not seven, …” While I’d love to see them win 8 titles, it sounds a bit (a lot) unlikely. But I can’t complain about their record so far. Winning 2 titles in 3 finals’ appearances isn’t bad at all. But what does this have to do with baseball umpires? Let’s get back to OR for a moment.

A couple of years ago, I wrote a post about scheduling baseball umpires. In that same article I co-authored with Hakan Yildiz and Michael Trick, we talked about a problem called the Traveling Umpire Problem (TUP), which doesn’t include all the details from the real problem faced by MLB but captures the most important features that make the problem difficult. Here’s a short description (detailed description here):

Given a double round-robin tournament with 2N teams, the traveling umpire problem consists of determining which games will be handled by each one of N umpire crews during the tournament. The objective is to minimize the total distance traveled by the umpires, while respecting constraints that include visiting every team at home, and not seeing a team or venue too often.

And when I say difficult, let me tell you something, it’s *really* hard to solve. For example, there are 16-team instances (only 8 umpires) for which no feasible solution is known.

Two of my Brazilian colleagues, Lucas de Oliveira and Cid de Souza, got interested in the TUP and asked me to join them in an effort to try to improve the quality of some of the best-known solutions in the TUP benchmark. There are 25 instances in the benchmark for which we know a feasible solution (upper bound) and a lower bound, but not the optimal value. Today, we’re very happy to report that we managed to improve the quality of many of those feasible solutions. How many, you ask? I’ll let LeBron James himself answer that question:

“Not one, not two, not three, … not ten, … not eighteen, … not twenty-three, but

24 out of 25.”

OK, LeBron got a bit carried away there. And he forgot to say we improved 25 out of the 25 best-known lower bounds too. This means those pesky optimal solutions are now sandwiched between numbers much closer to each other.

Here’s the approach we took. First, we strengthened a known optimization model for the TUP, making it capable of producing better bounds and better solutions in less time. Then, we used this stronger model to implement a relax-and-fix heuristic. It works as follows. Waiting for the optimization model to find the optimal solution would take forever because there are too many binary decision variables (they tell you which venues each umpire visits in each round of the tournament). At first, we require that only the decisions in round 1 of the tournament be binary (i.e. which games the umpires will be assigned to in round 1) and solve the problem. This solves pretty fast, but allows for umpires to be figuratively cut into pieces and spread over multiple venues in later rounds. Not a problem. That’s the beauty of math models: we test crazy ideas on a computer and don’t slice people in real life. We fix those round-1 decisions, require that only round-2 variables be binary, and solve again. This process gets repeated until the last round. In the end, we are not guaranteed to find the very best solution, but we typically find a pretty good one.

Some possible variations of the above would be to work with two (or more) rounds of binary variables at a time, start from the middle or from the end of the tournament, etc. If you’re interested in more details, our paper can be downloaded here. Our best solutions and lower bounds appear in Table 10 on page 22.

We had a lot of fun working on the TUP, and we hope these new results can help get more people excited about working on this very challenging problem.