The Coral Gables campus of the University of Miami is slowly transitioning into a smoke-free campus. (I can’t wait for that to happen.) Presently, there are a number of designated smoking areas (DSAs) around campus and nobody is supposed to smoke anywhere else. Here’s a map of campus with red dots representing DSAs (right-click on it and open it in a new tab to see a larger version):

Unfortunately, enforcement of this smoking policy is nowhere to be seen. The result? Lots of students smoking wherever they want and, even worse, smoking while walking around campus, which is a great way to maximize their air pollution effect. Don’t you *love* people who live in the universe of me, myself, and I? But let me stop ranting and return to operations research…

As someone who does not enjoy (and is allergic to) cigarette smoke, I started thinking about how to use OR to help with the enforcement effort. Let’s say there will be an enforcer (uniformed official) whose job is to walk around campus in search of violators. Based on violation reports submitted by students, faculty and staff, the University can draw a second set of colored dots, say black, on the above map. These black dots represent the non-smoking areas in which violations have been reported most often. For simplicity, let’s call them violation areas, or VAs.

In possession of the VA map, what is the enforcer supposed to do? You probably answered “walk around campus visiting each VA”. If you’re now thinking about the Traveling Salesman Problem (TSP), you’re on the right track. The enforcer has to visit each VA and return to his/her starting point. However, this is not quite like a pure TSP. Let me explain why. First of all, unlike the pure TSP, the enforcer has to make multiple passes through the VAs on a single day. Secondly, it’s also likely that some VAs are more popular than others. Therefore, we’d like the enforcer to visit them more often. Finally, we want the multiple visits to each VA to be spread throughout the day. With these considerations in mind, let me define the *Smoking Policy Enforcement Problem* (SPEP): We are given a set of locations on a map. For each location , let be the minimum number of times the enforcer has to visit during the day, and let be the minimum separation between consecutive visits to location . In other words, each time the enforcer visits , he/she has to visit at least other locations before returning to . The goal is to find a route for the enforcer that satisfies the visitation requirements ( and ) while minimizing the total distance traveled.

After a few Google searches, I discovered that the SPEP is not a new problem. This shouldn’t have come as a surprise, given the TSP is one of the most studied problems in the history of OR. The article I found, written by R. Cheng, M. Gen, and M. Sasaki, is entitled “*Film-copy Deliverer Problem Using Genetic Algorithms*” and appeared in Computers and Industrial Engineering 29(1), pp. 549-553, 1995. Here’s how they define the problem:

There are a few minor differences with respect to the SPEP. In the above definition, for every location . What they call is what I call , and they require exactly visits, whereas I require at least visits.

I wasn’t aware of this TSP variant and I think it’s a very interesting problem. I’m happy to have found yet another application for it. Can you think of other contexts in which this problem appears? Let me know in the comments.

Tallys, a possible alternative to the TSP is to solve this enforcement problem as a set covering problem (if there are unlimited number of enforcers available) or as a max covering problem (if there is a fixed number of enforcers available). Coverage could be defined as how far an enforcer can see from his monitoring location.

Thaddeus: thank you for the comment. Indeed, this is another way to address this problem: by we re-casting it as a variant of the Art Gallery Problem (http://en.wikipedia.org/wiki/Art_gallery_problem). This is a good example of how a given problem can be attacked from different perspectives. The decision maker has to make that call after considering all the intangibles that the mathematical models are not able to capture.