International Math Competition in Modeling 2009

The International Math Competition in Modeling (IMCM or MCM) took place 3 weeks ago over a period of 96 crazy hours.  However, I must say that the team clicked pretty well this year.  I think the latest any of us stayed up to was 2:30AM and we usually got together to work at like 11AM.  This competition, I was the dedicated programmer from the start.   We chose problem A (the traffic circle problem) after about an hour and a half brain storming about each, feeling a bit overwhelmed about each and then taking a vote on the confidence and enthusiasm for each problem.

Prior to this vote we had been leaning toward the cellphone problem.  The traffic circle problem seemed overwhelming.  However, on a plus side having taking Math 425 Numerical Analysis last semester, I already had written Matlab source code for solving the traffic PDE in which cars circled as if in a traffic circle.  Having this code as well as the higher overall enthuasism for the traffic circle problem led us to choose it.

One noteworthy observation was that as soon as we settled down on the traffic circle problem we checked Wikipedia and a reference to the IMCM 2009 was already on the page.

Unlike my previous competitions, as soon as we decided on a problem we ran off to the library.  It was helpful that one of the members of the team had a car so we could quickly go between the math and engineering libraries before they closed for the evening.  We ended up grabbing about 10 books, many of which we used, some of which we didn’t use, over the duration of the competition.

Later that evening I sat down with my old source code and began modifying it in order to make it fit the problem better.  However, all my changes were surface changes.  I needed to be able to simulate cars entering at certain roads and leaving at others.  Moreover, cars only want to get off at their exit so I had to develop a way to track the cars from each entrance around the circle.  I ended up creating multiple circles, for each exit, and summing their traffic together to get the total traffic around the circle.  The traffic circle PDE was then applied to the sum of circles.  Next I calculated the velocity at each “point” around the circle and then moved the cars in each of the circles the distance corresponding to that velocity multiplied by the timestep.  Moreover, the code also needed to check to make sure the density of the summed circle did not exceed the maximal density for the circle.  If it did, I just had the extra cars pile up behind the maximal density points.  This “piling up” was then analyzed in order to maintain the proper proportion of cars in each circle.

Then using these exit circles, the code allows a certain proportion of the density of cars trying to leave at each exit to leave the circle.  Thus, we can simulate cars getting stuck in the circle causing them to loop around.

We then also simulate the entry roads to the traffic circle since the circle can often cause a backup on the roads.  We then also simulated traffic control elements at the entry point of each road into the circle.  We used greedy/nongreedy yield and stop signs to simulate the circle being yielded/stopped and the entries yielded/stopped respectively.  We also simulated several traffic lights: fixed duration and local density optimizing.

We ended up running several basic setups in order to gauge the effectiveness of each traffic control element.  However, we did not have enough time to collect results for more complicated circle designs.  It did not help since the runtime often exceeded 5 minutes on my 3.8 Ghz overclocked Core 2 Duo system.  For several of the runs we had to increase the time resolution since it was possible for the cars to “jump” over their exits without even being allowed to leave.

If you have more interest in our work check out our paper.  Out of everything with the competition, as a team we feel like our title could not have been better: A Straightfoward Solution to a Roundabout Problem.

Overall, the competition was enjoyable and there is certainly room for improvement for the future.  However, this past competition was definetly the most enjoyable in which I have participated.