## The Mathematical Contest in Modeling Turns 20

October 26, 2004

James Case

Early in February, 613 three-student teams assembled on their home campuses for the 20th annual Mathematical Contest in Modeling. As usual, the students had the choice of two problems, one discrete and the other continuous. The problems were posted on the Web on the evening of Thursday, February 5, and solutions were due by Monday evening, February 9.

Slightly more than a third of the teams elected to work on problem A, the continuous problem. The judges rated three of the problem A solutions outstanding and 24 meritorious; 50 received honorable mention. The other 401 teams chose to work on problem B, the discrete alternative. Of the problem B papers, four were deemed outstanding, 38 were designated meritorious, and 109 received honorable mention.

The A problem concerned the commonly held belief that the thumbprints of all humans who have ever lived are distinguishable from one another. The students were asked to develop and analyze a model that would enable them to assess the probability that the foregoing belief is correct, and then to compare the odds of misidentification by fingerprinting against the odds of misidentification by DNA analysis.

Problem B grew from the observation that customers at amusement parks are often obliged to wait in line for two hours or more before embarking on the most popular rides, and that many such parks have begun to implement express pass systems to shorten their waiting lines. Such passes are typically dispensed by machines, which issue tickets entitling the bearer to embark on a designated ride at or before time T, provided only that he or she arrive at the point of departure no later than time t < T. Since t typically precedes T by no more than an hour, such tickets can save park goers substantial amounts of time and aggravation.

The sheer variety of express pass systems in use at different parks, and the number of parks that still don't offer them, suggest a lack of consensus as to the magnitude of the associated costs and potential benefits, not to mention confusion as to the most effective mode(s) of operation. The teams were asked to design a model that could be used to address such questions. In their solutions, the students raised a bewildering variety of related questions, of which the following are only a few: How many rides should be included in the express pass system? Should the tickets be sold or given away? How long should the [t,T] intervals be? How many such intervals should be available to a given customer? How many such tickets should a customer be allowed to hold simultaneously? Should the park be allowed to overbook popular time slots? Under what circumstances should customers be compensated for non-performance on the part of the park?

The outstanding papers on the A problem were from teams at the University of Colorado at Boulder, which also received the MAA award; Harvey Mudd College, which also received both the INFORMS and the SIAM awards; and University College, Cork. For the B problem, the outstanding papers were from Harvard (the MAA winner), the University of Washington at Seattle, the University of Colorado at Boulder (the SIAM winner), and Merton College, Oxford (the INFORMS winner).
Both of the SIAM winners presented their prize-winning papers in a special session at the 2004 SIAM Annual Meeting in Portland.

This year, for the first time, an additional prize was awarded. The Ben Fusaro award-named for the contest founder-recognizes the papers, one for each problem, that best exemplify the following characteristics:

• The paper presents a high-quality application of the complete modeling process.
• The team has demonstrated noteworthy originality and creativity in their modeling effort to solve the problem as given.
• The paper is well written, in a clear expository style, and is a pleasure to read.

The first recipients of the Fusaro award were a team from Central Washington University for the A problem and a team from MIT for the B problem.

James Case writes from Baltimore, Maryland.

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