# Mathematical Contest in Modeling, 1996

## The event

The Mathematical Contest in Modeling is an annual event, sponsored by the Consortium for Mathematics and Its Applications, in which students at colleges and universities all over the world are asked to develop and analyze mathematical models of open-ended, practical problems for which no direct solutions are known. Participants work in teams of three and are permitted to use libraries, computers, and other inanimate sources of knowledge and inspiration. The organizers of the contest propose three problems; over the three days of the contest, each team selects one of these problems, designs, implements, and analyzes a model, and writes a substantial report presenting its results.

The twelfth Mathematical Contest in Modeling was held on February 9-12, 1996. Grinnell's teams this year were: Robert Ciraldo 1996, Ioannis Sarafidis 1996, and Chris Akritides 1997, and John Liu 1997, Adam Sales 1996, and Ivan Sykes 1997. The faculty sponsor was Tom Moore.

## The problems

Here are the problems that COMAP posed this year. Both of Grinnell's teams elected to work on Problem B.

### Problem A

The world's oceans contain an ambient noise field. Seismic disturbances, surface shipping, and marine mammals are sources that, in different frequency ranges, contribute to this field. We wish to consider how this ambient noise might be used to detect large moving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, develop a method for detecting the presence of a moving submarine, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed frequency and amplitude.

### Problem B

When determining the winner of a competition like the Mathematical Contest in Modeling, there are generally a large number of papers to judge. Let's say there are P = 100 papers. A group of J judges is collected to accomplish the judging. Funding for the contest constrains both the number of judges that can be obtained and the amount of time that they can judge. For example, if P = 100, then J = 8 is typical.

Ideally, each judge would read each paper and rank-order them, but there are too many papers for this. Instead, there will be a number of screening rounds in which each judge will read some number of papers and give them scores. Then some selection scheme is used to reduce the number of papers under consideration: If the papers are rank-ordered, then the bottom 30% that each judge rank-orders could be rejected. Alternatively, if the judges do not rank-order the papers, but instead give them numerical scores (say, from 1 to 100), then all papers falling below some cut-off level could be rejected.

The new pool of papers is then passed back to the judges, and the process is repeated. A concern is that the total number of papers that each judge reads must be substantially less than P. The process is stopped when there are only W papers left. These are the winners. Typically for P = 100, W = 3.

Your task is to determine a selection scheme, using a combination of rank-ordering, numerical scoring, and other methods, by which the final W papers will include only papers from among the best 2W papers. (By best, we assume that there is an absolute rank-ordering to which all judges would agree.) For example, the top three papers found by your method will consist entirely of papers from among the best six papers. Among all such methods, the one that requires each judge to read the least number of papers is desired.

Note the possibility of systematic bias in a numerical scoring scheme. For example, for a specific collection of papers, one judge could average 70 points, while another could average 80 points. How would you scale your scheme to accommodate for changes in the contest parameters (P, J, and W)?

## Results

We received the COMAP's report of the results of the competition on April 15, 1996. Each of Grinnell's teams received an Honorable Mention.

This year, 393 teams submitted complete entries. Nine of these entries were judged Outstanding and fifty-four Meritorious. One hundred and fourteen entries received Honorable Mentions in COMAP's report. The remaining 216 teams were classified as Successful Participants.

This document is available on the World Wide Web as

http://www.math.grinnell.edu/mcm-1996.xhtml