Mathematical Contest in Modeling, 2005

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 four 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 twenty-first Mathematical Contest in Modeling was held on February 3-7, 2005. Grinnell fielded three teams this year:

The problems

Here are the problems that COMAP posed this year. The team comprising Daren Brantley, Andrew Rinne, and Sheng Wang chose to work on problem A, and the other two teams on problem B.

Problem A: Flood planning

Lake Murray in central South Carolina is formed by a large earthen dam, which was completed in 1930 for power production. Model the flooding downstream in the event there is a catastrophic earthquake that breaches the dam.

Two particular questions:

Problem B: Tollbooths

Heavily-traveled toll roads such as the Garden State Parkway, Interstate 95, and so forth, are multi-lane divided highways that are interrupted at intervals by toll plazas. Because collecting tolls is usually unpopular, it is desirable to minimize motorist annoyance by limiting the amount of traffic disruption caused by the toll plazas. Commonly, a much larger number of tollbooths is provided than the number of travel lanes entering the toll plaza. Upon entering the toll plaza, the flow of vehicles fans out to the larger number of tollbooths, and when leaving the toll plaza, the flow of vehicles is required to squeeze back down to a number of travel lanes equal to the number of travel lanes before the toll plaza. Consequently, when traffic is heavy, congestion increases upon departure from the toll plaza. When traffic is very heavy, congestion also builds at the entry to the toll plaza because of the time required for each vehicle to pay the toll.

Make a model to help you determine the optimal number of tollbooths to deploy in a barrier-toll plaza. Explicitly consider the scenario where there is exactly one tollbooth per incoming travel lane. Under what conditions is this more or less effective than the current practice? Note that the definition of “optimal” is up to you to determine.

Results

This year, 828 teams submitted complete entries. Thirteen of these entries were judged Outstanding; one hundred eleven others, Meritorious. Two hundred eighty-four received Honorable Mentions in COMAP's report. The remaining 420 teams were classified as Successful Participants.

All three of Grinnell's teams received the Successful Participant ranking.

This document is available on the World Wide Web as

http://www.math.grinnell.edu/mcm-2005.xhtml