# Mathematical Contest in Modeling, 2007

## 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-third Mathematical Contest in Modeling was held on February 8-12, 2007. Grinnell fielded two teams this year:

• Yuan Hu 2008, Trang Nguyen 2008, and Eric Omwega 2008 (faculty sponsor: Karen Shuman)
• B. Tolga Oztan 2010, Jiabei Pan 2010, and Nikoloz Sakvarelidze 2010 (faculty sponsor: Karen Shuman)

## The problems

Here are the problems that COMAP posed this year. Hu, Nguyen, and Omwega chose to work on Problem A, Oztan, Pan, and Sakvarelidze on Problem B.

### Problem A: Gerrymandering

The United States Constitution provides that the House of Representatives shall be composed of some number (currently 435) of individuals who are elected from each state in proportion to the state's population relative to that of the country as a whole. While this provides a way of determining how many representatives each state will have, it says nothing about how the district represented by a particular representative shall be determined geographically. This oversight has led to egregious (at least some people think so, usually not the incumbent) district shapes that look unnatural by some standards.

Hence the following question: Suppose you were given the opportunity to draw congressional districts for a state. How would you do so as a purely baseline exercise to create the simplest shapes for all the districts in a state? The rules include only that each district in the state must contain the same population. The definition of simple is up to you; but you need to make a convincing argument to voters in the state that your solution is fair. As an application of your method, draw geographically simple congressional districts for the state of New York.

### Problem B: The Airplane Seating Problem

Airlines are free to seat passengers waiting to board an aircraft in any order whatsoever. It has become customary to seat passengers with special needs first, followed by first-class passengers (who sit at the front of the plane). Then coach and business-class passengers are seated by groups of rows, beginning with the row at the back of the plane and proceeding forward.

Apart from consideration of the passenger' wait time, from the airline's point of view, time is money, and boarding time is best minimized. The plane makes money for the airline only when it is in motion, and long boarding times limit the number of trips that a plane can make in a day.

The development of larger planes, such as the Airbus A380 (800 passengers), accentuate the problem of minimizing boarding (and deboarding) time.

Devise and compare procedures for boarding and deboarding planes with varying numbers of passengers: small (85-210), midsize (210-330), and large (450-800).

Prepare an executive summary, not to exceed two single-spaced pages, in which you set out your conclusions to an audience of airline executives, gate agents, and flight crews.

An article appeared in the NY Times Nov 14, 2006 addressing procedures currently being followed and the importance to the airline of finding better solutions. The article can be seen at: http://travel2.nytimes.com/2006/11/14/business/14boarding.html

## Results

This year, 949 teams submitted complete entries. 14 of these entries were judged Outstanding; 122 others, Meritorious. 255 received Honorable Mentions in COMAP's report. The remaining 558 teams were classified as Successful Participants.

Both 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-2007.xhtml
```