# Mathematical Contest in Modeling, 2002

## 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 eighteenth Mathematical Contest in Modeling was held on February 7-11, 2002. Grinnell's teams this year were:

• Theo Beasley-Henderson 2003, Carl Long 2002, and Greg McGraw 2002 (faculty sponsor: Royce Wolf)
• Jon Wellons 2004, Andrew Maginniss 2005, and a student who wishes to remain anonymous (faculty sponsor: Royce Wolf).
• Ming Gu 2003, Ananta Tiwari 2004, and Rajendra Magar 2003 (faculty sponsor: Mark Montgomery).
• Greg Dyer 2004, Dawa Lama Sherpa 2003, and Devdatta Kulkarni 2005 (faculty sponsor: Mark Montgomery).

## The problems

Here are the problems that COMAP posed this year:

### Problem A: Wind and waterspray

An ornamental fountain in a large open plaza surrounded by buildings squirts water high into the air. On gusty days, the wind blows spray from the fountain onto passersby. The water-flow from the fountain is controlled by a mechanism linked to an anemometer (which measures wind speed and direction) located on top of an adjacent building. The objective of this control is to provide passersby with an acceptable balance between an attractive spectacle and a soaking: The harder the wind blows, the lower the water volume and height to which the water is squirted, hence the less spray falls outside the pool area.

Your task is to devise an algorithm which uses data provided by the anemometer to adjust the water-flow from the fountain as the wind conditions change.

### Problem B: Airline overbooking

You're all packed and ready to go on a trip to visit your best friend in New York City. After you check in at the ticket counter, the airline clerk announces that your flight has been overbooked. Passengers need to check in immediately to determine if they still have a seat.

Historically, airlines know that only a certain percentage of passengers who have made reservations on a particular flight will actually take that flight. Consequently, most airlines overbook -- that is, they take more reservations than the capacity of the aircraft. Occasionally, more passengers will want to take a flight than the capacity of the plane leading to one or more passengers being bumped and thus unable to take the flight for which they had reservations.

Airlines deal with bumped passengers in various ways. Some are given nothing, some are booked on later flights on other airlines, and some are given some kind of cash or airline ticket incentive.

Consider the overbooking issue in light of the current situation:

• Less flights by airlines from point A to point B
• Heightened security at and around airports
• Passengers' fear
• Loss of billions of dollars in revenue by airlines to date

Build a mathematical model that examines the effects that different overbooking schemes have on the revenue received by an airline company in order to find an optimal overbooking strategy, i.e., the number of people by which an airline should overbook a particular flight so that the company's revenue is maximized. Insure that your model reflects the issues above, and consider alternatives for handling bumped passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings and analysis.

## Results

Both the team of Ming Gu, Ananta Tiwari, and Rajendra Magar and the team of Theo Beasley-Henderson, Carl Long, Greg McGraw earned the rating of Honorable Mention from COMAP's judges. Both the team of Jon Wellons, Andrew Maginniss, and the anonymous student and the team of Greg Dyer, Dawa Sherpa, and Devdatta Kulkarni were ranked as Successful Participants.

This year, 522 teams submitted complete entries. Ten of these entries were judged Outstanding; eighty-six, Meritorious. One hundred twenty-one received Honorable Mentions in COMAP's report. The remaining 305 teams were classified as Successful Participants.

This document is available on the World Wide Web as

```http://www.math.grinnell.edu/mcm-2002.xhtml
```