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Math Models Examine the Effectiveness of Car-Sharing

By Lina Sorg and Karthika Swamy Cohen

When mass-produced cars hit the market in the early 20th century, car ownership in developed countries soared. Now, as drivers increasingly feel the economic burden of buying and owning cars, the market is undergoing a different kind of shift. Car-sharing, a form of short-term car rental frequently used for commuting, is experiencing unprecedented popularity; it offers consumers environmentally-friendly transportation without the complications and expenses of car ownership and maintenance.

But what factors determine the possible success of car-sharing in a given city, particularly from the perspective of auto and auto-sharing companies? Car-sharing’s worth—for both customers and auto corporations—is contingent upon the amount of time an individual spends behind the wheel and the daily mileage. Options for car-sharing depend on customer need, and range from roundtrip and one-way sharing to multiple ownership of a single car.

The various factors that influence car-sharing’s success made it a perfect topic for the eleventh annual Moody’s Mega Math (M3) Challenge, an applied mathematics contest in which participating teams of high school juniors and seniors across the United States address a realistic problem in 14 hours using math. Sponsored by The Moody’s Foundation and organized by SIAM, the Internet-based Challenge invites the top six teams to present their work to a panel of judges and compete for scholarships at Moody’s Corporation headquarters in New York City. 

“Moody’s Mega Math Challenge provides students with an outlet to investigate real-world problems while doing mathematics they understand, and that they already have in their toolbox,” said judge Ben Galluzzo (Shippensburg University), who served as a panelist at this year’s final presentations, which took place on April 25. “And that’s what really makes me excited about the contest, that you have this opportunity to take part in something that’s different from the classroom. It gives you a taste of what real-world math would be like.”

Galluzzo and fellow judges Katie Fowler (Clarkson University) and Karen Bliss (Virginia Military Institute), who was also on the panel, wrote this year’s car-sharing problem. The problem asked participants to classify U.S. drivers based on their extent of car usage and create a model to determine which of four given car-sharing options—roundtrip sharing, one-way sharing with manual car repositioning, one-way sharing with designated stations, and multiple ownership of cars—would work best in different locations. The problem also asked students to consider the impact of future advances, such as self-driving and alternative-energy vehicles, on car-sharing.

Mathematical models, which use various mathematical techniques to measure real-world situations and relay relevant conclusions, can identify the types of car-sharing that would be most beneficial based on population density, traffic, geographical location, and demographics.

“Car-sharing is a very timely and complicated problem, in that it brings in economics as well as urban studies, and many different parts of math,” said Steven Strogatz, a SIAM Fellow and speaker at the awards ceremony. “I saw students using probability theory, some geometry, some calculus, so it’s reflective of the challenges of math modeling if the students do go on to professions that use math.” 

To tackle the complicated Challenge question, the first-place team from Saint John’s School in Houston, Texas, created a function that determines the expected number of miles driven per day based on the population density and number of driving hours for multiple regions. The students then produced a normal distribution around the expected average value for the number of miles driven per day, and integrated a weighted cumulative density function of that distribution over time. This allowed them to categorize drivers as low, medium, or high car users based on hours and daily mileage. 

The team validated its model by testing in two starkly different regions—New York City and suburban Englewood Cliffs, NJ—and demonstrating that the former had a larger proportion of cars moving shorter distances while the latter had a larger proportion moving longer distances. Next, they determined which of the four given types of car-sharing would work best in four pre-determined cities.

“We took about 200,000 data points from a traffic survey in 2009 and found data about population density in the four cities we were assigned to analyze,” said Margaret Trautner, member of Saint John’s winning team. “We also used a lot of data about how humans move around in general, like how fast people walk, how much they walk per day, and how close they live to different stations.”

The team determined the “price” of car-sharing for a user based on both financial and opportunity cost, or time spent by a user in combination with the value of a user’s time. Charting cost versus user salary, the students factored in an individual’s salary and specific situation (such as how long and how frequently he/she would need a car) to determine which car-sharing option would be most appropriate for a given user. This user-benefit model, combined with the population density of a given region, then estimated the number of potential car-share users in each region. The revenue and expenditure per user for each business model, along with the potential number of users in each region, provided an estimate of a car-sharing company’s expected profit.

Government websites that analyze traffic, provide health-related information about how far people walk per day, and offer housing analysis were helpful in perfecting the model, said Trautner. Among the four cities specified by the problem, the winning team found that Richmond, VA, and Poughkeepsie, NY, would be most profitable for car-sharing, Richmond because it has more individuals overall that can afford car-sharing and Poughkeepsie because it has more individuals within a given area. The other two cities—Riverside, CA, and Knoxville, TN—wouldn’t fare as well. The students determined that the best business models were the free-floating and one-way models. While the round-trip Zipcar model yields a relatively equal profit, an individual is much more likely to use a one-way car. The team then adjusted this for usage, cost, and revenue to consider the effects of alternative energy vehicles and self-driving cars.

While Trautner, who stumbled across the Challenge during an online search  for scholarships, has never used car sharing herself, she was quick to note its appeal. “Cities are growing a lot, so a lot of people can’t afford vehicles, but they have to get around huge cities,” she said. “I’m from Houston, and it’s a huge city. We don’t have public transportation that works very well, so everyone has to have some sort of vehicle they can drive around in.”

The champion team from Saint John’s School—which will split $20,000 in scholarship money—consisted of seniors Nancy Cheng, Eric Gao, Daniel Shebib, and Anirudh Suresh, in addition to Trautner. The team, selected from over 1,100 teams across the country, was deservedly elated. 

"We were looking at a picture of last year’s finalists winning the championship, and it feels kind of reminiscent of that situation. Being in their shoes is a really incredible thing,” said Anirudh Suresh. “It’s also really cool that we’re from Texas, since the other top teams are from the east coast area. To see a team from this region opens up the possibility for future years.”

This was both the first team from Saint John’s to ever compete in the M3 Challenge and the first team from Texas to win the competition.

Dwight Raulston, coach of the champion team and instructor of mathematics and English, praised the Challenge’s creativity and positive impact on students. “You’re taking ideas apart and putting them together in different ways, providing your own unique contribution, and you get something out of it,” he said. “I think this is, in a sense, an artistic endeavor. You’re creating something for other people to use and enjoy, analogously to how artists create art.”

Visit the M3 Challenge playlist on SIAM’s YouTube channel for contest videos.

Lina Sorg is the associate editor of SIAM News.
Karthika Swamy Cohen is the managing editor of SIAM News.