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Executive Summary  We live in an era of unprecedented mobility—vehicles are much more affordable than they were at their inception in the early 20th century, and public transport provides an easy and economical means of travel for those without a personal vehicle. The latest trend in the transportation industry is that of car-sharing. Realizing that purchasing and owning a personal vehicle can be unnecessarily expensive, individuals are starting to turn to cheaper and more distributed means of paying for private vehicle transport. In order to help illuminate various aspects of the car-sharing process, our team developed mathematical models that address some of the main factors influencing car-sharing companies’ decisions. First, we developed a model that determines the proportion of drivers that fit into categories--low, medium, and high—for both hours driven per day and miles driven per day. We realized that much of the information regarding these two factors depended greatly on the amount of traffic in an area or city, which subsequently depended on the population density of that region. Hence, we created a function that gives the expected number of miles driven in a day based on the population density of the city or region and the number of hours driven in a day. We then placed a normal distribution around this expected average value and integrated a weighted cumulative distribution function of that distribution over time to get a table of proportions of drivers in each category. Next, we tested our model in two regions, New York City and Englewood Cliffs, a small suburban locale. Our model produced logical results in that it predicted a majority of cars moving shorter distances in New York City and a majority of cars moving longer distances in the less densely populated town of Englewood Cliffs. We were also asked to create a model to rank four potential business plans for car-sharing companies in four different cities. We found an equation to model a “price” for the consumer that included both financial cost and opportunity cost, which represents a combination of time spent and the value of that time. We graphed the cost versus user salary for each of 4 different consumer scenarios to determine which potential business plan would be most beneficial given a user’s salary and scenario. This user-benefit model incorporates the population density of a region to give the quantity of users for a car-sharing business in that region. We then calculated the company’s revenue and cost per user for each business model and combined these calculations with the number of users in a region to get the expected profit. We applied this to the four cities and ranked them. This analysis would be highly beneficial to any car-sharing company wishing to expand to a new urban location. Finally, we were asked to consider the effects of alternative energy vehicles and selfdriving vehicles on the car-sharing market. We altered our model from Part II to adjust for the changes in usage, cost, and revenue to show the effects of these future changes. Any company wishing to develop a car-sharing business should consider these insights and future changes in order to keep their service relevant in the fast-paced world of automobile technology.

Car(e) to Share? A Mathematical Anaylsis of the Car-Sharing Industry