The science of bike sharing

University braniacs use math to solve bike sharing availability problems.

A century and half of rail freight operations research might also be applied to management of bicycle sharing systems.

Back in the late 20th Century, when America mostly still cared about higher education, I took an introductory course in Operations Research. Operations Research (OR) uses math to manage systems. At the most basic level, OR quantifies individual tasks in an effort to improve productivity. Today, it’s what enables companies like UPS — which aggressively uses OR to improve the flow of packages while reducing costs — to “run the tighest ship in the shipping business.” OR is used extensively in any transportation intensive business.

And now, Dr. Tal Raviv and Prof. Michal Tzur from Tel Aviv University’s Department of Industrial Engineering propose mathematical models to improve the way bikes are allocated at bike sharing stations.

Today, many users of the busiest bike share systems experience frustration with unavailable bikes at their starting locations and unavailable slots at their destinations. Managers work with what they have, anticipating demand while moving bikes around the system.

“These stations are managed imperfectly, based on what the station managers see. They use their best guesses to move bikes to different locations around the city using trucks,” explains Dr. Raviv. “There is no system for more scientifically managing the availability of bikes, creating dissatisfaction among users in popular parts of the city.”

Raviv and Tzur normally develop mathematical models for railroads and other shipping methods, using formulas like the below to optimize railroad car use and avoid having empty cars sitting idle at one end of the country while cargo sits waiting at the other end because of a lack of rail cars.

operations research railroads

Taipei bike share

An environmentalist, Dr. Raviv wants to see more cities in America adopt the bike-sharing system and worked to adapt rail car optimizations to the problem of bike share systems. Dr. Raviv, Prof. Tzur and their students have created a mathematical model to predict which bike stations should be refilled or emptied — and when that needs to happen. In small towns with 100 stations, mere manpower can suffice, they say. But anything more and it’s really just a guessing game. A computer program will be more effective.

The researchers are the first to tackle bike-sharing system management using mathematical models and are currently developing a practical algorithmic solution. “Our research involves devising methods and algorithms to solve the routing and scheduling problems of the trucks that move fleets, as well as other operational and design challenges within this system,” says Dr. Raviv.

The researchers presented their proposal last November in Austin at INFORM 2010, an international Operations Research conference, and plan to implement their OR for bikes software at a bike share pilot program in Tel Aviv. The city government in Tel Aviv plans a bike share with 1500 bike distributed around 150 rental kiosks.


  1. I’m guessing that the complicated looking equations just say to identify the busier pickup and dropoff locations and spend the most person-hours shuttling the bikes from one to the other.

    Or just price one way trips accordingly. If a rental is normally $1, then charge $2 if you are leaving the bike in a spot too many people drop off, or nothing if you return the bike to a place picked up from too often. The $2 rental is essentially going towards paying someone to shuttle bikes. The $0 rental is justified because someone is doing the bike moving for you.

  2. The formulas actually come from a preso Raviv did on *ahem* minimizing fueling costs for locomotives, given different fuel prices at different locations and constraints like fuel tank size, routing and so forth.

    So it’s a lousy illustration for this article, but I figure nobody would actually know the difference.

  3. Indeed, as the editor commented, the mathematical model is related to locomotive fueling problem. See
    In case you are interested in a formulation of the bike-sharing model you can see our working paper in
    The idea of pricing incentives is very popular in air transportation but it is almost irrelevant mass transit transportation systems such as bike-sharing. The vast majority of the users in such system are regular commuters who use it in on their way to work or school. It is very unlikely that this demand can be diverted to more desirable routes or times (from the system’s perspective).
    There are two crucial factors in the success of a bike-sharing program:
    1. A large an dense network of stations must be deployed in order to exploit economies of scale and network
    2. An effective repositioning mechanism should be operated in order to meet the highly fluctuating and asymmetric demand

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