Minimizing Multimodular Functions and Allocating Capacity in Bike-Sharing Systems

The growing popularity of bike-sharing systems around the world has motivated recent attention to models and algorithms for the effective operation of these systems. Most of this literature focuses on their daily operation for managing asymmetric demand. In this work, we consider the more strategic question of how to (re-)allocate dock-capacity in such systems. We develop mathematical formulations for variations of this problem (service performance over the course of one day, long-runaverage performance) and exhibit discrete convex properties in associated optimization problems. This allows us to design a practically fast polynomial-time allocation algorithm to compute optimal solutions for this problem, which can also handle practically motivated constraints, such as a limit on the number of docks moved in the system. We apply our algorithm to data sets from Boston, New York City, and Chicago to investigate how different dock allocations can yield better service in these systems. Recommendations based on our analysis have been adopted by system operators in Boston and New York City. Beyond optimizing for improved quality of service through better allocations, our results also quantify the reduction in rebalancing achievable through strategically reallocating docks.