Minimizing Multimodular Functions and Allocating Capacity in Bike-Sharing Systems

Abstract

Abstract. 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.

Discrete Convex Analysis and Its Applications in Operations: A Survey

“…Freund et al. (2017) show by definition that for each random sequence X , is multimodular in , and thus so is , or equivalently is ‐convex by Example 2.2. Based on this observation, Freund et al.…”

“…Based on this observation, Freund et al. (2017) provide a steepest descent algorithm which outputs an optimal reallocation within γ iterations and a polynomial scaling algorithm.…”

“…In the dock reallocation problem, define fðxÞ ¼ min Shioura (2018) shows that f(x) is M-convex, which means that problem (23) can be reformulated as a special case of problem (MML1). For the general problem (MML1), Shioura (2018) provides a variant of the steepest decent algorithm, and shows that a minor modification of his algorithm turns out to be the steepest descent algorithm provided in Freund et al (2017). Moreover, Shioura (2018) shows that problem (MML1) can be reduced to an unconstrained M-convex function minimization problem which can be solved by a fast proximity scaling algorithm.…”

A co-evolutionary matheuristic for the car rental capacity-pricing stochastic problem

“…Bike-sharing, for example, has been a key driver of research on managing capacity to meet demand better, considering variations throughout time and space. Relevant works have focused on capacity reallocation, such as Freund, Henderson, and Shmoys (2017) , yet only a few works have focused on the role of pricing in influencing demand, like Chemla, Meunier, Pradeau, Calvo, and Yahiaoui (2013) . Nevertheless, this mobility system shows some differences to the car rental business, which hinder the direct application of the developed techniques, such as the homogeneity of the fleet, the design of the repositioning schemes and the motivations (and consequent distribution) of demand, among others.…”

Separable Convex Resource Allocation Problem With L1-Distance Constraint

“…Let us consider a situation where the original allocation of a resource is given by a vector y and we are required to re-allocate the resource. In such a situation it is often the case that we have a constraint that a new allocation x is close to the original allocation y, which can be represented by the L 1 -distance constraint ∥x − y∥ 1 ≤ K. Indeed, such a constraint is used by Freund et al [4] in the formulation of the bike allocation problem in a bike sharing system.…”