1. Introduction
Bike-sharing systems (BSSs) have been widely adopted around the world, serving as “last-mile” connections to public transit networks for urban residents. As a typical environmentally sustainable travel mode, bike-sharing can largely contribute to the mitigation of traffic congestion, improvement of air quality, and reduction of carbon emissions in urban areas. Zhang and Mi [
1] report that bike-sharing in Shanghai decreased carbon emission by 25,240 tons in 2016. Mobike, the largest Chinese BSS operator, claimed in 2017 that the total carbon emission reduction enabled by all of its users had reached 540,000 tons, which was measured by the total distance traveled by the “Mobikers” [
2].
In recent years, a relatively new model of BSS, known as the free-float bike-sharing (FFBS) system has increasingly gained its popularity, especially in many Chinese cities. As opposed to the traditional station-based bike-sharing (SBBS) systems, where bikes are stored at specific kiosks (see
Figure 1a for a representative system, Citi Bike in New York City), FFBS systems are dockless, and allow users to pick up and return bikes anywhere within their service region. In particular, owing to advanced information technology, the Chinese market has witnessed a surge in launches of privately operated FFBS systems since 2015. According to Chinese Ministry of Transport, by 2017 the number of dockless bikes and their users have reached 16 million and 130 million, respectively [
3]. Mobike
https://www.mobike.com (
Figure 1b) is a leading FFBS system operator, and has expanded its presence in numerous domestic and overseas cities.
Despite the convenience and flexibility provided to users and its contribution to the sustainability of urban transportation, FFBS systems also face numerous challenges. Owing to the fluctuating and asymmetric demand for rides throughout the day, the spatial distribution of bikes is highly imbalanced [
4]. It is commonly observed that parking areas and even sidewalks near bus stations are packed with bikes (
Figure 2 shows a bus stop blocked by an excessive number of bikes parked there), while the availability of bikes is rather low in some other locations. It is necessary for the operator to relocate bikes across areas at appropriate times to rebalance the system. A widely adopted rebalancing tactic is the operator-based approach, characterized by a fleet of trucks and staff dedicated to manually transferring bikes across different regions. However, this approach is often implemented at certain times during the day, so that bike numbers cannot be adjusted in real time. It is associated with considerable costs and an environmentally unfriendly operating mode since truck routing is extensively used in rebalancing.
In this study, we consider a different strategy for rebalancing BSSs, namely a user-based approach where the system operator can provide users with monetary incentives to induce repositioning activities that are beneficial for the system balance, such as changing the destinations of certain rides. Such an approach has in fact already been practiced in BSSs that are in operation, such as Vélib in Paris [
5] and Citi Bike [
6]. Mobike also offering cash rewards to encourage users to move bikes from low-traffic areas to high-traffic spots like a subway stop or business district [
7]. Given the increasing practical adoption of user-based rebalancing and the existing gap in research on this for BSS, we set out to address research questions regarding when system operators should provide customer incentives, and how large these incentives should be.
One problem faced by system operators, particularly those newly starting their businesses in certain cities, is the lack of information on demand and customer behaviors [
8], e.g., their arrival rates, usage patterns, and willingness to accept the incentives. This problem occurs fairly commonly, given the recent rapid expansion of BSSs in China. Inferring this information from historical operational data in other cities, however, may not be effective. This is because that customers can be heterogeneous, and different incentive plans result in varied estimations, which further affect the estimation quality and consequent decision-making.
From a theoretical perspective, methodologies such as robust optimization (RO) can be deployed to resolve problems of limited demand information. The key idea of RO is to identify an uncertainty set of the unknown model parameters, and then optimize against worst-case realizations within this set [
9,
10,
11]. However, for our problem settings where there exist various stochastic components regarding demand patterns, the structure of uncertainty set is very complicated and prevents the derivation of concise analytical results by the RO approach. In contrast, we adopt the ranking and selection (RS) method, which is a data-driven approach in the simulation optimization (SO) field, with the advantage of determining the optimal solution with a certain probability by means of limited experiments.
For example, when a BSS operator enters a new market, limited information exists regarding customer demand and behavior, and it is difficult to construct an RO model. However, with the RS approach in our study, no assumptions are made regarding the factors such as customer types, arrival rates, and ride times. We assume that a firm can only observe real-time data of these quantities. In certain cases, these variables can be estimated using historical data, which is somewhat useful for theoretical analysis, and offers certain benefits. However, the main drawback of this method is that the quality of estimation may significantly affect the results. Furthermore, when a firm enters a new market, there is little possibility of obtaining similar historical data, while the RS approach fully utilizing the realized system outcomes data can be more suitable in such a setting.
Focusing on a typical FFBS system in the urban area, we describe its operational processes in detail, including the system’s network structure, system users’ demand characteristics, how the incentive provision interacts with ride patterns, and the associated costs incurred. The operator’s profit maximization problem is formulated with a service-level constraint. That is, the system operator will guarantee a desirable service availability to customers. Owing to its complexity, it is challenging to solve this optimization directly, and we use discrete event simulations to compute the profit and deploy RS, which is an extensively adopted SO approach, to derive an optimal incentive plan.
This paper makes the following contributions to the literature on BSS. First, we develop a network model to describe the operational processes of a BSS, and describe the manner in which an incentive program can influence customer travel behavior, and thus the number of bikes across different areas of the system. A formal profit optimization model with a constraint on the customer service level is constructed to reflect the operator objective. Second, considering the limited information problem in decision making, we present an RS procedure as the main methodology to determine the optimal incentive plan, which is feasible even when the model’s random variables can follow any type of distribution. RS takes full advantage of the observed real-time demand and significantly shortens the time required to derive the optimal incentive plan with a certain probability. This is very helpful for system operators with little relevant historical data, which is common in practice. To the best of our knowledge, we are among the first to tackle the BSS rebalancing problem with SO approach such as RS. Third, through extensive numerical experiments, the validity and effectiveness of our procedure are tested against different model variants, parameter settings, and customer behavior scenarios. We find that the system operator can identify the optimal incentive plans with reasonable sample sizes.
The remainder of this paper is organized as follows. We review related literature on the BSS rebalancing problem, simulation optimization and RS approach in
Section 2. In
Section 3, we construct a network model for the FFBS system, and describe its operational processes in detail, particularly the manner in which the incentive program facilitates the system balance through its interactions between the operator and customers. The operator profit optimization problem is also formally presented. The discrete event simulation process and the RS procedures are demonstrated in
Section 4, followed by comprehensive numerical studies for testing the validity and effectiveness of our procedures, as well as several further extensions, in
Section 5. Finally, we conclude our paper in
Section 6.
2. Literature Review
Our paper is closely related to two research streams: studies on rebalancing operations in bike-sharing systems, and the RS method in SO literature.
The study of BSS has been an emerging research area in recent years. Demaio [
12] and Shaheen et al. [
13] discuss the history and future directions of BSS in detail. Readers are referred to [
14,
15,
16] for more recent reviews. Gavalas et al. [
17] examine existing literature concerning the design and management of vehicle-sharing systems, and classify them into strategic, tactical, and operational levels, based on the planning horizon. Under a similar framework, Laporte et al. [
18] provide an extensive literature review of shared mobility systems. Various topics are discussed regarding the economic, societal and environmental issues of BSS, such as its externalities on the environment and public health [
19], user behavior characteristics [
20,
21], host city resident acceptability patterns [
22,
23], strategies to improve its performance, profitability, marketing appeal [
24,
25,
26] and its role in normalising cycling [
27]. In the remainder of this section, we mainly review the literature on rebalancing operations given is closer relevance to our work.
BSS rebalancing approaches can be classified into operator-based and user-based strategies, where the former corresponds to the optimal dispatching of trucks to reposition bikes across different areas in a system, and the latter emphasizes directing customers to select more appropriate origins/destinations to realize a more balanced system (see [
28] for a specific subdivision of rebalancing strategies). Certain studies consider
static rebalancing, which is generally carried out at night when few customers are using the system. Chemla et al. [
29] treat this as a pickup and delivery problem and present efficient algorithms. See [
30,
31,
32,
33,
34], among others, for further research in this stream. In
dynamic rebalancing, which is often performed during the daytime, operator transfers and user trips occur simultaneously, so more real-time information must be taken into account. Representative literature in this stream includes [
5,
35,
36,
37,
38].
Although various user-based rebalancing schemes exist, such as system regulation [
39] and parking reservation policies [
40,
41], offering users monetary incentives is the most commonly studied scheme in the literature, and we do not distinguish between incentive-based and user-based approaches in this paper. In this line of research, Febbraro et al. [
42] first propose the use of a fare discount to motivate users to change their destinations. In addition, Pfrommer et al. [
43] examine the combination of dynamic truck routing and an incentive scheme design in bike redistribution, using model-based predictive control principles. Singla et al. [
4] extend their work by incorporating a budget constraint on incentive provision and a learning procedure regarding a user utility function in optimal pricing into the model. [
44] propose a deep reinforcement learning algorithm to solve a incentive-based rebalancing problem, which consider both spatial and temporal features in the system. Other related research includes [
45,
46,
47,
48,
49,
50]. Almost all of these studies conduct analyses on SBBS systems. Reiss and Bogenberger [
51] study a user-based approach to a FFBS system, where the authors discuss the advantages and applicable scenarios for both operator-based and user-based relocation strategies for the FFBS system in Munich, and validat their results through an empirical analysis using GPS-booking data. However, in contrast to their analysis based on certain given relocation policies, our study constructs a more detailed quantitative model, and derives the optimal incentive scheme.
Given the complexity of our model due to the various stochastic components therein, and the assumption that limited information is available regarding customer demand, it is challenging to obtain analytical results of the optimal incentive plan. Consequently, we employ RS from the SO field to address the problem. A SO problem is a nonlinear optimization problem in which the objective function is defined implicitly through a simulation and can only be observed with error [
52]. The problem is frequently seen in various real-world applications such as manufacturing line design, inventory control, healthcare system operations [
53]. Readers are referred to [
54,
55] for more detailed introduction to SO. RS is a widely used approach in solving SO problems with finite solutions. It aims to select the superior option from a set of systems, which has the largest mean value under the sequential analysis framework. As opposed to the fixed sample size analysis, where a certain amount of data is required prior to carrying out the analysis, RS uses the information from every new sample, and can save numerous samples with a similar probability of correct selection (PCS). Essentially, there are two approaches in RS: the frequentist and Bayesian approaches. The frequentist approach can select the optimal system satisfying a required PCS (see [
56] for a basic introduction). The Bayesian approach focuses on maximizing the posterior PCS given a fixed total sample size. Comprehensive literature reviews on RS are provided in [
57,
58].
Using the frequentist approach, Bechhofer and Sobel [
59] first propose a procedure based on the indifference-zone parameter
, where
is the smallest difference of concern. Paulson [
60] improves these results by introducing the sequential analysis method. To improve the performance, Fabian [
61] develops a tighter bound of selection. Based on these advancements, Kim [
62] presents an effective procedure called the Kim-Nelson (KN) method. In addition to focusing on improving the bound, Hong [
63] employs a variance-dependence sampling method to reduce the required sample size.
In this study, the best incentive plan must be selected under a service-level constraint. Most RS procedures adopt a non-constrained optimization framework, such as the KN procedure [
62]. Andradottir and Kim [
64] and Hong et al. [
65] present two types of procedures, taking the constraint into consideration. In this paper, we employ the AK+ procedure described in [
64] to solve our incentive plan selection problem. We provide a brief description of the AK+ procedure in
Section 4.
4. Procedures for Optimization Based on RS
In this section, we reformulate the model in
Section 3 under the SO framework, and present the related RS procedures. We can rewrite the optimization model as
is a finite set in which all possible solutions are included, represented by the product of sets
.
is the node set, so
in the model in
Section 3.
denotes all possible incentive values.
are finite sets denoting the feasible solutions for both
and
, and
denotes all stochastic variables in the expression of
. It should be noted that these sets contain a finite number of elements, corresponding to discretized feasible solutions. This is quite reasonable, because in practice incentives can only be provided with certain values (for example, if the charge per ride is
$1, the incentive can be
$0.1,
$0.2, etc.), while the number of bikes is naturally an integer. We add a constraint
for each feasible solution
j, which is to ensure the optimal solution derived can provide the desirable service level. Now, we can deploy the RS procedure to determine the optimal solution numerically. In a more general manner and using the terminologies in the RS field, we let
k be the number of elements in
and
i be an arbitrary element in
. Then, the feasible set can be presented abstractly as
, and we want to determine
The RS approach aims to select the optimal solution from a set of solutions, and it belongs to the field of sequential analysis theoretically. As opposed to the fixed sample size analysis, RS compares contending solutions following every observation. In this manner, it can take advantage of each observation. If the statistical variable reaches a given threshold, we can declare the results with a required probability for correct selection. Sequential analysis usually can curtail the comparison process and save numerous samples. We use the AK+ procedure presented by [
64] as the main RS algorithm.
AK+ is a fully sequential ranking and selection procedure which can solve constrained SO problem. Before the invention of AK+ procedure, most ranking and selection procures were focusing on unconstrained SO problems. AK+ procedure can choose the system with best or worst performance under certain constraint among a set of systems. In our model setting, it means that the procedure can choose the best incentive plan within those plan candidates that can achieve desired service level. Besides, as proved in Theorem 4 in [
64], AK+ procedure can guarantee that the probability of correct selection is larger or equal to
, which is given artificially. In our problem, it means that we can choose the best incentive plan with a desired probability of correct selection. This is the reason why the chosen plan is optimal.
We provide two procedures for conducting SO by means of RS. Procedure 1 provides an illustration of the daily operational process of the FFBS system, with which one sample consisting of a profit and service level is generated. Procedure 2 is our main procedure, describing the selection of the optimal solution through sequential sampling and comparison. Because the system operator has no information regarding the statistical distribution of the stochastic components in the model, it is necessary to obtain
samples to estimate related information. This procedure will determine the feasibility of solutions and compare the performances of different solutions simultaneously. In the feasibility check, solutions that cannot satisfy the service level constraint will be eliminated. In the comparison step, inferior solutions with a low average daily profit will be deleted. Using these two steps, we can select the optimal system with the highest average daily profit and required service level. Note that for Procedure 1, since it focuses on a single ride with pre-determined origin and destination, we omit the corresponding subscripts somewhere for brevity. For Procedure 2, given that it is mainly base on AK+ procedure [
64], we omit some unnecessary details in its specifications.
4.1. Procedure 1 (Daily Operations Procedure)
Setup. Set the incentive I, the initial amount of bikes for each node, price per ride p, unit penalty cost , and the probability of incentive acceptance . Generate the customer arrival time and customer type for each customer , according to and , where denotes origin and destination and . M is the total number of customers, satisfying . Let the customer counter .
Single ride profit calculation. For customer
m with customer type
and arrival time
, if
, then let
. If
, first determine whether the following three conditions to trigger an incentive program hold:
,
, and
, where
denotes the other nodes in the same region as
d, which can be an alternative destination. If one of the three conditions holds, generate
; otherwise, set
. If
P is generated and
, then set
. If
, set
. The system dynamics of bike numbers follow the description in
Section 3.1. Let
.
Profit calculation. If , return to Single ride profit calculation. Otherwise, calculate the total cost C, including the overage and truck-based relocation costs. The total profit for a single day .
4.2. Procedure 2 (Customer Incentive Plan Procedure (CIPP))
Setup. Select the initial sample size
, overall confidence level
, service level
, and indifference zone parameters
and
. Solving
and
from the following equations:
The two equations are necessary for the proof of the probability of correct selection in our RS procedures, see [
64] for more details. Define
and function
, which will be used in the following procedures.
Initialization. Let
be the initial contending plan set. Let
and
, where
F and
denote the set of feasible solutions and the set that records the solutions inferior to solution
i, respectively. Run
Procedure 2 times and obtain observations
and
,
from each system (feasible solution)
i. Define a counter
, which is the variable recording the sample size needed for the procedure. For system
i, calculate its service level variance
for
with
where
is the initial sample mean of service level. For all system pairs
, calculate the profit variance
between systems by means of
Feasibility check. For , if , then move i from to F. For all l in , eliminate l from or F. If , let .
Comparison. For any pair of
i and
and
,
,
, and
If , then eliminate l from or F. If , add index l to .
Stopping rule. If and , then stop and select the system with an index in F as superior. If , then declare that no feasible system exists. Otherwise, take one more observation of all systems in or F by Procedure 2, set , and return to Feasibility check.
6. Conclusions
In this study, we present a rebalancing scheme based on customer incentives for free-float bike-sharing systems, an environmentally sustainable travel mode that gains its increasing popularity. We describe the dynamic operational processes of the system under a continuous-time setting, with the network structure and demand patterns specified in detail. A stochastic optimization model incorporating a service level constraint is constructed, which describes how monetary incentives change customers’ riding behavior and influence the revenue accumulation/cost incurrence of the system operator. Given that the problem is often faced by firms with limited demand information so a clear understanding of the stochastic components in the model is unavailable, we adopt the RS approach as the main methodology, which is widely used in solving simulation optimization problems. Using our proposed procedure, the system operator can select the optimal incentive plan with a reasonable sample size. By means of comprehensive numerical studies, we determine that the optimal customer incentive-based rebalancing plan can not only significantly improve the average daily profit, but can also achieve a higher service level. The overall performance of the procedure, including its validity and effectiveness, is tested for various parametric settings and model variants, demonstrating the wide applicability of our approach across different scenarios.
It should be pointed out that in order to provide a concise description of the complicated system and to facilitate the tractability of the problem, several somewhat strong assumptions are adopted in this paper. For example, we assume the cost/demand parameters and the incentive plans provided are node-independent and stationary in time. These settings may to some extent limit the practical applicability of our model. Notwithstanding, our study is among the first attempts to employ simulation optimization methodologies such as RS, in solving rebalancing decisions in FFBS systems. Our results imply that well system performance can be attained even if little information on demand is available, which can also provide insights for incentive-based rebalancing in the real world. Given the increasing attention paid to FFBS systems by both researchers and practitioners, we believe there are abundant research opportunities in the future, and we list several below.
First, one direction for future research may focus on more general extensions of our model, which can incorporate non-stationary demand process, node and region-specific incentive plans, etc. Second, most of the parameter settings in our numerical investigations can reflect the operational practice, but are still mostly artificial. Given that empirical analysis based on data from real-world BSSs are critical and has also been used in existing studies, another possible extension can be case studies with computational procedures adjusted to a large-scale model, which will increase the applicability of our methodology. Finally, apart from including company profit and customer service level into our decision model, given that the critical role played by FFBS systems in reducing carbon emission and air pollution, we may consider some environmental performance measures in future research.