Integrated rebalancing and faulty bike recycling in dockless bike-sharing systems: a hybrid metaheuristic approach with dynamic clustering

Abstract

Bike-sharing systems have become increasingly popular worldwide due to their affordability, efficiency, health benefits, and environmental friendliness. However, dockless bike-sharing systems face significant bike imbalance and faulty bike accumulation problems that negatively affect service quality, revenue, and user safety. This article addresses the integrated rebalancing and faulty bike recycling problem using a vehicle routing framework with partitioned vehicle capacity. For small-scale validation, the model is solved exactly using the CPLEX optimizer tool on 38 Nanjing stations. To enable large-scale applicability, we develop a hybrid genetic algorithm (GA)–adaptive large neighborhood search (ALNS) metaheuristic with cluster-first, route-second decomposition. Experiments show near-optimal solutions (gaps <3.2%) for 200-station instances within practical timeframes. Dynamic clustering reduces routing distance by 6.5% versus static approaches, while faulty bike integration adds only 8–12% computational overhead yet eliminates separate collection trips. Results confirm the framework effectively minimizes user dissatisfaction and operational costs while ensuring efficient faulty bike recycling.

1 Introduction

Bike sharing is one of the fastest-growing new modes of transport in the world, with more schemes opening every year (Caulfield et al., 2017). It allows people to get/rent a bicycle when needed at their nearest location at one of many rental stations scattered around the city. In July 2014, Metro Bike LCC (Du and Cheng, 2018) observed that 721 cities had a public bike-sharing system, with a total of approximately 814,000 bicycles, and 228 more were planned or under construction. These systems range from less than one hundred bicycles in small towns to many thousands in cities such as Paris (20,600), Hangzhou (78,000), or Wuhan (90,000). The bike-sharing system allows people to use bicycles for a short time and, after using a bicycle, return it to any other station in the city. This system adds to environmental sustainability and encourages people to find a socially equitable mode of transportation. Such systems also serve to complement other modes of mass transit systems by mode sharing.

Worldwide, there were 535 schemes by April 2013, in 49 countries (Borgnat et al., 2011). Bike network design (Caggiani and Ottomanelli, 2012), bike demand analysis (Cervero et al., 2009), bike network flow analysis (Chemla et al., 2013), bike service-level analysis (Contardo et al., 2012), and many other issues are important research topics. Dockless bike-sharing systems in recent years have emerged in China and many countries with the development of the mobile Internet and global positioning systems (GPS) becoming increasingly affordable. This system can effectively solve the last-mile problem in multimodal transportation. In China, bicycle usage is higher than in many other countries, which results in exclusive use of bike sharing. Compared to traditional bike sharing, bike-sharing systems without docks are more flexible and comfortable for users. The wide distribution of bicycles provided by bike-sharing systems reduces travel time and distance from a traveler to a bicycle docking station. Users register themselves in the app while using a bicycle for the first time, scan their identification, and give a deposit. Then, the map displays all the bicycles around users and locates the nearest bicycle. A quick response code is scanned by the user, and within 10 s, they can hear the click and depart.

To address the inherent imbalances in bike availability and ensure user satisfaction, operators deploy a fleet of dedicated service vehicles, commonly light trucks equipped with bicycle racks, to transport bicycles between stations. These rebalancing vehicles collect surplus bikes from stations with excess inventory and deliver them to stations with deficits, thereby correcting the imbalances caused by asymmetric travel patterns. The efficient routing and loading/unloading scheduling of these trucks constitutes the core of the rebalancing problem. It is not necessary for the trucks to visit all bike stations, which is a clue to the fact that the problem involves selecting and determining the stations where bike numbers fluctuate. Because of the pick-up and drop-off quantities, in deterministic fluctuation and shortage, the bicycle rebalancing problem is more complex and operationally challenging than the vehicle routing problem (VRP) (Dell'Amico et al., 2014) and the classical traveling salesman problem (TSP) (Erdoğan et al., 2014). Unlike VRP, in which decisions are purely about routing, the rebalancing problem involves (1) determining the quantity of bikes to load or unload at each station based on inventory levels and demand forecasts, (2) managing vehicle capacity for both rebalancing and faulty bikes simultaneously, and (3) minimizing user dissatisfaction due to unmet demand, introducing a service-level objective beyond simple distance minimization. Fricker and Gast (2016) call this the rebalancing problem.

With the rapid development of Internet technology and mobile payments, free-floating bike sharing (FFBS) or dockless bikes began to appear in China in March 2016 (Biehl et al., 2018). In 10 years, China built the world’s leading classical city docked bikeshare (BS) market. However, the dramatic growth of the new innovative dockless bikeshare system has quickly overtaken the BS market. Private operators introduced dockless bikeshare in China in 2016, and since then, it has dramatically spread across mainland China and fundamentally changed the way people travel and therefore their lives. Literature on bike-sharing systems focuses on traditional bike-sharing systems with docking stations, keeping in mind the development policies, safety issues, operation scales, and the operator and benefits to the environment (Gu et al., 2019). To reduce the operating cost and improve the user experience for bicycles, many researchers have studied and are studying the rebalancing position. The first literature study was done by Froehlich et al. (2008) to introduce rebalancing operations for a bike-sharing system and proposed Tabu search algorithms for solving it. Schuijbroek et al. (2017) and Ghosh et al. (2017) presented unified dual-bounded service-level constraints that add inventory flexibility and vehicle routing for static rebalancing in bike-sharing systems.

Hampshire and Marla (2012) worked on developing an optimization formulation to reposition bikes using vehicles, taking into consideration the routes for vehicles and future expected demand to improve bike-sharing availability and reduce the use of private vehicles. A heuristic algorithm for solving the dynamic rebalancing problem with multiple vehicles was proposed by Ho and Szeto (2014), who presented a dynamic pricing strategy that encouraged users to return bikes to empty stations.

Dockless bike-sharing systems have attracted attention recently. Reiss and Bogenberger (2015) and Larsen (2013) created a demand model to obtain an optimal distribution of bikes within the operating area, based on a detailed GPS-data analysis for the bike-sharing system. Caggiani et al. Lin et al. (2011) proposed a method to use the revenue collected by a congestion price policy to implement a bike-sharing system.

Nair et al. (2013) noted that the strongest predictor for bicycle use is the availability of a bike at the nearest and closest place. Effective faulty bike-sharing recycling (FBSR) operations are critical to ensuring bicycle availability. However, literature on faulty bike-sharing recycling is limited.

Ignoring the stochastic nature of the demand, the problem of static bicycle rebalancing can be classified as a variation of the pickup and delivery problem (PDP), a type of problem that has attracted considerable attention in recent years. A study on static PDP was done by Pfrommer et al. (2014), who classified these problems according to various parameters. A single-commodity PDP was studied by Benchimol et al. (2011), Raviv and Kolka (2013), Chemla et al. (2013), Reiss and Bogenberger (2015) without time constraints, with the goal of minimizing the total travel distance of the vehicle as it completed a prescribed rebalancing task. A branch-and-cut algorithm for solving a relaxation of the problem was presented by Chemla et al. (2013) and Romero et al. (2012), which resulted in obtaining a feasible solution via Tabu search. A stochastic programming approach was used by Nair and Miller-Hooks Sayarshad et al. (2012) for handling rebalancing planning in shared mobility systems.

The bike-sharing system behavior and the effect of load-balancing and rebalancing strategies on the performance of BSS are studied by Shu et al. (2013) and Szeto et al. (2016) based on empirical analysis. Along with a self-balancing mechanism, deploying trucks to rebalance bicycles is necessary to overcome the shortage and fluctuations in an asymmetric system. To study the system’s behavior and the effect of various load-balancing strategies on their performance, they conclude that in asymmetric systems, rebalancing of bicycles by trucks is necessary even when an incentive mechanism to self-balance is put in place. Vogel and Mattfeld (2010) present a stylized model to assess the effect of dynamic rebalancing efforts on service levels. Their model is useful for strategic planning but is not detailed enough to support rebalancing operations.

In the rebalancing operation of bicycles, no time constraint is imposed on the rebalancing, so deviation from the target level is not allowed. This study was extended by Erdoğan et al. (2014) by allowing the final inventory at each station to be within a pre-specified interval instead of at a given target value. Loading and unloading bicycles in trucks takes much of the rebalancing time, so the inclusion is crucial to correctly represent the rebalancing problem and rebalancing plan.

To determine the expected bicycle shortage (by number of bicycles) at a station as a function of inventory level at the beginning of the day, which gives an idea of the dynamics of demand at that location, a study conducted by Raviv and Kolka (2013) and Alvarez-Valdes et al. (2016). Liu et al. (2025) developed an adaptive large neighborhood search (ALNS)-based dynamic rebalancing framework that considers spatio-temporal clustering to maximize operational profit while preventing bike oversupply and user dissatisfaction. Wu et al. (2024) recently highlighted that rebalancing and recycling operations are typically treated as independent tasks in existing literature, with most studies addressing them separately rather than jointly optimizing both processes. Their method used the rates of two independent non-homogeneous Poisson demand streams for users desiring to hire bicycles and users trying to return bicycles at a single station, during some planning period, as input. However, their model did not consider a multi-station system. It used a single-station system demonstration through simulation, which did not perfectly specify the target levels.

While individual shared-bike trips are typically short and occur within limited geographic areas, the operational network of a bike-sharing system encompasses the entire city. There are several compelling reasons why large-scale network solutions are essential. First, the scale of real-world systems is substantial. For instance, Shanghai’s major operators manage more than 50,000 dockless bikes spread across the entire metropolitan area, while New York City’s CitiBike system operates more than 1,000 stations with 25,000 bikes. These networks cannot be decomposed into independent small problems without losing operational efficiency.

Second, the benefits of integration, particularly the simultaneous collection of faulty bikes during rebalancing operations, increase with network size. In a 50-station system, the marginal benefit of integration may be modest, but in a 500-station system with a 2% faulty bike rate (10 faulty bikes per 500), eliminating separate collection trips yields substantial savings. Third, large-scale solutions enable operators to achieve economies of scale in fleet management. A single optimized route serving multiple zones reduces total vehicle distance and labor costs compared to separate routes for each small area.

Fourth, the spatial distribution of demand creates interactions across the city. A bike shortage in one area may be resolved by surplus from a distant area, requiring a holistic network view rather than localized solutions. Finally, the dynamic nature of demand with peaks at different times in different parts of the city requires a system-wide optimization to effectively respond to changing conditions. Small-scale solutions that ignore these interdependencies may achieve local optimality but fail to provide system-wide efficiency.

Thus, while individual trips may be short, the operational network requires large-scale solutions that consider the entire system’s dynamics. Our cluster-first approach bridges the gap by enabling large-scale optimization while maintaining solution quality.

While Wu et al. (2024) and Du and Cheng (2018) have explored integrated rebalancing and recycling, our approach fundamentally differs in three critical aspects:

First, methodologically, prior integrated models rely on exact mixed-integer programming (MIP) formulations that become computationally intractable beyond 50 stations. Our hybrid GA–ALNS with dynamic clustering achieves near-optimal solutions for 200+ stations, representing a 4× scale improvement.

Second, operationally, existing studies treat vehicle capacity as a single aggregate constraint. Our partitioned capacity model (Q_k^R for rebalancing, Q_k^F for faulty bikes) enables truly simultaneous operations, whereas prior work implicitly assumes sequential loading/unloading.

Third, strategically, prior clustering approaches are static and geography-based. Our dynamic clustering jointly optimizes cluster count, centroids, and assignments based on expected inter-cluster traversal costs, which is a feature absent from existing integrated studies. This addresses the fundamental trade-off between cluster count and routing efficiency that prior work ignores.

Thus, our contribution does not merely combine known techniques but also addresses the scalability, operational, and strategic limitations of existing approaches through novel methodological integration.

Recent advances in bike-sharing optimization have increasingly focused on integrating operational decisions with emerging technologies and sustainability considerations. Li et al. (2024) developed a deep reinforcement learning framework for dynamic rebalancing that adapts to real-time demand fluctuations, although their approach treats faulty bike management as a separate post-processing step. Zhang and Wang (2025) proposed a bi-level programming model for dockless bike-sharing systems that simultaneously optimizes station placement and rebalancing routes, yet they acknowledge the limitation of not considering bike degradation and faulty bike collection in their framework.

The integration of faulty bike recycling with rebalancing operations has gained attention only recently. Wu et al. (2024) presented a joint optimization model for inventory routing that considers broken bike recycling in bike-sharing systems, establishing a foundation for integrated approaches. However, their model relies on exact solution methods that are computationally intractable for networks exceeding 50 stations, and they do not address dynamic clustering to adapt to spatial demand patterns. Similarly, Chen et al. (2025) explored multi-period rebalancing with vehicle capacity constraints but treated faulty bike collection as an independent maintenance operation scheduled during low-demand periods, missing potential synergies from simultaneous operations.

Methodologically, recent studies have explored various metaheuristic approaches for large-scale bike-sharing problems. Liu et al. (2025) proposed an ALNS-based dynamic rebalancing framework incorporating spatio-temporal clustering to maximize operational profit, demonstrating the value of clustering for large-scale problems. However, their clustering approach is static and does not dynamically adapt based on expected inter-cluster traversal costs. Park and Lee (2026) introduced a hybrid genetic algorithm with local search for static rebalancing, achieving near-optimal solutions for up to 100 stations, but their framework does not accommodate the simultaneous collection of faulty bikes, nor does it address the partitioned vehicle capacity requirement for handling two distinct types of bicycles.

Recent empirical studies have also highlighted operational challenges in dockless systems. Wang et al. (2024) conducted a comprehensive analysis of faulty bike accumulation patterns in major Chinese cities, finding that faulty rates can exceed 2% during peak usage periods and that delayed collection significantly impacts user satisfaction and safety. Their findings underscore the practical importance of integrating faulty bike management within daily rebalancing operations rather than treating it as a separate maintenance activity.

Despite these advances, several gaps remain. First, existing integrated models are limited to small-scale instances due to computational constraints, restricting their practical applicability. Second, dynamic clustering for large-scale problems has not been jointly optimized with the integrated rebalancing-recycling problem, particularly considering the trade-off between cluster count and vehicle routing efficiency. Third, partitioned vehicle capacity management, where vehicles simultaneously carry both rebalancing bikes and collected faulty bikes, has not been adequately addressed in the literature. Our study directly fills these gaps by proposing a scalable hybrid metaheuristic with dynamic clustering that jointly optimizes rebalancing and faulty bike recycling, providing both methodological innovation and practical applicability.

While existing studies on dockless bike-sharing rebalancing have primarily relied on exact MIP formulations that are limited to small-scale instances, and recent integrated models (Wu et al., 2024; Chen et al., 2025) have not addressed the simultaneous optimization of dynamic clustering with partitioned vehicle capacity management, our research addresses the pressing need for scalable solution approaches for large real-world networks. This article introduces a novel solution framework combining a cluster-first, route-second strategy with a hybrid metaheuristic to efficiently solve the integrated rebalancing and faulty bike recycling problem. By decomposing the large-scale problem into manageable clusters and employing an adaptive large neighborhood search (ALNS) algorithm for route optimization, we provide a practical approach that maintains high solution quality while achieving significant computational efficiency. To the best of our knowledge, this is among the first studies to jointly optimize the dynamic clustering process with simultaneous faulty bike collection in dockless systems, directly responding to recent calls for integrated operational strategies in bike-sharing literature (Liu et al., 2025; Park and Lee, 2026).

To answer these questions, the objectives of this study are (1) to develop an integrated mathematical model for simultaneous rebalancing and faulty bike recycling with partitioned vehicle capacity; (2) to design a dynamic clustering approach that optimizes cluster count, centroids, and station assignments based on expected inter-cluster traversal costs; (3) to develop a hybrid GA–ALNS metaheuristic for efficient large-scale solution; and (4) to validate the proposed framework using real-world data from Nanjing and benchmark it against state-of-the-art methods.

2 Literature review

This section provides a thematic review of the literature on bike-sharing operations, organized by problem type, solution methods, and research gaps.

2.1 Classical bike-sharing rebalancing problems

The static bicycle repositioning problem (SBRP) has been extensively studied since the early 2010s. The seminal work of Chemla et al. (2013) formulated the SBRP as a single-commodity pickup and delivery problem with the objective of minimizing total travel distance. This formulation was extended by Erdoğan et al. (2014) to allow final inventory levels within a pre-specified interval rather than fixed target values, providing greater operational flexibility. Dell'Amico et al. (2014) developed a comprehensive mathematical formulation and benchmark instances that became standard in the field.

A major methodological shift occurred with the introduction of multi-vehicle models. Schuijbroek et al. (2017) proposed a unified framework that incorporated dual-bounded service-level constraints and vehicle routing decisions simultaneously. Their work demonstrated that integrating inventory management with routing decisions yields superior solutions compared to sequential approaches.

2.2 Faulty bike management in bike sharing

Faulty bike management has received limited attention compared to rebalancing. Early studies recognized the operational challenge of faulty bikes but treated it as a maintenance problem separate from rebalancing (Froehlich et al., 2008). Operators typically scheduled dedicated maintenance tours during low-demand periods to collect and repair faulty bikes.

The first integrated approach was proposed by Wu et al. (2024), who developed a joint optimization model for inventory routing considering broken bike recycling. Their model demonstrated that integration could reduce total costs by 12–18% compared to separate operations. However, their solution method relied on exact optimization and was limited to instances with fewer than 50 stations. Chen et al. (2025) extended this work by incorporating multi-period planning but still required solving MIP formulations that become intractable for larger networks.

2.3 Solution methods for bike-sharing optimization

Solution methods fall into three approaches. Exact methods (branch-and-cut, branch-and-bound) are suitable for small instances (<50 stations) and provide optimality guarantees (Dell'Amico et al., 2014; Erdoğan et al., 2014). However, their exponential complexity makes them impractical for real-world networks. Metaheuristics (Tabu search, ALNS, and genetic algorithms) have achieved near-optimal solutions for up to 100 stations (Liu et al., 2025; Park and Lee, 2026). Recent hybrid approaches combine multiple metaheuristics to leverage complementary strengths.

2.4 Clustering approaches in bike sharing

Clustering has emerged as a decomposition strategy for large-scale problems. Static clustering based on geographical proximity has been used to partition networks into manageable subproblems (Schuijbroek et al., 2017). However, static approaches do not account for demand patterns or traversal costs. Dynamic clustering, introduced by Liu et al. (2025), adapts cluster boundaries based on spatio-temporal demand patterns. Our work extends this concept by jointly optimizing cluster count and station assignments based on expected inter-cluster traversal costs.

2.5 Research gaps and our contributions

Based on this review, we identify four critical gaps: (1) Existing integrated models are limited to small instances due to computational constraints; (2) Dynamic clustering has not been jointly optimized with integrated routing decisions; (3) Partitioned vehicle capacity management has not been addressed; (4) Scalable solution methods for large-scale integrated problems are lacking. Our contributions directly address these gaps through a hybrid GA–ALNS metaheuristic with dynamic clustering, achieving near-optimal solutions for large-scale instances while maintaining computational efficiency.

2.6 Justification for large-scale network solutions

The spatial interdependence of bike-sharing networks is well-documented in the literature. Fishman (2016) demonstrated that demand-supply imbalances in bike-sharing systems are inherently city-wide, requiring repositioning across non-adjacent zones. Caggiani et al. (2018) quantified this phenomenon in Milan, finding that approximately 38% of rebalancing trips involve stations in non-adjacent zones. Similarly, Wang and Szeto (2018) observed that 45% of peak-hour repositioning in Beijing requires moving bikes across distances exceeding 5 km, far beyond local zone boundaries. These empirical findings confirm that cross-zone flows are not rare exceptions but constitute a substantial proportion of rebalancing operations, warranting city-wide rather than localized solutions.

3 Problem statement

This section formally defines the integrated rebalancing and faulty bike recycling problem addressed in this article.

3.1 Definition

The vehicles have a time budget T for completing their tours, including travel time between stations, loading/unloading time for rebalancing bikes (L and U minutes per bike), and handling time for faulty bikes (t minutes per bike). The travel time between stations i and j is given by c_ij · a_ij, where c_ij is the base travel time, and a_ij is a traffic congestion factor.

3.2 Objectives

3.3 Assumptions

3.4 Key decisions

3.5 Problem complexity

The faulty bike recycling process in our integrated framework operates through a systematic four-stage procedure that seamlessly coordinates with rebalancing operations. First, faulty bike identification is enabled through the IoT-enabled monitoring system installed on each bicycle, which continuously tracks battery status, GPS signal strength, QR code scan success rates, user-reported issues via mobile applications, and usage patterns (bikes idle beyond 7 days are flagged for inspection), generating a prioritized list based on fault severity, duration, station importance, and quantity of faulty bikes at each station before daily operations commence. Second, the collection process begins when vehicles depart the depot with partitioned cargo areas, the main compartment carrying rebalancing bikes and a dedicated partition reserved for faulty bike collection. Upon arriving at each station, operators sequentially unload rebalancing bikes to meet demand, collect all pre-identified faulty bikes using handling time t per bike, load surplus operational bikes, and update station inventory to ensure faulty bikes are removed from the system. Third, transportation and capacity management is governed by Equations 9,10, which enforce that cumulative faulty bikes and rebalancing bikes never exceed their respective partition capacities at any route segment, with physical separation maintained between partitions to prevent cross-contamination and facilitate efficient depot unloading. Meanwhile, the model dynamically adjusts the balance between both cargo types as the vehicle progresses through its route. Fourth, return and processing occur when vehicles return to the depot with remaining rebalancing bikes for redistribution and all collected faulty bikes, which are unloaded and categorized for immediate repair (minor issues), scheduled maintenance (major repairs), or recycling and parts recovery (beyond repair). This integrated approach eliminates the need for separate collection trips by enabling simultaneous operations. Vehicles can carry rebalancing bikes from depot to deficit stations, collect surplus bikes from excess stations, and gather faulty bikes from all visited stations in a single route, reducing total vehicle distance by 15–25% compared to sequential operations, with computational overhead of only 8–12% for managing the partitioned capacity constraints. This approach delivers substantial operational efficiency gains while maintaining high service quality and user safety.

4 Mathematical modeling

4.1 Mathematical formulation

Table 1 presents all sets, parameters, and decision variables used in the formulation. Explanations are provided immediately after each equation that follows the table.

4.1.1 Objective function

The objective function (Equation 1) minimizes three cost components: user dissatisfaction, operational costs, and faulty bike handling costs:

where the first term represents user dissatisfaction, the second term represents travel costs, the third term represents loading/unloading costs, and the fourth term represents faulty bike handling costs. This formulation ensures that all three operational priorities are balanced simultaneously.

4.1.2 Route constraints

Constraints expressed by Equations 2–7 ensure vehicle route feasibility and continuity:

Explanation: This constraint ensures that if a vehicle serves station j, it must arrive from some station i.

Explanation: This constraint ensures that if a vehicle arrives at station i, it must depart to some station j.

4.1.3 Capacity constraints

Constraints expressed by Equations 8–10 manage the partitioned vehicle capacity:

Explanation: This constraint forces the vehicle to collect all faulty bikes from each served station.

Explanation: This ensures faulty bikes carried never exceed the faulty bike partition capacity Q_k^F.

Explanation: This ensures rebalancing bikes carried never exceed the rebalancing partition capacity Q_k^R.

4.1.4 Inventory constraints

Constraints expressed by Equations 11–16 manage station inventory and vehicle load balance:

4.1.5 Sub-tour elimination and depot constraints

Constraints expressed by Equations 17–19 ensure valid tours and depot conditions:

4.1.6 Non-negativity and time constraints

Constraints expressed by Equations 20–28 enforce feasibility:

The proposed GA–ALNS algorithm begins with an initialization phase in which an initial population of feasible routes is generated using a constructive heuristic, with each solution encoded as a permutation of station visits along with the corresponding loading and unloading quantities, and fitness evaluated using the objective function (Equation 1). The genetic algorithm phase then employs tournament selection to identify parent solutions, applies order crossover (OX) for route permutations to generate offspring, and introduces diversity through swap, insertion, and inversion mutation operators while preserving the best 10% of the population through elitism. Subsequently, the adaptive large neighborhoods search component intensifies the search by applying destroy operators including random removal, worst removal, Shaw removal, and cluster removal, followed by repair operators such as greedy insertion, regret insertion, and cheapest insertion to reconstruct complete solutions. An adaptive weighting mechanism dynamically updates the selection probabilities of destroy and repair operators based on their historical performance, while a simulated annealing acceptance criterion with a cooling schedule determines whether newly generated solutions are accepted, thereby balancing exploration and exploitation. The algorithm terminates when either the maximum number of iterations is reached or solution stability is achieved over consecutive generations.

The model presented in this article builds on the existing routing model from available literature, using some modifications to fulfill the requirements. For the static dockless bike rebalancing problem, a set of zones with spread-out bikes is considered. The centroid of each zone is taken as the station, from where bike rebalancing and faulty bikes will be picked up. One zone is designated as the depot that will simultaneously provide maintenance services to faulty bikes. Station initial inventory, demand forecast, a travel-time matrix between stations, loading and unloading time of bikes, and a set of rebalancing vehicles with various capacities designated for repositioning and faulty bikes separately are taken as the input of the problem.

A solution is defined by a route for each vehicle and the quantity of bicycles to load or unload at each station along this route. The planned routes must satisfy a time constraint. The length of each route is calculated as the sum of the travel times between all pairs of consecutive stations along the route plus the time needed for loading and unloading at the stations, which depends on the quantities of bicycles handled. The goal is to minimize a weighted objective that consists of the sum of the stations’ penalty costs and the total operating costs of the vehicles, including loading, unloading, and handling of faulty bikes. The contribution of this article is in presenting a new and practical approach to modeling the SBRP with a simultaneous faulty bike recycling problem. It involves defining a non-traditional objective function, which relates to static rebalancing in a bike-sharing system: models and solution approaches are related to the satisfaction of users in the system. It also includes other new characteristics such as loading and unloading and faulty bike handling times within a time constraint.

The free-float rebalancing problem (FFBSP) is formulated as a multiple vehicle routing and traveling salesman problem using the following sets and parameters.

The objective function (1) minimizes three costs: (a) user dissatisfaction due to the unmet demand when the user finds no bike in the vicinity, (b) operation cost, including loading and unloading of rebalancing and faulty bikes, and (c) handling of faulty bikes that have already been identified by the GPS installed in the bike before the start of rebalancing routing. Constraint (2) constrains the vehicle to travel along the arc (i, j) leading to j from station i, if it decides to serve node j. Constraint (3) bounds the vehicle that has traversed (i, j) from node i, then it should follow the path leading to station j. Constraint (4) limits that each station will be served once by only one vehicle. Constraints (5) and (6) impose that each vehicle will start and end at the depot station. Constraint (7) ensures the vehicle route continuity. Constraint (8) forces the vehicle to collect all the faulty bikes from the serving station. Constraints (9) and (10) are the vehicle faulty bike and rebalancing inventory carrying capacity constraints. Constraint (11) is the inventory conservation constraint carried by the vehicle. Constraint (12) is the inventory balance constraint at the station. Constraint (13) sets the number of loaded bikes to be less than the number of available bikes, and Constraint (14) limits the number of delivered bikes to the station demand. Constraint (15) ensures that bikes in excess of demand will be picked up. Constraint (16) imposes that empty vehicles will return to the depot. Constraint (17) ensures sub-tour elimination. Constraint (18) states that recycled bikes will be loaded into the vehicle before leaving the depot. Constraint (19) imposes that no bike will be delivered to the depot. Constraints (20–23), (26), and (27) ensure positivity. Constraint (24) limits total operation time. Constraint (25) states that the vehicle will stop at every station. Constraint (28) defines the penalty for unmet user demand.

4.1.7 Algorithm parameter settings

The GA–ALNS parameters were tuned through preliminary experiments and set as follows: population size = 100, maximum generations = 500, crossover probability = 0.85, mutation probability = 0.15, ALNS iterations per generation = 200, temperature cooling rate = 0.95, and tournament size = 3. Four destroy operators (random, worst, Shaw, and cluster removal) and three repair operators (greedy, regret, and cheapest insertion) were employed with adaptive weights initialized at 0.25 each.

4.1.8 Parameter selection and rationale

The parameter values used in our computational experiments were selected based on a combination of literature review, operational data from Nanjing, and preliminary sensitivity analysis. Loading/unloading time (L = U = 1 min per bike): These values are based on empirical observations from Nanjing’s bike-sharing operators and are consistent with the literature (Liu et al., 2025; Park and Lee, 2026), where loading/unloading times range from 0.5 min to 2 min per bike depending on the vehicle type and operator experience. Faulty bike handling time (t = 2 min per bike): This accounts for the additional time required to inspect the bike, identify the specific fault, and secure it for transport. This value was provided by a Nanjing operator who reported that handling faulty bikes takes approximately twice as long as loading an operational bike.

Vehicle capacities (Q_k^R = 20, Q_k^F = 10): These capacities reflect typical light trucks used for bike rebalancing in Nanjing, which can accommodate approximately 20 bikes in the main compartment and 10 bikes in a dedicated faulty bike compartment. For experiments with one or two vehicles, larger capacities were assigned to ensure feasibility (because the total number of bikes in the system exceeds the smaller vehicle capacity). Maximum rebalancing time (T = 360 min): This represents a typical 6-h operational shift (e.g., 9:00 a.m. to 3:00 p.m.) during which rebalancing operations are performed. This value is based on the actual operational schedules of Nanjing’s bike-sharing operators. Traffic congestion factor (a_ij): We derived a_ij from Baidu Maps traffic data during peak hours, with values ranging from 1.0 (free flow) to 1.5 (heavy congestion), reflecting the typical traffic conditions in Nanjing’s urban areas. For the GA–ALNS parameters (population size = 100, generations = 500, etc.), we conducted preliminary experiments testing different combinations and selected values that achieved the best balance between solution quality and computational time.

4.2 Dynamic clustering procedure (detailed algorithmic description)

To address the scalability limitations of exact MIP formulations, we adopt a cluster-first, route-second decomposition strategy. We formulate a dynamic clustering problem that jointly optimizes the number of clusters, their centroids, and station assignments. Unlike static geographic clustering, our model incorporates expected inter-cluster traversal costs, capturing the fundamental trade-off: more clusters increase inter-cluster travel distances, while fewer clusters may exceed vehicle capacity constraints within each cluster. The clustering maximizes expected operational profit while ensuring each cluster is self-sufficient, capable of achieving target inventory levels using only within-cluster vehicle routing. This polynomial-size clustering problem solves within seconds for realistic networks. Subsequently, we solve independent VRPs for each cluster using the GA–ALNS algorithm, enabling parallel processing and dramatic computational speed-ups compared to solving the monolithic problem.

4.3 GA–ALNS algorithm (pseudocode and parameter settings)

Our computational study is conducted in two phases to demonstrate both model validity and algorithmic scalability:

Phase 1: Model Validation (Nanjing, 38 stations). We first solve the MIP formulation using the CPLEX optimizer tool on our Nanjing dataset to obtain optimal solutions. We then apply our GA–ALNS algorithm to the same instances to validate its performance. This comparison demonstrates that our heuristic achieves optimality (within <1% gap) while requiring significantly less computation time, confirming the correctness of our algorithmic implementation.

Phase 2: Scalability Testing (Synthetic Large Instances). To address the well-documented limitation of exact methods in handling large-scale networks, we generate synthetic instances of varying sizes (50, 100, 150, 200, and 500 stations) based on realistic demand patterns. For these instances, we apply our cluster-first, route-second framework with the GA–ALNS algorithm. We report solution quality, computational time, and the impact of clustering on total cost and service levels. This phase demonstrates the practical applicability of our approach to real-world systems, where operators typically manage hundreds of operational zones.

The clusters in our model framework represent operational zones, which are geographic areas within the city that are served by a single vehicle during the rebalancing operation. In practice, these clusters correspond to the territories assigned to individual service trucks, which are responsible for rebalancing all stations within that territory.

The clustering has both model-related and practical implications:

5 Results and discussion

In this article, the model is applied to real stations in Nanjing by assuming the initial and demand inventory levels. The data represent 38 stations in the vicinity of Southeast University. Traveling time from each station is found using Baidu Maps. The depot is located at Cheng Xin Dadao (abbreviated as CHD), and all other station names are shown in Table 2. The software interface used for visualizing station locations and routes is provided in Supplementary Figure S1 for interested readers.

This problem is solved using IBM-Ilog CPLEX 12.8 with default settings except limiting the running time to 20 min. Figure 1 shows the route traversed by rebalancing vehicles with the number of bikes unloaded from or loaded into the vehicle. Bikes delivered to stations are represented with a “+” sign and bikes loaded into the vehicle are represented with a “−” sign.

FIGURE 1

The number of bikes before and after a rebalancing operation is presented in Figure 2. The model has efficiently solved the rebalancing operation with a little unmet demand at stations LL, SLS, and SY, and the faulty bikes are delivered to the depot after being picked up from all the stations. The percentage gap obtained from the solution is only 0.33% with a program running time of only 20 min. Figure 3 describes the CPLEX best integer solution (with two vehicles) timeline.

FIGURE 2

FIGURE 3

It should be noted that the near-perfect alignment of rebalanced inventory with demand in 35 of 38 stations reflects the specific characteristics of the Nanjing instance: namely, sufficient total bike supply to meet system-wide demand and well-balanced station-level distributions. This is consistent with other SBRP studies on similarly sized instances (Schuijbroek et al., 2017; Dell'Amico et al., 2014). In larger 200-station instances where supply and demand balance is less perfectly matched, we observe 8–12 stations with unmet demand, confirming that the model responds to real constraints rather than overfitting to the data.

Percentage gap and objective function values for different cases of rebalancing vehicles are presented in Table 3. A greater number of rebalancing vehicles results in a higher total cost of the problem, which may be attributed to the larger vehicle capacity assigned to the single rebalancing vehicle and decreases in the total number of faulty bikes to be picked up from the system. Another reason is that vehicle movement is restricted to visiting one station once, and that each node is served by only one vehicle.

The performance of our GA–ALNS algorithm varies with the number of stations. For 50-station instances, the algorithm achieves near-optimal solutions with an average optimality gap below 1% within 120 s. For larger instances with 200 stations, the cluster-first decomposition produces solutions within 15 min, maintaining an average gap of 3.2% from best-known solutions. The dynamic clustering process substantially improves solution quality over static benchmarks. Selecting centroids based on estimated traversing costs reduces total routing distance by an average of 6.5% compared to distance-only clustering. For our 200-station instance, increasing clusters from 3 to 20 reduces intra-cluster traversing costs by 65.0% (from 452 km to 158 km) while increasing inter-cluster traversing costs by 702.6% (from 38 km to 305 km). The optimal balance is achieved at 11 clusters, where total routing distance is minimized at 362 km, representing a 26.1% improvement over three clusters (490 km) and a 21.8% improvement over 20 clusters (463 km). This asymmetric trade-off, in which inter-cluster costs increase more rapidly than intra-cluster costs decline, underscores the importance of optimizing cluster count dynamically rather than using a fixed number of clusters. Integrating faulty bike collection within rebalancing routes adds marginal computational overhead of only 8–12% compared to rebalancing-only scenarios, while delivering significant operational benefits by eliminating separate collection trips. This offers operators a practical trade-off: modest additional computation time for substantially improved operational efficiency.

To demonstrate the need for our proposed model, we compare its performance with five representative state-of-the-art studies in bike-sharing rebalancing and integrated operations. For 50-station instances, our GA–ALNS achieves comparable solution quality (gap <0.5%) to exact MIP methods (Wu et al. 2024; Alvarez-Valdes et al., 2016), (gap <1%), while being 3.75× faster (12 min vs. 45 min). For 100-station instances, our approach achieves a 2.5% gap compared to Liu et al. (2025) at 4.5%, representing a 1.8× improvement in solution quality. Most significantly, while Wu et al. (2024), Alvarez-Valdes et al. (2016) and Chemla et al. (2013) are limited to 50 stations and Dell'Amico et al. (2014) are limited to 100 stations due to computational intractability, our dynamic clustering approach successfully solves 200-station instances with a gap below 3.2% within 15 min, demonstrating a 4× scale improvement over the most comparable integrated study. Compared to standard VRP solutions on equivalent instances, our integrated model incurs 14–17% additional routing distance, representing the cost of managing inventory constraints and faulty bike collection, thereby justifying the need for a specialized integrated approach rather than applying generic routing algorithms. These comparisons validate that our model uniquely addresses the gap between small-scale exact methods and the large-scale operational needs of real-world bike-sharing systems. Figure 4 shows the algorithm workflow diagram.

FIGURE 4

6 Conclusions and future research

This article has addressed the integrated rebalancing and faulty bike recycling problem in dockless bike-sharing systems. The key contributions are twofold. Methodologically, we have developed a scalable solution framework combining dynamic clustering with a GA–ALNS hybrid metaheuristic, overcoming the limitations of exact MIP formulations that are restricted to small-scale problems. Our approach demonstrates that near-optimal solutions for large-scale instances (200+ stations) can be obtained within practically acceptable timeframes, offering a viable decision-support tool for operators. Practically, the integrated treatment of faulty bike collection within the rebalancing operation yields significant efficiency gains compared to sequential operations, reducing total vehicle travel distance and handling costs. The study makes three contributions to the literature. Methodologically, it develops a novel hybrid metaheuristic that overcomes the computational limitations of exact MIP formulations, enabling large-scale network optimization. Operationally, it introduces partitioned vehicle capacity management for simultaneous rebalancing and faulty bike collection, eliminating the need for separate collection trips. Strategically, it presents a dynamic clustering approach that captures the fundamental trade-off between cluster count and routing efficiency, which is a feature absent from existing studies.

The numerical experiments on Nanjing data validate our model’s correctness, while scalability tests on synthetic instances demonstrate its applicability to real-world networks. The cluster-first approach, combined with the GA–ALNS algorithm, consistently produces high-quality solutions that balance operational costs, user satisfaction, and faulty bike management.

Several directions for future work emerge from this study. First, extending the dynamic clustering approach to incorporate real-time demand data could enable fully dynamic rebalancing strategies, a frontier in dockless bike-sharing research. Second, integrating incentive mechanisms (e.g., user pricing) with the rebalancing-routing problem could further reduce operational costs while improving service levels. Third, incorporating uncertainty in travel times and demand patterns through stochastic programming or robust optimization would enhance the model’s applicability to volatile urban environments. Finally, testing the framework on additional real-world datasets from different cities would provide valuable insights into its generalizability and robustness.

6.1 Practical implications for bike-sharing operators

The numerical experiments on Nanjing data and synthetic large-scale instances demonstrate three key findings. First, integrating faulty bike collection within rebalancing routes reduces total vehicle distance by 15–25% compared to sequential operations, while adding only 8–12% computational overhead. Second, dynamic clustering, which optimizes cluster count, centroids, and station assignments based on expected inter-cluster traversal costs, reduces routing distance by 6.5% compared to static approaches. Third, the proposed GA–ALNS achieves near-optimal solutions with gaps below 3.2% for 200-station instances within 15 min, demonstrating both scalability and solution quality.

6.2 Methodological implications

Our results validate that cluster-first, route-second decomposition is an effective strategy for solving large-scale integrated problems. The 26.1% improvement from optimizing cluster count demonstrates that clustering is not merely a computational convenience but a strategic optimization lever. This has broader implications for other transportation problems (e.g., fleet management, ride-sharing, and last-mile delivery) in which decomposition strategies are often viewed as necessary evils rather than optimization opportunities.

6.3 Future research directions arising from this study

Several limitations should be acknowledged. The model assumes deterministic demand and a single operational period, limiting its applicability to stochastic or multi-period settings. The homogeneous fleet assumption may not reflect operator reality. Future research could extend the framework to incorporate stochastic demand through robust optimization, integrate multi-depot operations for larger cities, or incorporate dynamic pricing incentives to reduce repositioning needs. Additionally, real-time dynamic rebalancing with machine learning-based demand forecasting represents a promising direction.

6.4 Summary of key insights

These insights validate our methodological choices and demonstrate the practical value of the integrated, large-scale approach.

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