**1. Introduction**

The station-free bike sharing system (BSS), also known as the free-floating or fourth generation BSS, is a new type of public bike system that has been widely deployed in China since 2017 and expanded to other countries. In this system, bikes can be selected using private apps and parked in the appropriate places. Comparing the traditional station-based BBS, the station-free BSS can expand the bike sharing service with lower cost as the high initial capital investment required for the docking stations is not needed. Due to the freedom and convenience the BSS provides, it has attracted a large number of consumers requiring "last mile" transportation.

However, rapid growth has vastly outpaced the immediate demand and overwhelmed Chinese cities, where infrastructures and regulations were not prepared to handle a sudden flood of millions of shared bikes [1]. In many cities, adequate parking facilities for bikes are not available, city o fficials lack

the regulation experience for this mode of transportation, and normal social behaviors have not been established [2]. It is very common to have more than one operating company for the station-free BSS in a city. From an operational perspective, the most important goal is to occupy the market, which is why many companies would prefer to provide more bikes and exceed the demand [3]. On the other hand, large bike fleets are associated with a waste of resources because many bikes would remain idle for long periods, making the system ine fficient.

In response to these problems, more and more Chinese cities such as Shanghai, Hangzhou, Guangzhou, Shenzhen, have banned the addition of further shared bikes [4]. A series of strict regulations for bike share providers are being implemented in China and European countries, including orderly parking, quality and timely maintenance of the bikes, license system for operators and fleet size control [5,6]. However, a fundamental unsolved problem is determining how many station-free bikes are su fficient to e ffectively meet the needs of users?

The Bike Sharing Planning Guide provides guidelines for the fleet size of a station-based BSS, which are 10–16 stations per km2, 10–30 bicycles for every 1000 residents within coverage area, and 202.5 docking spaces for every bicycle [7]. However, they are for the station-based BSS, not the station free BBS. Moreover, these quantities are rough indications and mainly depend on the characteristics of city built-environment, such as land use, population density, and road conditions.

To determine the optimal fleet size and distribution of station-free BBS, this paper proposes a heuristic bike optimization algorithm (HBOA) considering the e fficiency of bike cycling. It can be an objective basis for city related departments to issue the total control policy and be applied to design the layout of bikes in small-scale spatial units to improve the system's e fficiency.

### **2. Literature Review**

In past decades, many studies have focused on three main issues related to public bike systems with docking stations: the spatial structure of a city [8–11], the inflow and outflow of vehicles at each station [12–14], and the rebalancing of the vehicles among stations [15–18]. In a station-based BSS, the supply of the vehicles must be compatible with the scale of the fixed stations. Once the construction of the stations is complete, the system is di fficult to change. Therefore, extensive research on station-based BSSs has focused on the locations and capacities of stations to optimize the e fficiency of these systems [19–22].

Station-free BSSs completely di ffer from station-based BSSs. The characteristics of a station-free BSS allow the system scale to be enlarged by providing many vehicles without station capacity constraints. Because vehicle parking is scattered and the spatial distribution is changing all the time, the demand for rebalancing might increase in some cases, and predictions of potential imbalances are relatively complex. There are no predetermined stations in these systems, so scheduling schemes are often unclear, even if the real-time parking distribution is known. Furthermore, sometimes relocation occurs based on spur-of-the-moment changes without following a specific strategy [23].

Most research on relocation in station-free BSSs has extended the ideas and methods applied to stationed-based BSSs, and studies have focused on the e ffects of urban features [24,25], spatiotemporal patterns of biking behavior [26,27], and relocation or rebalancing of shared bikes [28–31]. For example, by setting virtual tra ffic zones, each tra ffic area is treated as a bike sharing station, and the first distribution and relocation scheme of the BSS are designed according to the demand model combined with the vehicle outflows and inflows in the tra ffic zone. Some studies have proposed algorithms to achieve e fficient relocation strategies for stationed-based BSSs from both static and dynamic perspectives [32,33]. Other studies used OD matrix data from bike sharing companies to analyze and simulate bike sharing travel patterns [34]. In another study, the demand was forecasted with deep learning methods to predict the gap between the inflow and outflow of sharing bike trips at a TAZ [35]. These studies based on virtual stations have helped simplify the analysis process, but they fail to take full advantage of the unique use characteristics of free-floating BSSs to a large extent. First, due to the randomness of parking with no docking stations, it is di fficult to set a fixed TAZ for relocation. In the division of virtual tra ffic zones, zones that are too large may not reflect the reality of operation, and zones that are too small will make relocation complicated. Second, a very important di fference between a stationed-based BSS and station-free BSS is that the chain of travel can more easily occur at a smaller scale because of the spontaneous usage in the station-free BSS.

Due to the large number and usage frequency of shared bikes, the randomness of shared bikes movement and spacing is high. From the perspective of complex systems, the behaviors of users can be regarded as a self-organizing process. On the one hand, the hidden reasons behind user behaviors are worth studying compared to the inherent system randomness. On the other hand, it is important to identify which factors in the complex system are critical to the self-organizing process. For example, Chen et al. simulated the interactions between supply and demand based on agent-based modeling and suggested that the key aspects of the sustainable development of the bicycle-sharing market are twofold: the reliability of the supply must be improved, and the uncertainty in the demand must be reduced. Standardizing the distribution of shared bikes and fixing their locations could solve the disorder issue associated with excessive supply [36]. Vazifeh et al. proposed a solution to address the minimum fleet-size problem at the urban scale for the general case of taxi trips based on the demand mobility [37]. This study combined applied mathematics and graph algorithms from computer science field and transformed the minimum fleet problem into a minimum path coverage problem based on the directed graphs, which led to breakthroughs in operational e fficiency. If the chain of travel is considered, it is possible to optimize and simplify the relocation of bikes and improve the e fficiency of the station-free BSS. However, unlike taxis, the principle of shared bikes is that individuals can use bikes "as-needed" by finding the surrounding bikes instead of dispatching vehicles on demand. Taxi drivers can actively choose the optimal route, but a shared bike must be selected by a user according to the location and parking time and is controlled by the user.

Therefore, based on the construction of a shared bike trip chain with actual riding data for a certain period of time, this paper develops a heuristic algorithm to determine the optimal demand for public bikes with little operation intervention required. This method is then applied for multi-company cycling data analysis in the megacity of Shenzhen, China. The results indicate that the algorithm can reveal the mobility patterns of shared bikes and provide useful information for shared bikes to improve the use e fficiency at the city scale.
