# Decision System Based on Markov Chains for Sizing the Rebalancing Fleet of Bike Sharing Stations
## 5 Discussion  
Citi Bike New York was used to evaluate the proposed strategy. The raw data, which consisted of the number of available bikes at each station, was processed, to determine the station load for modeling purposes. Given the large geographical area covered by the BSSs and the specific traffic limitations, it was not feasible to implement a centralized rebalancing strategy. Therefore, a cluster-based approach was explored, to assess the effectiveness of the proposed model. In this context, a cluster refers to a collection of bike stations that are highly dynamic and located within a defined area. BSS cluster behavior identification is relevant because it enables the assessment of rebalancing needs. The cluster behavior is defined as a set of states and the transition rates in-between. Each state corresponds to the overall criticality degree, depending on how many bike stations are unable to provide the resources required by the customers. Determining the fleet size for a set of bike stations is important for efficient operations. A fleet size that is too big can result in high acquisition and maintenance costs. On the other hand, a fleet size that is too small can lead to slow reaction times to customer needs, ultimately resulting in customer loss. To achieve the right balance, a method was developed based on the cluster steady state and the fleet rebalancing cost. The model effectiveness was demonstrated on two subsystems. Each subsystem had four distinct states corresponding to an overall criticality degree as follows: [0%; 25%), [25%; 50%), [50%; 75%), [75%; 100%]. The first subsystem presented had 20 bike stations and had, in total, 206 selected days for identification and validation purposes, while the second had 16 bike stations and 174 selected days. The second subsystem posed particular challenges because of the observed transitions between states that had intervals that were not neighboring: [0%; 25%)\(\rightarrow\)[50%; 75%) and [25%; 50%)\(\rightarrow\)[75%; 100%]. The evaluations performed for both subsystems analyzed passed. The proposed model underwent a theoretical evaluation strategy that involved using a few months of data to identify a model. This was followed by another few months of data, with the goal of identifying an equivalent model. The equivalence was determined by ensuring that the deviation between the two models was less than 3%. To assess the deviation, each transition rate from one model was compared to the corresponding transition rate from the other model. For both subsystems, the theoretical evaluation criterion passed by obtaining maximum deviations of 2.47% and 2.28%, respectively. Within the practical evaluation strategy, the model together with the fleet sizing method were applied to the same subsystems. The selected rebalancing application was tested, to ensure that it operated within the maximum travel time restriction (\(\Gamma=282\) s) to each bike station during the rebalancing trip. The practical evaluation was also successful, due to the fact that all the underlying transition rates were within bounds derived from the mentioned studies of real world conditions of BSSs (25.5 km/h truck velocity; 2 km distance between stations). Additionally, the model effectively reached another operational goal by lowering the duration of the bike station unworking times, to a noticeable extent. Furthermore, in the context of user expectations, it demonstrated efficacy in decreasing the rate of lost customers, which is noteworthy, albeit without ensuring complete reduction, which is a limitation. The model is, in theory, applicable to other BSSs because its viability is not conditioned by the specific properties of the studied system. This means that any BSS with a subset of bike stations that frequently experience load imbalances could use the model effectively.