## 5 Conclusions  
In this study, we devised a comprehensive methodology aimed at optimizing the travel paths of multiple management vehicles to address the bike rebalancing problem. Our approach encompasses three key components: probabilistic demand forecasting, determining optimal inventory level, and solving pick-up and delivery problems for multiple management vehicles with multi-objectives. Recognizing that bike demand is interdependent between stations, owing to users' ability to walk to nearby stations when their preferred station is empty, appropriate demand  
Figure 8: Pareto frontiers for pick-up and delivery operations of bike-sharing systems ((a) and (b) represent two selected solutions, the paths of which are displayed in the Figure 9). Figure 9: Travel paths of two vehicles for the solutions (a) and (b) ((a) and (b) refer to the solutions shown in Figure 8). forecasting necessitates grouping bike stations based on geographical proximity. To this end, we employed a k-means algorithm to cluster bike stations within walking distance, and utilized the DeepAR approach for demand forecasting for each cluster. The strength of DeepAR lies in its capacity to provide a probabilistic distribution of demand, which is vital for determining the optimal service level and the associated target inventory using a news vendor model. The bike-rebalancing problem essentially entails pick-up and delivery tasks for multiple management vehicles that transfer bikes between clusters with excess and deficit capacities. Given the operation of multiple vehicles, it is imperative to determine the optimal paths for these tasks. Two primary performance metrics guide the bike-rebalancing problem: minimizing travel distance and minimizing the gap between the target and final inventories after pick-up and delivery operations. The multi-objective nature of our problem complicates the solution process and poses challenges in achieving optimal results. Initially, we modeled the problem using an integer programming approach with a multi-objective framework (referred to as MOOP--Multi-Objective Optimization Problem). However, the MOOP yields only a single solution and does not guarantee its optimality. To address this limitation, we developed a hybrid genetic algorithm (HGA) that mixes a genetic algorithm and a mixed integer programming model. The former tries to find the shortest path, and the latter tries to find the optimal allocation of bikes in the given paths. The application of the proposed methodology to real bike-sharing service data from the Mapo-gu district of Seoul City, South Korea, yielded balanced travel routes for management vehicles, considering both the travel distance and the gap between target and final bike inventories post rebalancing. The frontiers provide flexibility in decision-making process. For example, if there is an upper limit to the distance or to the net balance deviations, it is possible to choose an appropriate set of travel routes which meets the condition. This paper acknowledges certain limitations when applied to real-world scenarios. First, the central point of each station cluster, which is derived from the computed central coordinates of actual bike stations, has not been validated as a feasible location for bike drop-off and pick-up. Second, the travel paths between these cluster centers are calculated using simple Euclidean distances, without accounting for actual road conditions. Additionally, traffic conditions and alternative routes are not considered in the path calculations. Consequently, these limitations may hinder potential operational cost savings when implementing the proposed methodology. A practical implementation is necessary to identify and address any challenges that may arise in real-world applications. Computational efficiency is another issue in the implementation of the algorithm in real-world scenarios. Due to the significant computational burden, the proposed algorithm may not be appealing to decision makers.