## 1 Introduction  
Bicycle-sharing systems (BSSs) offer an eco-friendly and accessible mode of transport, contributing to a decrease in greenhouse gas emissions, reduction in energy consumption, and enhancement of societal benefits in urban settings. BSSs improve the linkages between public transport facilities by providing an effective means of covering the initial and final segments of urban journeys, appealing to both residents and tourists. BSSs are divided into two main operational types: deck-based (DBSSs) and free-floating (FBSSs). DBSSs operate with designated stations to pick up and return bicycles, whereas FBSSs offer flexibility for users to access and leave bikes at undetermined locations, removing the need for fixed docking points (bike stations). The singular directionality of BSS rentals necessitates a redistribution system, termed rebalancing (DeMaio [1]), to ensure the effective distribution of bicycles. Rebalancing strategies for DBSSs rely on predictive models to estimate the future demand at each station, whereas FBSSs' rebalancing may operate based on regional restrictions. For instance, Seoul might see FBSS approaches leaning towards the employment of electric bikes or scooters in specific zones, reducing the need for rebalancing efforts. In this regional restriction setting, if someone attempts to move outside the area, the vehicle sounds an alarm and slows down. This study aims to craft vehicle routing algorithms for multiple management trucks to support the collection and redistribution of bicycles in line with predictive demand models for DBSSs. Leveraging precise demand forecasts, the process of reallocating bikes involves logistical intricacies, where vehicles transport bikes from saturated stations to those with a deficiency, ensuring route cost effectiveness all day, thereby seamlessly integrating inventory management with vehicle routing optimization principles. For a precise and actionable forecasting method regarding demand distribution within DBSSs, we employ DeepAR ([PERSON] et al. [2] and [PERSON] et al. [3]), which is based on autoregressive recurrent networks. This method enhances the prediction accuracy of future bike-rental demand by analyzing historical usage patterns and integrating the inherent variability and uncertainties in user behavior. Following the establishment of forecasted demand distributions, we determine the ideal target bike inventory levels using a periodic inventory review model. This systematic approach regularly evaluates inventory requirements based on the anticipated demand, ensuring that stations are adequately stocked to meet user demand without oversupply. To operationalize the set of target inventory levels, we devise a multi-objective optimization framework utilizing a mixed-integer programming (MIP) technique. This framework is designed to achieve the multi-objectives of minimizing the travel distance for bike redistribution vehicles and reducing the deviation between the targeted and actual final inventory levels. By identifying the most efficient redistribution routes, this approach effectively tackles the logistical hurdles of matching inventory with forecasted demand, thereby achieving an equilibrium between resource distribution efficiency and inventory precision. It is worth noting that the multi-objective optimization approach based on MIP, as commonly implemented in commercial solvers such as CPLEX, may not always guarantee the identification of the optimal solution for every problem instance owing to the underlying algorithms. To address this issue, we propose a hybrid genetic algorithm (HGA) that combines the adaptive search efficiency of genetic algorithm with the precision of MIP. This approach leverages the exploratory capabilities of the GA while using MIP for solution refinement and alignment with the optimization criteria. Our hybrid strategy aims to enhance solution optimality by generating a superior Pareto frontier when tackling complex optimization tasks in the bike-rebalancing problem of BSSs.