## 1 Introduction  
Bike sharing systems are public transportation systems adopted by many different municipalities to reduce traffic congestion and pollution and provide more sustainable means of transport within smart cities. Additionally, compared to the use of private cars, the adoption of bicycles combined with other means of public transport (bike-and-ride) offer environmental benefits and a number of social benefits [1]. Such systems are widely adopted: more than 700 systems are active world-wide [2; 3]. In the context of smart cities and Sustainable Development Goals (SDG), understanding the key factors towards large scale adoption by users is fundamental to reduce air pollution and improve quality of life in cities [4; 5; 6], according to SDG 11 and 13. In fact, additionally to environmental benefits and a reduced amount of pollution, Ref. [7] quantified health and economical benefits which arose in European cities with large adoption of bicycles. Despite the benefits, such systems require investments from companies and municipalities to be effective and successful. The documented case of the city of Seattle [8] underlines the importance of proper design and the need of maintenance to provide good quality of service. In this context, we highlight the relevance of our work in developing a framework to solve the bike rebalancing problem in bike sharing systems, which can be categorized into two main types: (i) free-floating and (ii) station-based systems (also referred to as dock-based systems). In free-floating systems users can find and drop bicycles at almost any location within a delimited operational area in the city, whereas station-based systems require the user to pick up and drop bicycles in docking stations, specifically located within the administrative boundaries of the municipality. The main advantage of free-floating services is that users can freely drop bicycles everywhere, especially in case their destination is in an area far from points of interests and in outskirts of the city, but the disadvantage is that when the user needs to pick up a bicycle, none may be available in his/her vicinity. Station-based bike sharing systems may instead provide more reliability since there are several stations spread across the city in the most relevant and popular locations. The main disadvantage of such systems is that upon the arrival at the designated destination, the station may be full, thus requiring the user to reach another (nearby) station to drop the rented bicycle and eventually incurring in higher costs due to the additional trip, or the station from which the trip starts may be empty, requiring the user to walk to another station within walking distance or to use other means of transport. Station-based bike sharing systems thus require rebalancing operations to maintain users' satisfaction and limit the inabilities to use a station, either for pick-up or drop of bicycles. However, station-based bike sharing services have a great advantage compared to free-floating competitors: easier maintenance operation planning due to the presence of a limited number of stations in fixed locations. The Bike sharing Rebalancing Problem (BRP) consists of using a set of vans to move bicycles across different stations for rebalancing purposes, avoiding situations in which stations cannot be used as starting or destination points due to them being empty or full, respectively. The BRP problem can be tackled in two different ways, either via static rebalancing or dynamic rebalancing. Static rebalancing strategies are performed when the system is offline or when the system is barely used, i.e., during night time, whereas the dynamic rebalancing strategy consists of moving bicycles across the stations when the system is operating and is used by the users. Dynamic rebalancing operations need to be performed in a short amount of time to avoid large amount of bicycles being unavailable to users due to operators moving them across different stations.