## 1 Introduction  
In recent years, bicycle sharing systems (BSSs) have been implemented in many cities around the world to ease traffic jams in town centers, to reduce CO\({}_{2}\) emissions and to improve public health [1]. A BSS is composed of many dispersed automatic rental ports with rental bicycles. People can rent a bicycle at one of the many ports and return it to a different port. In BSSs, when many bicycles are used, the number of bicycles at each port becomes unbalanced, because almost all users conduct not only round trips but also one-way trips. To overcome this problem, capacitated vehicles restore the required number of bicycles to the rental ports. Recently, many researchers have investigated the optimum rebalancing of the bicycles with BSSs. [PERSON] et al. (2013) [2] proposed a branch-and-cut-algorithm for finding an optimal solution to the static rebalancing problem using a single vehicle with up to 100 ports. The results of their numerical experiments demonstrated that their method is very efficient up to 60 ports [2]. [PERSON] et al. (2013) [3] proposed a construction method and a local search method for the bike sharing system routing problem. They reported that for up to 50 ports, their proposed method performed better than their previous method [3]. In real-life situations, the number of ports is more than 100, making it almost impossible to restore the bicycles using a single vehicle. To address this issue, a bike repositioning problem with multiple vehicles has been proposed [4; 5; 6]. [PERSON] et al. (2012) [4] proposed heuristic algorithms using the actual path distance and a maximum of 11 vehicles to demonstrate that shorter travel distances are obtained when the actual path distance is used instead of the Euclidean distance [4]. [PERSON] et al. (2013) [5] presented two mixed-integer linear program formulations, with the aim of minimizing the total travel cost and penalty cost. They obtained good solutions for the Paris and Washington D.C. problems, using two vehicles, with an instance size up to 60 ports and 104 ports, respectively [5]. [PERSON] et al. (2017) [6] proposed a hybrid large neighborhood search method and tested instances with up to 518 ports using a maximum of three vehicles. They conducted experiments and compared their solution with the method proposed by [PERSON] et al. (2015) [7] and they obtained better solutions than [PERSON] et al. (2015) [7]. In Refs. [4; 5; 6], it is assumed that the vehicles do not need to visit all ports. In this case, an operator cannot notice a broken bicycle or any trouble at the port. To address the problem of visiting all ports using multiple vehicles, [PERSON] et al. (2014) [8] proposed four mathematical formulations, built a set of benchmark instances and solved them using the branch-and-cut algorithm. An optimal solution was obtained with one of the proposed formulations for the 49 instances out of 65. Research on the bicycle rebalancing problem, while producing interesting results, does not consider time limit constraints [8]. Without a time limit constraint, the operations differ in the amount of work required. In the case of the static bicycle rebalancing problem, the bicycle relocation must be performed within a limited time after the end of the daily BSS. Table 1 summarizes the literature related to the static bicycle rebalancing problem. As for other related works, [PERSON] et al. (2011) [9] proposed a static bicycle repositioning problem, discussed the complexity of the bicycle relocation process and provided an integer linear programming problem and an approximation algorithm. [PERSON] et al. (2016) [10] proposed an approach for determining the tours while considering the bicycle demand. It is difficult to directly compare these related studies because of the different constraints that have been considered. This is also mentioned in [PERSON] et al. (2014) [11]. The research on BSS is not only about the relocation of bicycles, but also about the further development of BSS.