## 2 Literature Review
### Pick-Up and Delivery Vehicle Routing Problem (PDVRP) and Multi-Objective Optimization  
To meet the unbalanced target inventory level for each station, a set of capacitated vehicles travels to redistribute bikes while minimizing the total cost. This problem is classified as a one-commodity PDVRP that determines the optimal delivery or routes from one or several depots to a certain number of customers, subject to additional constraints. In the field of logistics and transportation networks, the PDVRP is a critical problem that has been widely studied in the literature; see [21; 22] for a detailed review. Using existing routing models in the literature, [PERSON] et al. [23] presented two MIP formulations and developed several valid inequalities to strengthen these formulations. A numerical study demonstrated that their formulations are effective in large instances with two vehicles and about 100 stations. [PERSON] et al. [24] determined the service-level requirements at each station and obtained optimal vehicle routes for inventory rebalancing. Since finding an exact optimal solution is practically intractable, they provided an efficient heuristic algorithm for the clustered problem. Most of the models for PDVRP in BSS are aimed at the optimization of a single objective, such as minimizing the total costs or the maximum distance. However, multi-objectives must be also considered in public sectors such as BSS. [PERSON] et al. [25] developed a bi-objective optimization solution for public BSSs to minimize the mean number of rejected requests for renting bikes and free docks for returning bikes per satisfied demand. [PERSON] et al. [26] dealt with a particular case of the static bike rebalancing problem (SBRP), in which only a single vehicle is available. Their objective was to find the lowest-cost route that meets the demand of all stations and does not violate the minimum (zero) and maximum (vehicle capacity) load limits along the tour. There is a substantial amount of literature on the PDVRP of the bike rebalancing problem using meta-heuristic algorithms. Given the large number of bicycle stations involved, determining optimal travel paths for multiple management vehicles with an exact solution approach is not trivial. Additionally, the multi-objective nature of the problem has led to extensive use of meta-heuristics in the literature. [PERSON] et al. [27] presented a method for rebalancing the loads of bike-sharing system (BSS) stations under conditions of dynamic behavior of the bike-rebalancing problem. Their study developed a two-step method to perform effective rebalancing operations in bike sharing. The core elements of their method are a fuzzy-logic-controlled genetic algorithm (GA) for bike station prioritization and an inference mechanism aimed at assigning stations and trucks. Their model aims to maximize efficiency, which is a type of productivity measured by the number of bikes transported per unit of travel distance. [PERSON] et al. [28] proposed a decomposition-based multi-objective evolutionary algorithm to address the bike-rebalancing problem. Their model seeks to minimize truck operation costs, which include travel-distance costs and other operational expenses, and to maximize overall satisfaction profit, a complex metric based on the necessary demand estimated from historical and predicted pick-up and return data. In another domain of PDVRP using meta-heuristics, [PERSON] et al. [29] applied a multi-objective genetic algorithm to address the open green vehicle routing problem. This problem involves delivering services or products from a single depot to customers across various geographical locations, aiming to minimize both travel distances and carbon emissions. The authors first used a k-means algorithm to cluster customer regions, which is similar to the prioritization of bike stations in the aforementioned studies.