## 3 Model and Methodologies
### Rebalancing Phase
#### 3.2.2 Hybrid Genetic Algorithm(HGA)  
An HGA is proposed to address the PDVRP in this study, which combines GA and MIP techniques. The overall algorithm is illustrated in Figure 1. The GA generates and updates the feasible paths (chromosomes) for the management vehicle(s). Then, the MIP for bike allocation, presented below, determines the optimal allocation of bikes to the clustered stations based on the optimal target inventory level obtained from Equation (2). In this phase, the optimal solutions for allocation are saved in memory to retrieve the resultant value in the future when the same path pops up in the course of the evolutionary process. After obtaining the optimal solutions, the deviation in the net balance for each path is calculated. The path length is directly derived from the chromosome. These values collectively constitute a fitness function used to determine an efficient frontier for each generation. A superior frontier is selected through evolutionary iterations. Moreover, the surviving chromosomes comprising the final frontier constitute the solution to the problem when the termination conditions are satisfied. Let \(r_{i}\) be the \(i\)-th station listed in the route, \(d_{i}\) be the quantity delivered to station \(i\in N\), and \(b_{i}\) be the onboard inventory carried by the management vehicle at station \(i\in N\). The MIP model for the bike allocation problem is formulated as follows:  
Min \[\sum_{i=1}^{n}(s_{i}^{+}+s_{i}^{-})\] (26) s.t. \[b_{0}=C\] (27) \[d_{0}=0\] (28) \[y_{i}=I_{r_{i}}+d_{i}\quad\forall i=1,\ldots,n\] (29) \[b_{i}=b_{i-1}-d_{i}\quad\forall i=1,\ldots,n\] (30) \[b_{i}\leq C\quad\forall i=0,\ldots,n\] (31) \[y_{i}-T_{i}=s_{i}^{+}-s_{i}^{-}\quad\forall i=1,\ldots,n\] (32) \[s_{i}^{+}\geq 0,s_{i}^{-}\geq 0\quad\forall i=1,\ldots,n\] (33)  
The objective function in Equation (26) quantifies the overall discrepancy between the target and deployed inventory levels across all the rental stations. This model determines the number of bikes to be picked up or delivered to each bike station (denoted by \(t_{i}\)) along a given path. Despite having an optimal target inventory for each station, the capacity of  
Figure 1: Structure of the proposed hybrid genetic algorithm. each vehicle (denoted as \(C\)) and the predetermined travel sequence prevent the fulfillment of the target inventory. Constraint (27) ensures that the vehicle starts each trip fully loaded with \(C\) bicycles, where \(b_{0}\) denotes the number loaded at the dummy station. Constraint (28) prevents any deliveries to the dummy station. Inventory changes at each station are governed by Constraint (29), which calculates the updated stock levels based on delivery quantities. Constraint (30) determines the number of bikes loaded onto the vehicle upon departure from station \(i\), whereas Constraint (31) enforces the vehicle's capacity limit for \(C\) bikes to prevent overloading. Constraint (32) is equivalent to Equation (24) in MIP. To uphold mathematical validity, Constraint (33) ensures that all variables used in the deviation calculations remain non-negative. Among the various multi-objective genetic algorithms in the literature, the nondominated sorting genetic algorithm (NSGA-II; [31]) is applied to the problem in this study. NSGA-II is fast and suitable for solving bi-objective problems [36; 31]. We only explain NSGA-II briefly here; for more details, please refer to [37; 31]. It first groups the population into sets of nondominated members by comparing the chromosomes, starting from the top level. The population for the next generation is then selected from the nondominated groups, progressing from top to bottom. When only a portion of a group must be selected owing to population size limitations, chromosomes that are not located in crowded regions in the solution space are prioritized for selection. [PERSON] et al. [31] call this process crowd-distance sorting.