## 4 Computational Experiments
### Optimal Solutions from HGA for PDVRP of BSS  
In this subsection, we present the optimal solutions obtained from the HGA for inventory-rebalancing problems in BSSs. Inventory rebalancing involves the PDVRP of bikes through vehicle management. Consequently, these solutions comprise traveling paths along with the respective quantities of bikes to be picked up or delivered. We apply the HGA to address the PDVRP in the BSS of Mapo-gu district in Seoul, South Korea, which encompasses 126 bike stations. Users access real-time bike inventory information for all stations via a mobile application, allowing them to locate bikes within walking distance. This indicates that it is more valid to perform demand forecasting in cluster units that group stations within walking distance rather than on an individual bike station basis. Moreover, forecasting the demand for station clusters assists in mitigating the forecasting errors on a station basis. The bike stations are clustered using a well-known machine learning technique, the K-means algorithm. Figure 6 depicts 40 clusters, with stations of the same color indicating membership in the same cluster group. The maximum distance between the furthest stations within all clusters is approximately 800 m, which is within an acceptable walking range. The essential goal of bike rebalancing is to enhance bike utilization by ensuring that an adequate number of bikes are available when required. This task resembles the news vendor problem, as it involves balancing the costs associated with underestimating (resulting in bike shortages) and overestimating (leaving bikes idle) demand. It is assumed that bike shortages lead users to opt for buses with comparable fares, thereby potentially increasing carbon emissions. Factoring in the carbon emission price as of January 2024, the daily cost of inventory shortage is calculated to be KRW 922.3761. Conversely, when bikes remain unused at stations, there is an inventory surplus cost of KRW 304.0613, which accounts for excess bike investment, attached smart devices, and insurance premiums. To optimize the service levels, Equation (2) is used to determine a target service level of 0.752. This decision is informed by probabilistic demand forecasting, which allows setting the target inventory  
Figure 6: Bike station clusters in Mapo-gu district, Seoul, Korea (The dots in different colors represent different clusters). at the 75.2 percentile of demand calculated from the probability distribution for each time period within each cluster. In our forecasting using the DeepAR methodology, we selected a family of normal distributions because it showed the least errors in predicting bike user demand for the station clusters. This suggests that the actual demand is highly likely derived from normal distributions, although the parameters of these distributions vary over time and between clusters (for more details, please refer to [3]). For each station, the 75.2 percentile value of the corresponding normal distribution is set as the target inventory and the initial inventory is obtained from actual data gathered from the bike-sharing app. The HGA is applied to solve the PDVRP, utilizing the target inventory for each station cluster as bikes within the 75.2 percentile of user demand. In the Mapo-gu district, where two management vehicles operate, two distinct travel routes are generated. These routes share the same origin and destination points, and the two points refer to a single management warehouse in the Mapo-gu district. The objectives are to minimize the longer travel distance between the two routes and to minimize the sum of the absolute net demand deviations. The absolute net demand deviation is calculated as the difference between the target inventory and the ending inventory after rebalancing by management vehicles. The two management vehicles must visit a total of 40 points. Therefore, terminating the HGA after only 1000 generations, as applied in the sample problem sets, is too early for the solution to converge.