## 6 Numerical Experiments
### Effectiveness of the Proposed Optimization Framework  
In this section, we examine the effectiveness of the proposed optimization framework by accessing the level of consistency (LOC, denoted as \(\Omega^{s}\)) between the optimal locations identified by (1) the optimization framework and (2) the PM model for a specific selection \(s\in S\). The value of \(\Omega^{s}\) is determined as follows. First, we define two vectors \(\mathbf{A}^{s}=\left[a_{j}^{s},\forall j\in I\right]\) and \(\mathbf{A}^{s}=\left[a_{j}^{*},\forall j\in I\right]\). \(a_{j}^{s}\) equals 1 if candidate \(j\) is matched with at least one passenger (\(\sum\limits_{i\in I^{s}}x_{ij}^{s}>0\)) in the optimal solution for selection \(s\), and 0 otherwise; \(a_{j}^{*}\) equals  
Figure 10: The spatial distribution of the selected locations of the parcel lockers obtained by the optimization framework.  
Figure 9: The total frequency of being matched with passengers for all the candidate parcel locker locations.  
1 if candidate \(j\) is selected by the optimization framework as one of the optimal locations, and 0 otherwise. Then, \(\Omega^{s}\) can be expressed as follows:  
\[\Omega^{s}=\frac{\sum\limits_{l\in J}\left(1-\left|a_{j}^{s}-a_{j}^{s}\right| \right)}{P} \tag{6}\]  
One should see from Equation (6) that \(\Omega^{s}\in[0,1]\). A larger \(\Omega^{s}\) indicates a higher LOC, which means that the optimal locations obtained by the proposed optimization framework align more closely with those selected by the PM model in selection \(s\).  
We computed \(\Omega^{s}\) under \(N-P\) combinations, where \((N,P)=\) (100,50), (90,40), (80,40), (70,30), (60,30), and (50,20) for a broad coverage of parameters, and hence this represents a more comprehensive analysis. For all \(N-P\) combinations, \(S\) was set to 2000.  
Table 3 displays the mean, maximum, minimum, and standard deviation of \(\Omega^{s}\) across all the 2000 selections for each \(N-P\) combination. Interestingly, we can observe that the mean of \(\Omega^{s}\) increases as \(N\) and \(P\) grow, despite a slight drop from 0.313 to 0.296 as \((N,P)\) changes from (50,20) to (60,30). In practice, the mean LOC indicates the overall performance of the optimal parcel locations determined by the optimization framework. The overall upward trend suggests that as the passenger demand for ride-pooling and the number of locations grow, the performance of the proposed optimization framework improves, with more locations selected by the framework aligning with the optimal locations in the random selections. The performance improvement can also be supported by the changes in the maximum, minimum, and standard deviation of \(\Omega^{s}\). Both the maximum and minimum values of \(\Omega^{s}\) increase as \((N,P)\) shifts from (50,20) to (100,50). While the maximum \(\Omega^{s}\) shows a relatively modest and fluctuating increase from 0.600 to 0.680, the minimum \(\Omega^{s}\) experiences a dramatic rise, nearly sevenfold, from 0.05 to 0.340. The sharp rise in the minimum \(\Omega^{s}\) suggests that when both the number of randomly selected trips and the planned number of locations are sparse, the optimal locations can vary significantly between different selections. This variation is further evidenced by the decreasing standard deviation of \(\Omega^{s}\) as \((N,P)\) increases from (50,20) to (100,50). As \(N\) and \(P\) continue to grow, it is anticipated that the performance of the proposed optimization framework will further improve.