## 6 Numerical Experiments
### A Ranking-Based Classification of Candidate Parcel Locker Locations  
We continue to explore the properties of the proposed optimization framework by examining how candidate locations' rankings change in response to varying \(P\), based on which we propose a ranking-based classification of the candidate parcel locker locations. We are particularly interested in how the ranking of a candidate location change as \(P\) increases within the same set of samples. To see this, we set \(N=100\) and performed sample generation \(S=2000\) times. Then, we let \(P\) vary from 10 to 100 at an interval of 10 to simulate the increasing locker locations density. For each resulting \(N-P\) combination, we ran the proposed optimization framework and recorded the rankings of all the 421 candidate \begin{table}
\begin{tabular}{l c c c c} \hline \hline _(N,P)_ & **Mean** & **Maximum** & **Minimum** & **Standard Deviation** \\ \hline (50,20) & 0.313 & 0.600 & 0.050 & 0.081 \\ (60,30) & 0.296 & 0.500 & 0.067 & 0.061 \\ (70,30) & 0.395 & 0.667 & 0.167 & 0.069 \\ (80,40) & 0.459 & 0.650 & 0.250 & 0.060 \\ (90,40) & 0.459 & 0.650 & 0.275 & 0.060 \\ (100,50) & 0.512 & 0.680 & 0.340 & 0.053 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Information of LOC \(\Omega^{s}\) in different cases. locations. Based on the results, we classified all the candidate locations into five types according to how their rankings changed as \(P\) increased, namely (I) the stable candidates, (II) the rising candidates, (III) the falling candidates, (IV) the concave candidates, and (V) the convex candidates. To avoid a cumbersome presentation, we select two representatives from each type of candidate and present their ranking changes against \(P\) in Table 4. We remind the readers that the candidate indices listed in the first column of Table 4 serve solely as identifiers and are not indicative of their ranking. It is noteworthy that the rankings of all the listed candidates in Table 4 tend to stabilize as \(P\geq 60\). This is because as more locations are permitted to be selected in the PM model, each passenger is increasingly likely to be matched with a location that minimizes their detour distance. In fact, one can tell from the formulation of the PM model that the optimal location for every passenger is deterministic if all the candidates can be selected (i.e., \(P=421\)) and is not affected by the other passengers since we did not consider capacitated parcel lockers and the queuing time for parcel pick-up. The only situation in which a passenger is not matched with his/her optimal location is when that location is excluded from the optimal solution to the PM model due to the limited \(P\). Therefore, we also note that the classification we discuss in this section is based on the pre-stabilized rankings of the candidates (i.e., we only observe the ranking trends when \(P\leq 60\)), as the discussion would be meaningless if all the rankings were stabilized. To facilitate our analyses, we introduce two terms for each candidate \(j\in J\), namely (1) the frequency that candidate \(j\) appears in the optimal solutions across the \(S\) selections (denoted as \(f_{j}\)) and (2) the average number of passengers matched with candidate \(j\) across the \(S\) selections if \(j\) is selected (denoted as \(m_{j}\)). Then, the total frequency for candidate \(j\) to be selected across the \(S\) selections can be expressed as \(f_{j}m_{j}\). It is important to note that the reasons behind the emergence of the five types of candidate location are less straightforward since the change in the ranking of candidate \(j\), as \(P\) increases, is jointly resulted from both the change in its ranking and that in the other candidates' rankings, which are further determined by the changes in \(f_{j}\) and \(m_{j}\) (and thus \(f_{j}m_{j}\)). For example, if the ranking of candidate \(j\) rises (e.g., see candidates 81 and 139 in Table 4), more locations can be selected due to the increase in \(P\), and \(j\) appears in the