## 6 Numerical Experiments
### A Ranking-Based Classification of Candidate Parcel Locker Locations
\begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Candidate Index**} & \multicolumn{10}{c}{\(P\)} \\ \cline{2-11}  & **10** & **20** & **30** & **40** & **50** & **60** & **70** & **80** & **90** & **100** \\ \hline \multicolumn{11}{c}{Type I: Stable} \\
87 & 2 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
138 & 3 & 2 & 2 & 3 & 4 & 4 & 4 & 4 & 3 & 3 \\ \hline \multicolumn{11}{c}{Type II: Rising} \\
81 & 47 & 16 & 11 & 12 & 8 & 8 & 9 & 8 & 8 & 8 \\
139 & 107 & 69 & 47 & 35 & 22 & 20 & 20 & 20 & 20 & 20 \\ \hline \multicolumn{11}{c}{Type III: Falling} \\
99 & 7 & 19 & 23 & 24 & 27 & 30 & 28 & 28 & 27 & 27 \\
193 & 5 & 6 & 16 & 22 & 25 & 26 & 26 & 26 & 28 & 28 \\ \hline \multicolumn{11}{c}{Type IV: Concave} \\
146 & 52 & 17 & 21 & 25 & 33 & 42 & 42 & 42 & 42 & 42 \\
194 & 19 & 10 & 8 & 7 & 11 & 16 & 16 & 15 & 15 \\ \hline \multicolumn{11}{c}{Type V: Convex} \\
137 & 8 & 21 & 20 & 16 & 14 & 14 & 14 & 14 & 14 & 14 \\
158 & 79 & 129 & 104 & 95 & 78 & 73 & 73 & 72 & 73 & 70 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Five types of candidate locations according to their rankings against \(P\). optimal solutions to the PM model in more samples. By contrast, the falling ranking of candidate \(j\) (e.g., see candidates 99 and 193 in Table 4) may imply that although \(j\) is not the optimal location to many passengers in the samples, it is selected as a compromise when the value of \(P\) is small. As \(P\) keeps increasing, fewer passengers are matched with \(j\), and hence its ranking drops. Rather than diving into the reasoning behind the five types of candidate, we can draw implications from the classification itself. Firstly, planners should carefully determine the optimal number of candidate locations for parcel locker deployment, recognizing that there exists a threshold value for \(P\). In our example, this threshold value is 60, below which the rankings of the candidates are affected by the deployed parcel lockers, while surpassing this threshold scarcely alters the candidate rankings, but certainly increases the parcel locker deployment costs. Selecting more locations beyond the threshold may not substantially reduce the passengers' detour times, but will likely contribute to higher costs. Furthermore, it is clear that the threshold value is dependent on the spatial distribution of both trip demand and the potential parcel locker locations. This signifies that the proposed optimization framework can be used to quantify the threshold for any given set of trips and candidate locations. Secondly, the necessity of choosing an appropriate \(P\) arises not only from the presence of the threshold, but also from the fact that the optimally selected locations may vary as \(P\) changes. From a temporal perspective, if the parcel lockers are deployed in batches (i.e., \(P\) gradually increases as time elapses), then there are either newly included or abandoned locations as \(P\) increases. Two instances can be found in Table 4. Candidate 139 (type II) is not selected until \(P\geq 40\), while candidate 137 (type V) is always selected, except when \(P=20\). Therefore, if \(P\) gradually increases from 10 to 50 in several periods, candidate 139 is only opened when \(P\) equals 40 and 50, and candidate 137 needs to be temporarily shut down when \(P=20\).