## 6 Numerical Experiments
### Sensitivity Analyses
#### 6.5.2 The Rankings of Candidate Locations  
Introducing the minimum distance \(R\) between any pair of selected candidate locations (indexed with \(j\) and \(j^{\prime}\)) is equivalent to adding the following non-linear constraint into the PM model:  
\[y_{j^{\prime}}y_{j^{\prime}}R_{jj^{\prime}}\leq R,\ \forall j,j^{\prime}\in J\ \text{and}\ j\
eq j^{\prime} \tag{7}\]  
The parameter \(R_{jj^{\prime}}\) stands for the Euclidean distance between locations \(j\) and \(j^{\prime}\). The constraint can be linearized by defining an auxiliary variable \(z_{jj^{\prime}}\) and replacing Constraint (7) with the following set of linear constraints:  
\[z_{jj^{\prime}}\leq y_{j},\ \forall j,j^{\prime}\in J\ \text{and}\ j\
eq j^{\prime} \tag{8}\]  
\[z_{jj^{\prime}}\leq y_{j^{\prime}},\ \forall j,j^{\prime}\in J\ \text{and}\ j\
eq j^{\prime} \tag{9}\]  
Figure 11: Average minimum passenger detour distance against \(N\).  
Figure 12: Average minimum passenger detour distance against \(P\).  
\[z_{jj^{\prime}}R_{jj^{\prime}}\leq R,\ \forall j,j^{\prime}\in J\text{ and }j\
eq j^{\prime} \tag{10}\]  
\[z_{jj^{\prime}}\in\{0,1\},\ \forall j,j^{\prime}\in J\text{ and }j\
eq j^{\prime} \tag{11}\]  
With Constraints (8)-(11), we set \((N,P)=(100,10)\), (100,50), and (100,90), and used the same sample set \((S=2000)\) that appeared in Section 6.4 as inputs to the optimization framework. Note that the ranking results obtained in Section 6.4 are equivalent to those obtained with \(R=0\) since we did not specify \(R\) in Section 6.4. For each \(N-P\) combination, we let \(R\) vary from 1 km to 4 km at an interval of 1 km, and the corresponding rankings for all the candidates were obtained. To avoid a redundant presentation, we consider the rankings of the ten most frequently chosen candidates when \(R=0\) under each \(N-P\) combination and present their ranking changes against \(R\) in Table 5.  
Table 5 reveals that when \(P\) is relatively small, the rankings of the candidates in the table remain stable until \(R\) is sufficiently large. For example, all the rankings remain unchanged when \(P=10\) and \(R<3\). As \(P\) grows, more candidates become ranking-sensitive to \(R\). It is worth noting that we failed to obtain the rankings of stations when  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Candidate Index**} & \multicolumn{5}{c}{**R (km)**} \\ \cline{2-6}  & **0** & **1** & **2** & **3** & **4** \\ \hline  & & & \multicolumn{3}{c}{\(P=10\)} \\  
\(P=90\) and \(R=4\) (N/A is displayed) since the PM model cannot have any feasible solution. The change in ranking sensitivity is clearly due to the increase in the density of parcel lockers (i.e., the increase in \(P\)). In most cases, increasing \(R\) does not impact the optimal solution when \(P\) is small since the constraints are automatically guaranteed by the sparse distribution of parcel lockers. By contrast, a dense distribution of parcel lockers (i.e., \(P\) is large) implies that it is more likely for a chosen candidate location to violate Constraints (8)-(11), and it is therefore excluded. As a result, when \(R\) keeps increasing, we notice that fewer of the ten candidates remain in the top ten rankings as \(P\) increases, with the number decreasing from 9 to 6, and finally to 4.