# The Electric Vehicle Routing Problem with Time Windows, Partial Readers, and Parcel Lockers
## 6 Discussion  
We further conduct an evaluation to compare the results obtained from EVRPTW-PR and EVRPTW-PR-PL (with \(\delta_{L}=0.3\)) in terms of the cost ratio \"CR\" and the difference in the number of used vehicles \(\Delta|K|\). The cost ratio \"CR\" is computed as the ratio between the best cost value \(f_{b}\) of EVRPTW-PR and the best cost value \(f_{b}\) of EVRPTW-PR-PL. The difference in used vehicles \(\Delta|K|\) is calculated as the number of used vehicles \(|K|\) in EVRPTW-PR minus the number of used vehicles \(|K|\) in EVRPTW-PR-PL. Therefore, in Tables A8-A16, we recall the results of EVRPTW-PR instances in Section 5.3 and report terms \"CR\" and \(\Delta|K|\) in the right-most columns. A summary of the results of the average cost ratio \"CR\" and the average difference in the number of used vehicles for each subset is provided in Table 5. For the results from the large-sized instances, the patterns of \"CR\" and \(\Delta|K|\) are illustrated in Figure 7 by sorting the subsets in ascending order of \"CR\" values. The figure helps highlight some features, as follows. Figure 6: Search process of the proposed ALNS for solving instance c101. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Size**} & \multirow{2}{*}{**Subset**} & \multicolumn{3}{c}{**EVRPTW-PR-PL**} & \multicolumn{3}{c}{**EVRPTW-PR**} & \multirow{2}{*}{**CR**} & \multirow{2}{*}{\(\Delta|K|\)} \\ \cline{3-3} \cline{5-8}  & & I K I & \(\bar{f}_{b}\) & \(\mathrm{I}\,\mathrm{K}\) I & \(\bar{f}_{b}\) & & & \\ \hline \multirow{4}{*}{Small-sized} & 5C & 1.42 & 106.80 & 1.42 & 188.90 & 2.10 & 0.00 \\  & 10C & 2.08 & 156.10 & 2.08 & 301.49 & 2.17 & 0.00 \\  & 15C & 2.67 & 191.27 & 2.67 & 357.80 & 2.11 & 0.00 \\ \hline \multirow{4}{*}{Large-sized} & C-1 & 11.00 & 945.82 & 10.56 & 1005.62 & 1.08 & \(-\)0.44 \\  & C-2 & 4.00 & 526.97 & 4.00 & 629.82 & 1.20 & 0.00 \\ \cline{1-1}  & R-1 & 10.42 & 741.90 & 12.67 & 1222.96 & 1.66 & 2.25 \\ \cline{1-1}  & R-2 & 3.27 & 492.36 & 2.64 & 918.90 & 1.87 & \(-\)0.64 \\ \cline{1-1}  & RC-1 & 12.38 & 1030.82 & 12.75 & 1404.22 & 1.38 & 0.38 \\ \cline{1-1}  & RC-2 & 4.00 & 600.15 & 3.13 & 1142.23 & 1.89 & \(-\)0.88 \\ \hline \multicolumn{2}{c}{Average} & 5.69 & 532.47 & 5.77 & 796.88 & 1.72 & 0.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of the results of EVRPTW-PR and EVRPTW-PR-PL instances. * The implementation of parcel lockers in delivery networks significantly reduces travel costs for routing plans; i.e., the overall cost ratio is 1.72. Figure 8 provides routing plans obtained from instance c101 for EVRPTW-PR and EVRPTW-PR-PL, which illustrates the differences between the two routing plans. * Considering customer locations, subsets with randomly distributed locations (i.e., subsets (R) and (RC)) show higher \"CR\" values. For example, the \"CR\" value for subset C-1 is only 1.08, while those for subsets R-1 and RC-1 are 1.38 and 1.66, respectively. This implies that utilizing parcel lockers becomes more effective when the locations are randomly scattered. * Considering time windows, subsets with wider customer time windows (i.e., C-1, R-1, and RC-1 subsets) provide higher \"CR\" values than those with stricter time windows (i.e., C-2, R-2, and RC-2 subsets). This means that instances with wider customer time windows have more potential to find cost-effective solutions when implementing parcel lockers. * Values of \(\Delta|K|\) are not significant for all subsets; i.e., 0.07 vehicles on average. However, for subset R-1, the number of used vehicles is significantly reduced by 2.25 vehicles. This occurs when the customer locations are randomly distributed and the time windows are wider. The insights noted above can be valuable in reality for delivery companies when making decisions regarding the deployment of parcel lockers in their last-mile delivery networks, aiming to leverage the cost-saving benefits.