## 4 Numerical Experiments
### Algorithm Analyzes  
The normalized weights of shaking operators are the selection probability of the operator in iterations. It can reflect the ability of an operator to find an acceptable/better solution than the current/best solution in the past iterations. As shown in Figure 4, normalized weights of five shaking operators are tested on three different instance scales. Removal and insertion operators outperform the other three operators. The normalized weights tend to be equal at the later-stage iterations. That is because as the temperature cools down and the iteration goes on, operators are more difficult to obtain scores. Their weight will gradually approach the protection threshold. This feature increases the diversity in searching at the later stage of iterations.  
However, the high weight can not entirely indicate that an operator can find a new best solution more effectively. The new acceptable solution also can obtain scores. So the %\(\omega\)_IBest_/%_IU_sage_ ratio can reflect the ability of operators to find a new best solution, where %\(\omega\)_IBest_ is the percentage of an operator finding a new best solution times to the total best solution updated times, and %_IU_sage_ is the percentage of an operator is used times to total iterations. Table 1 shows the %\(\omega\)_IBest_/%_IU_sage_ ratio of shaking operators, which reflects the ability to find a new best solution in iterations.  
The insertion operator shows the most effective ability to find new best solutions, whereas the exchange and swap operators demonstrate the least impressive results. However, removal and insertion operators traverse their entire neighborhood and select the lowest operation cost solution, instead of exchange, swap, and relocation operators selecting randomly. The removal and insertion operators are employed to evaluate if a demand location needs to be replenished, which is much like the \"destroy-repair\" strategy in the adaptive large neighborhood search algorithm. The other three operators provide diversity, similar to the shaking strategy in traditional VNS. The results of Figure 4 and Table 1 show that, although the weights of exchange, swap, and relocation operators are much lower than removal and insertion operators, the %_U_IBest_/%_IU_sage_ ratio of them are 0.30, 0.29, and 0.42, respectively. That means these three operators also effectively find new best solutions by providing diversity.  
Figure 4: Normalized weights of shaking operators.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Instance** & **Removal** & **Insertion** & **Exchange** & **Swap** & **Relocation** \\ \hline n32 & 1.28 & 1.41 & 0.34 & 0.18 & 0.44 \\ n33a & 1.30 & 1.89 & 0.12 & 0.18 & 0.61 \\ n33b & 1.22 & 2.07 & 0.21 & 0.19 & 0.70 \\ n34 & 1.16 & 1.44 & 0.32 & 0.60 & 0.89 \\ n36 & 1.49 & 1.21 & 0.61 & 0.20 & 0.64 \\ n37a & 0.87 & 1.50 & 0.24 & 0.19 & 1.06 \\ n37b & 1.02 & 1.43 & 0.37 & 0.35 & 0.96 \\ n38 & 1.10 & 1.62 & 0.36 & 0.31 & 0.27 \\ n39a & 0.83 & 1.76 & 0.58 & 0.13 & 0.58 \\ n39b & 1.05 & 1.64 & 0.33 & 0.22 & 0.41 \\ n44 & 0.76 & 1.96 & 0.43 & 0.64 & 0.21 \\ n45a & 1.18 & 1.63 & 0.24 & 0.23 & 0.37 \\ n45b & 1.15 & 1.59 & 0.34 & 0.24 & 0.51 \\ n46 & 1.20 & 1.29 & 0.44 & 0.41 & 0.57 \\ n48 & 1.22 & 1.55 & 0.19 & 0.07 & 0.41 \\ n53 & 1.14 & 1.48 & 0.67 & 0.29 & 0.18 \\ n54 & 1.16 & 1.66 & 0.06 & 0.31 & 0.18 \\ n55 & 0.97 & 1.78 & 0.26 & 0.23 & 0.26 \\ n60 & 1.06 & 1.68 & 0.26 & 0.41 & 0.23 \\ n61 & 1.03 & 1.61 & 0.26 & 0.24 & 0.45 \\ n62 & 0.93 & 1.67 & 0.30 & 0.25 & 0.15 \\ n63a & 1.00 & 1.96 & 0.21 & 0.35 & 0.15 \\ n63b & 1.14 & 1.68 & 0.17 & 0.27 & 0.40 \\ n64 & 0.91 & 1.76 & 0.19 & 0.40 & 0.20 \\ n65 & 0.68 & 1.99 & 0.24 & 0.44 & 0.26 \\ n69 & 0.91 & 1.89 & 0.10 & 0.20 & 0.16 \\ n80 & 0.82 & 2.08 & 0.22 & 0.16 & 0.17 \\ Average & 1.06 & 1.68 & 0.30 & 0.29 & 0.42 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Performance of shaking operators (%_UBest_/%_IU_sage_ ratio).