## 4 Computational Experiments
### Computational Results
#### 4.2.1 Results on the 2L-VRP Benchmark Instances
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Inst.**} & \multicolumn{3}{c}{**Class 2**} & \multicolumn{3}{c}{**Class 3**} & \multicolumn{3}{c}{**Class 4**} & \multicolumn{3}{c}{**Class 5**} \\ \cline{2-13}  & **BKS** & **Cost** & **Gap** & **BKS** & **Cost** & **Gap** & **BKS** & **Cost** & **Gap** & **BKS** & **Cost** & **Gap** \\ \hline  
Ave & 762.35 & 767.93 & 0.59 & 754.24 & 761.43 & 0.90 & 770.08 & 780.34 & 0.96 & 713.70 & 720.21 & 0.90 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results for 2\(|\)SO\(|\)L 2L-VRP (Classes 2–5). “BKS” denotes the best-known solution found among Refs [8; 9; 12; 41]. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Inst. **} & \multicolumn{3}{c}{**Class 2**} & \multicolumn{3}{c}{**Class 3**} & \multicolumn{3}{c}{**Class 4**} & \multicolumn{3}{c}{**Class 5**} \\ \cline{2-13}  & **BKS** & **Cost** & **Gap** & **BKS** & **Cost** & **Gap** & **BKS** & **Cost** & **Gap** & **BKS** & **Cost** & **Gap** \\ \hline
1 & 278.73 & 278.73 & **0.00** & 284.23 & 284.52 & 0.10 & 282.95 & 282.95 & **0.00** & 278.73 & 278.73 & **0.00** \\
2 & 334.96 & 334.96 & **0.00** & 352.16 & 352.16 & **0.00** & 334.96 & 334.96 & **0.00** & 334.96 & 334.96 & **0.00** \\
3 & 384.93 & 384.93 & **0.00** & 385.32 & 385.32 & **0.00** & 362.41 & 364.45 & 0.56 & 358.40 & 358.40 & **0.00** \\
4 & 430.89 & 430.89 & **0.00** & 430.89 & 430.89 & **0.00** & 447.37 & 447.37 & **0.00** & 430.89 & 430.89 & **0.00** \\
5 & 375.28 & 375.28 & **0.00** & 379.94 & 385.69 & 1.51 & 383.88 & 383.88 & **0.00** & 375.28 & 375.28 & **0.00** \\
6 & 495.85 & 499.08 & 0.65 & 498.16 & 498.16 & **0.00** & 498.32 & 502.11 & 0.76 & 495.85 & 507.18 &are 0.59%, 0.90%, 0.96% and 0.90% for the instances in Classes 2-5, respectively. Under the 2\(|\)SR\(|\)L setting, the corresponding average percentage gaps are 0.46%, 0.83%, 0.66% and 0.68%, respectively. It is worth noting that the EATS-MOBFS method has reached the best-known solutions for almost 10 instances in each class. It infers that the proposed hybrid heuristic is efficient for solving the 2L-VRP under the 2\(|\)SO\(|\)L and 2\(|\)SR\(|\)L settings. As illustrated in Table 8, the results obtained by the proposed EATS-MOBFS under the 2\(|\)UR\(|\)L and 2\(|\)SR\(|\)L settings are compared with the existing heuristic approaches, which shows that the proposed approach is capable of producing comparable results under the 2\(|\)UR\(|\)L and 2\(|\)SR\(|\)L settings. Specifically, the overall performance of the proposed EATS-MOBFS method outperforms that of ACO under the 2\(|\)UR\(|\)L setting. In comparison to ACO, our approach is able to produce better solutions for 11 out of 20 instances. Under the 2\(|\)SR\(|\)L setting, our approach is superior to that of ACO and has produced comparable results compared with those obtained by BR-LNS. The proposed approach yields better solutions for 11 out of 20 instances compared with ACO. To sum up, the proposed approach produces satisfactory performances in solving the 2L-VRP under all four loading configurations, where the average percentage gaps between the solutions obtained by the EATS-MOBFS method and the best-known solutions are 0.97%, 0.48%, 1.14% and 0.64%, respectively. Particularly, our approach has reached the best-known solutions for 34, 35, 25 and 36 out of 80 instances under four settings, respectively. It can be observed that the average percentage gaps over the best-known solutions are smaller under the 2\(|\)UR\(|\)L and 2\(|\)SR\(|\)L settings compared with the 2\(|\)UO\(|\)L and 2\(|\)SO\(|\)L settings. Namely, the performance of our approach is more satisfactory under the 2\(|\)UR\(|\)L and 2\(|\)SR\(|\)L settings, while the performance is inferior to the existing state-of-art methods under the 2\(|\)UO\(|\)L and 2\(|\)SO\(|\)L settings. We can also observe that for some large-scale instances, our proposed approach shows a slight disadvantage over the existing methods.