## 4 Computational Experiments
### Computational Results
#### 4.2.2 Results of the 2L-VRPSC Benchmark Instances  
In this section, we extend the Class 2 2L-VRP benchmarks described in the previous section to generate a large stochastic instance pool. Specifically, four classes of stochastic 2L-VRPSC instances are generated by varying the probability that customers require service \(p=\{0.2,0.4,0.6,0.8\}\). Here, we only test the performance of the approach under the 2|UO|L setting, with the results presented in Table 9. To verify the necessity of considering customer uncertainty, we use the value of the stochastic solution (VSS) to measure the extra cost due to ignoring the customers' uncertainty. The VSS index is defined as the difference between the expected results of using the expected value problem solution (EEV) and the here-and-now solution (HN), where the value of HN is the objective value obtained by stochastic programming model and the value of EEV represents the expected cost of the scenarios ignoring the customers' uncertainty. See [43; 44; 45; 46] for more information about these indices.  
As shown in Table 9, the values of VSS, EEV and HN indices for each instance are given. It can be observed that the value of the VSS index for each instance is greater than zero, which is consistent with our expectation that ignoring the uncertain information raised in the distribution operations incurs extra costs. Meanwhile, the direct application of the distribution schemes ignoring uncertainty may lead to costly ad-hoc arrangements for re-routing, which may substantially increase the actual operating costs and time. On the contrary, the application of distribution schemes considering the customers' uncertainty is conducive to the distribution companies to gaining familiarity with the distribution route and can have a good estimate of the time used to reach each customer so as to improve customers' satisfaction and distribution efficiency.  
We can also observe that the average values of the VSS index are 73.6, 59.8, 40.5 and 18.6 under the probabilities of 0.2, 0.4, 0.6 and 0.8, respectively. A lower probability of the presence of customers results in a higher value for VSS, EEV and HN indices. These observations suggest that a lower probability of the presence of customers will aggravate the uncertainty of the distribution scheme, which incurs a significant increase in the  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{3}{c}{**0.2**} & \multicolumn{3}{c}{**0.4**} & \multicolumn{3}{c}{**0.6**} & \multicolumn{3}{c}{**0.8**} \\ \cline{2-13}  & **EEV** & **HN** & **VSS** & **EEV** & **HN** & **VSS** & **EEV** & **HN** & **VSS** & **EEV** & **HN** & **VSS** \\ \hline  
Ave & 887.0 & 813.4 & 73.6 & 861.4 & 801.7 & 59.8 & 827.2 & 786.7 & 40.5 & 785.0 & 766.4 & 18.6 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Comparative results of stochastic instances on 2|UO|L 2L-VRPSC under different probabilities that customers require service.  
transportation cost. Additionally, the increased number of customers in the distribution scheme causes a significant increase in the transportation cost when customer uncertainty is considered. In terms of computational time, in our experiments, the average computational time for all 2L-VRPSC instances under the deterministic and stochastic settings are 1019 and 1380 s, respectively. Particularly, the computational time is affected by the size of scenarios generated from the Monte Carlo simulation, which contributes to the trade-off between the model fidelity and computational burden.