## 5 Case Study
### Computational Results  
In Table 2, the efficacy of the HRES algorithm in solving both larger and smaller network cases surpasses that of CPLEX. The HRES algorithm demonstrates efficient convergence, reaching the final solution within a notably brief timeframe. In contrast, the solving time of CPLEX exhibits a rapid escalation, especially as the network scale increases. This advantage of the HRES algorithm becomes particularly pronounced for larger network cases, surpassing 12 nodes. For example, when assessing a network with 24 nodes, CPLEX requires 203.474 s for a single evaluation, whereas our method completes the same task in only 0.112 s. Although the results of the HRES algorithm may not be as optimal as those of CPLEX, the difference is not considered significant. The experimental outcomes affirm the effectiveness of the developed HRES algorithm. As shown in Table 3, the construction of underground networks significantly improves the resilience of the original ground network. The network resilience for nearly all nodes increased by at least 15%. The resilience of the ground network with 24 nodes increased by 25%. Figure 7 shows the ground-underground logistics system planning diagram corresponding to the 10-node network. The orange parts of the figure represent the nodes and arcs that are constructed. The gray parts represent the ones that are not built. The figure shows an underground logistics network plan with maximum resilience under budget constraints. Under this layout, the resilience of the ground network is improved by 25.1%. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Population size & 60 \\ Maximum number of iterations & 400 \\ Fixed construction costs for underground nodes & 2 \\ Unit construction costs for underground pipelines & \(1/\)km \\ Crossover rate & 0.95 \\ Mutation rate & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The key parameters of the computational experiment. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Node Number (N)** & **Approach** & **Resilience** & **CPU Time (s)** \\ \hline
5 & CPLEX & 1.0 & 0.148 \\  & HRES & 0.9163 & 0.001 \\  & CPLEX & 0.9698 & 6.251 \\
8 & HRES & 0.9122 & 0.003 \\  & CPLEX & 0.9108 & 31.542 \\
10 & HRES & 0.8708 & 0.011 \\  & CPLEX & 0.9999 & 61.321 \\
12 & HRES & 0.8855 & 0.029 \\  & CPLEX & 0.9442 & 68.406 \\
18 & HRES & 0.8014 & 0.068 \\  & CPLEX & 0.9851 & 203.474 \\
24 & HRES & 0.9209 & 0.112 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of HRES and CPLEX on small-sized instances. To test the operational effect of the GA-HRES algorithm, the convergence process is recorded as shown in Figure 8. The convergence process for each case is generally smooth and effective. In particular, for networks with five and eight nodes, not only does the process converge steadily, but the resilience values also evolve from initial to final results in a large scope. The results prove that the GA-HRES algorithm achieves excellent results. To explore the correlation between resilience values, budget, and underground pipeline capacity, a sensitivity analysis is conducted. As shown in Figure 9, within a certain range, the resilience value increases as the budget grows. However, once a certain threshold is  
Figure 8: Convergence of algorithm results. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Node Number (N)**} & **Original Network Resilience** & **Planned Network Resilience** & **Rate of Increase** \\ \hline
5 & 0.5083 & 0.8917 & 75.4\% \\
8 & 0.6205 & 0.8625 & 39.0\% \\
10 & 0.6915 & 0.8655 & 25.1\% \\
12 & 0.7758 & 0.8927 & 15.0\% \\
18 & 0.7372 & 0.7837 & 6.3\% \\
24 & 0.6548 & 0.8210 & 25.4\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of original and planned network resilience in small-sized instances. Figure 7: Planned underground logistics network. exceeded, the resilience value does not increase any further. For example, for a network of 10 nodes, the resilience value stays at 0.86 once the budget reaches the threshold of 250.