## 5 A Case Study in Engineering Application
### Control Effect Analysis for Resilient Recovery  
In this case study, the following disruption is considered. A fire occurs in nodes \(W_{2}\) and \(W_{4}\), causing a disruption of the contacts in W5-M1, W5-M2, W5-P1, W1-M11, W1-M15, W1-P3, W1-M16, and W1-P2. We assume that the control parameters before the disruption are \(g=2\), \(\mu=1.5\), \(\Gamma=diag(5\), 5, 5), \(B_{ik}=diag[-0.5\), \(-0.5\), \(-0.5\)], \(t=0.05\), and the corresponding parameters for the cost are \(K_{g}=0.15\), \(K_{\mu}=0.1\), \(K_{\Gamma}=0.1\), \(K_{Bik}=1\), \(times=T/t=6\). The above initial values continue to be used. Then, we get \(R(0.3)=0.7075\), cost \(=5.95\), \(R_{c}(0.3)=0.1189\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Network Structure** & **Remove Links** & **\(R(0.3)\)** \\ \hline \multirow{3}{*}{Case 1} & Backbone & 18 links, as in Table 2 & 0.7127 \\ \cline{2-5}  & Network 1 & W3-M3, W3-M5, W3-P1, W1-M11, W1-M17, W1-P3 & 0.7132 \\ \cline{2-5}  & Network 2 & W3-M8, W3-M12, W3-P2, W1-M10, W1-M16, W1-P2 & 0.7250 \\ \hline \multirow{3}{*}{Case 2} & Network 3 & P2-S5, P2-S6, P2-W4, W2-P1, W2-P6, W2-M8 & 0.7205 \\ \cline{2-5}  & Network 4 & P3-S10, P3-W2, P3-W3, W2-P2, W2-M9, W3-M12 & 0.7236 \\ \hline \multirow{3}{*}{Case 3} & Network 5 & W5-M1, W5-M2, W5-P1, W1-M11, W1-M15, W1-P3 & 0.7073 \\ \cline{2-5}  & Network 6 & W5-M3, W5-P3, P3-W4, P3-W3, P3-W2, P2-W1 & 0.7253 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Resilient recovery rate \(R(t)\) in the cases. There are five kinds of recovery strategies (as shown in Section 4.2) to choose from. To further verify the method that helps to choose the resilient recovery strategy, the following two scenarios are designed. Scenario 1: After the fire occurs, the emergency management ability, the strategy adjustment ability, the coupling node control ability, and the incentive mechanism of each enterprise remain unchanged. That is to say, the cost parameters of each strategy are not changed. We adopt the five strategies for the experiment. With different strategies, \(\Delta g=1\), \(\Delta\mu=1\), \(\Delta\Gamma=1\), \(\Delta B_{ik}=0.1\). As shown in Table 5, the recovery rate, the cost, and the control efficiency are calculated. In scenario 1, the control efficiency of emergency management measures is the highest. In strategy 5, recovering the backbone link \(W_{1}\)-\(P_{3}\) has higher control efficiency than recovering the redundant link \(W_{1}\)-\(P_{2}\), although the recovery cost is the same. Scenario 2: After the fire, the cost parameters of the incentive system are reduced rather than increased when employees voluntarily work overtime to restore enterprise operations. The emergency management capability, strategy adjustment capability, and coupling node control capability of each enterprise remain unchanged. That is to say, the corresponding cost parameters of each strategy stay constant. As shown in Table 6, the control efficiency of the incentive mechanism is the highest in scenario 2. From the above discussion, it can be seen that the proposed method can help supply chain managers to select the appropriate control strategies. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Strategy**} & \begin{tabular}{c} **Strategy** \\ **Parameter** \\ \end{tabular} & \begin{tabular}{c} **Cost** \\ **Parameter** \\ \end{tabular} & \
\begin{tabular}{c} **Cost** \\ **Parameter** \\ \end{tabular} &
\begin{tabular}{c} **Cost** \\ **Cost** \\ \end{tabular} & \(R\)**(0.3)** & \(R_{\text{C}}\)**(0.3)** \\ \hline Strategy 4 & Incentive mechanism & \(\Delta B_{ik}=0.1\) & \(K_{\text{BL}}=0.5\) & \(C=C_{0}=+K_{\text{BL}}\), \(\Delta B_{ik}\), \(\tau/l=3.95\) & 0.7264 & 0.1839 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Scenario 2. Figure 15: Comparison of the resilient recovery rate \(R(t)\).