## 6 Numerical Analysis
### Sensitivity Analysis  
This section analyzes the impact on the results by changing the input parameters. Each calculation only varies the value of one parameter, ranging from \(-50\%\) to \(+50\%\). Other parameters remain unchanged, and the changes in the results are observed. Since unit 2 and unit 3 are the same two units, their two parameters change simultaneously. Hereby, this part selects \(\theta_{1}\), \(\theta_{2}\), \(\mu\), \(c_{Set}\), \(c_{l}\), \(c_{R}\), \(c_{S}\), \(c_{P}\), \(c_{PM}^{(i)}\), \(c_{PM}^{(i)}\), a total of 10 parameters for analysis. The influence of parameter changes on lot size \(Q\), preventive maintenance threshold of \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Unit \(i\)** & \(\alpha_{i}\) & \(\beta_{i}\) & \(D_{f}^{(i)}\) & \(\epsilon_{PM}^{(i)}\) & \(\epsilon_{CM}^{(i)}\) & \(\lambda_{i}\) & \(\mu_{i}\) \\ \hline
1 & 1.4 & 2.8 & 10 & 1600 & 4700 & 1 & 0.5 \\
3 & 2.2 & 3.2 & 12 & 1400 & 4200 & 2 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of units. Figure 8: Iteration diagram. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(p_{r}\)(unit/day) & 200 & \(\beta\) & 1.26 \\ \(d_{\max}\)(unit/day) & 160 & \(\mu\) & 0.1 \\ \(\theta_{1}\) & 0.1 & \(c_{Set}\) (Yuan/each time) & 600 \\ \(\theta_{2}\) & 0.6 & \(c_{I}\) (Yuan/unit/day) & 0.5 \\ \(p_{0}\) & 0.004 & \(c_{R}\) (Yuan/unit) & 10 \\ \(\eta\) & 0.071 & \(c_{S}\) (Yuan/unit) & 20 \\ \(p_{0}\) & 0.0046 & \(c_{P}\) (Yuan) & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters of system. unit \(1\)\(D_{p}^{(1)}\), preventive maintenance threshold of unit 2 \(D_{p}^{(2)}\), and expected cost rate \(EC\) are shown in Figure 9.  
Figure 9a shows the influence of parameter changes on the optimal production lot size \(Q\). We can see that two parameters have significant effects on lot size. The first is the setting cost \(c_{Set}\). With the increase of setting cost, the quantity has an apparent upward trend. When the setting cost increases, the PM thresholds \(D_{p}^{(i)}\) increase to minimize the expected cost rate and reduce the frequency of maintenance operations. When the production time increases, the lot size increases significantly. The second is inventory cost \(c_{I}\). When inventory cost increases, the production lot size is reduced to minimize the expected cost rate. Figure 9b shows the influence of parameter changes on the preventive maintenance threshold of unit 1. \(D_{p}^{(1)}\) is sensitive to the ten parameters analyzed. The most sensitive is setting cost \(c_{Set}\) and PM cost of unit 2 \(c_{PM}^{(2)}\). Similar to Figure 9a, when setting cost increases, the PM threshold \(D_{p}^{(1)}\) increases to reduce maintenance frequency. When the PM cost of unit 2 increases, the threshold of unit 2 \(D_{p}^{(2)}\) increases to reduce the frequency of maintenance unit 2, and \(D_{p}^{(1)}\) decreases to decrease the expected cost rate. Figure 9c shows the influence of parameter changes on the preventive maintenance threshold of unit 2. \(D_{p}^{(2)}\) is similar to \(D_{p}^{(1)}\). The most sensitive is setting cost \(c_{Set}\) and PM cost of unit 1 \(c_{PM}^{(1)}\). When setting cost increases, the PM threshold \(D_{p}^{(2)}\) increases to reduce maintenance frequency. When the PM cost of unit 1 increases, the threshold of unit 2 \(D_{p}^{(1)}\) increases to reduce the frequency of maintenance unit 1, and \(D_{p}^{(2)}\) decreases to decrease the expected cost rate. Figure 9d shows the impact of parameters on the expected cost rate. It can be seen from the figure that the expected cost rate is more sensitive to setting cost \(c_{Set}\) and inventory cost \(c_{I}\). This is because the increase in setting cost causes a lot of changes in the production lot size, resulting in an increase in inventory costs and thus affecting the expected cost rate. Overall, the expected cost rate is still increasing.