## 4 Case Study
### Performance of Proposed Formulations  
To validate the effectiveness of the chance-constrained robust optimization model developed in Section 2.3, the IBM ILOG CPLEX 12.9.0 Solver was utilized to solve the Mk01A-Mk10A test cases. In this experiment, the processing times of operations are assumed to follow a normal distribution with a mean of \(P_{i}\) and a standard deviation of \(\sigma=0.1\times P_{i}\), where \(P_{i}\) represents the standard processing time from the original case. Furthermore, five uncertain scenarios with different confidence levels were selected, and the CPLEX solution results are shown in Table 6.  
This section will validate and analyze the proposed model and algorithm using Case MK05 as an illustrative example. Case MK05A comprises a total of 15 subassemblies and components. Figure 9 illustrates the assembly relationships among the components, while Table 7 presents the operations associated with each component. Given the overall large scale of the cases considered in this study, the specific available machines and processing times for each operation are not detailed.  
Furthermore, using Case MK05A as an example, this study validates the impact of processing time uncertainty on the performance of schedules. Figure 10 illustrates the variation of \(C_{max}\) as the confidence level of processing time ranges from 0% to 100%.  
Figure 8: Comparative analysis of the diversity of the two encoding schemes.  
In Figure 10, the dashed line represents the optimal solution when the processing time is fixed at a certain value, while the solid line indicates the trend of \(C_{max}\), as the confidence level of processing time varies from 0% to 100%. As shown in this figure, when the confidence level is 0, the robust optimization model is equivalent to the deterministic model, allowing for a higher probability of constraint violation. As the confidence level increases, the model accounts for a broader range of scenarios related to processing time uncertainty, necessitating a higher value of \(C_{max}\) to compensate for performance deviations caused by fluctuations in processing time, thereby ensuring that constraints are satisfied. In extreme cases, when the confidence level is equal to 100%, it means that the model considers all possible uncertainty scenarios related to the fluctuations in processing times. Given that the experiment assumes that processing times follow a normal distribution with the standard processing time as the mean, the closer the actual processing time is to the standard, the higher its probability of occurrence. According to Equation (13), the process of increasing confidence levels represents a transition from high-probability scenarios  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Cases**} & \multicolumn{5}{c}{**Confidence Levels**} & \multirow{2}{*}{**Runtime (s)**} \\ \cline{2-2} \cline{4-6}  & **0\%** & **20\%** & **40\%** & **60\%** & **80\%** & \\ \hline Mk01A & 78 & 81 & 85 & 96 & 113 & 30 \\ Mk02A & 87 & 90 & 98 & 118 & 137 & 30 \\ Mk03A & 263 & 277 & 287 & 316 & 399 & 120 \\ Mk04A & 135 & 138 & 141 & 147 & 159 & 120 \\ Mk05A & 223 & 230 & 238 & 243 & 262 & 120 \\ Mk06A & 130 & 138 & 146 & 157 & 185 & 120 \\ Mk07A & 176 & 188 & 201 & 230 & 323 & 120 \\ Mk08A & 505 & 584 & 631 & 752 & 982 & 300 \\ Mk09A & 546 & 566 & 622 & 701 & 832 & 300 \\ Mk10A & 419 & 452 & 496 & 550 & 733 & 300 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Solution results of test cases obtained by CPLEX.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Subassemblies**} & **Contained** & **Subassemblies** & **Contained** \\ /**Components** & **Operations** & **/Components** & **Operations** \\ \hline C1 & (101–106) & C9 & (16–24) \\ C2 & (81–85) & C10 & (58–66) \\ C3 & (86–93) & C11 & (51–57) \\ C4 & (94–100) & C12 & (10–15) \\ C5 & (40–45) & C13 & (67–73) \\ C6 & (1–9) & C14 & (33–39) \\ C7 & (46–50) & C15 & (74–80) \\ C8 &