## 4 Case Study  
### Test Instances and Parameters Setting  
Currently, a substantial number of benchmark test cases have been proposed for traditional job shop scheduling problems. However, to the best of our knowledge, there is no existing benchmark for the FAJSP that incorporates uncertain processing times. To evaluate the performance of the proposed model and methods, this section constructs ten benchmark FAJSP based on the Mk series of test cases proposed by [PERSON] [28] for flexible JSP. The ten original test cases from Mk01 to Mk10 are categorized into 3 groups according to the number of components: 10 components (corresponding to Mk01, Mk02, and Mk06), 15 components (corresponding to Mk03, Mk04, and Mk05), and 20 components (corresponding to Mk07, Mk08, Mk09, and Mk10). By combining the test cases from the three groups with different operation scales, new cases that include assembly structures are constructed. To facilitate differentiation, the constructed test cases are designated as Mk01A to Mk10A, and their scales are presented in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Instances**} & \multicolumn{2}{c}{**Number of**} & \multirow{2}{*}{**Number of Operations**} & \multirow{2}{*}{**Number of Machines**} \\  & **Subassembly/Components** & & \\ \hline Mk01A & 10 & 55 & 6 \\ Mk02A & 10 & 58 & 6 \\ Mk03A & 15 & 150 & 8 \\ Mk04A & 15 & 90 & 8 \\ Mk05A & 15 & 106 & 4 \\ Mk06A & 10 & 150 & 10 \\ Mk07A & 20 & 100 & 5 \\ Mk08A & 20 & 225 & 10 \\ Mk09A & 20 & 240 & 10 \\ Mk10A & 20 & 240 & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Characteristics of test instances. Figure 6: Gantt chart of a feasible solution based on interval scheduling. The performance of evolutionary algorithms is significantly influenced by parameter settings. To obtain the optimal combination of parameter levels, this paper conducts a statistical analysis of the algorithm's results under different parameter level combinations using orthogonal experiments and range analysis [29]. For the MAE algorithm, three key factors are considered: the maximum number of iterations per cycle (F1), the maximum allowable search attempts per cycle in tabu search (F2), and the maximum number of attempts for neighborhood switching in the path relinking operator (F3). Considering the impact of various parameter combinations on algorithm efficiency, each parameter is assigned three selectable levels within an acceptable time frame, as shown in Table 4.  
Orthogonal experiments are designed based on the selection of factors and levels and perform range analysis on the experimental results, as shown in Table 5. In this table, the K value represents the sum of experimental results when a factor is at a certain level. K avg denotes the average of that K value at each level. R represents the range value of the factor for the minimization problem studied in this paper. R is the difference between the minimum K avg and the maximum K avg of the factor, reflecting the degree of influence of the current factor on the experimental metrics. The average effect results under different factors can be visually represented in Figure 7. The optimal levels for each factor and the corresponding best parameter combination are as follows: F1 at L3 (100), F2 at L3 (200), and F3 at L2 (5). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Factors**} & \multicolumn{4}{c}{**Factor Level**} \\ \cline{2-5}  & **L1** & **L2** & **L3** \\ \hline F1 & 30 & 50 & 100 \\ F2 & 50 & 100 & 200 \\ F3 & 1 & 5 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The levels of parameters.