## 4 Case Study
### Comparative Analysis of Algorithms  
To further validate the effectiveness and advantages of the proposed MAE, two competitive optimization algorithms for the FAJSP were selected for comparison. Both the MAE and the aforementioned methods were applied to the Mk01A-Mk10A test cases and performance comparisons were conducted through experimental evaluation. The specific descriptions of the comparison algorithms are provided below:  
* **Genetic algorithm based on the earliest available machine priority Rule (GA-EAM)**: This algorithm is designed to generate an initial population for the operation sequence of the FAJSP and to perform genetic operations. During the decoding phase, the Earliest Available Machine (EAM) rule is applied for machine allocation in the operation sequencing chromosome encoding. Specifically, among all eligible machines for a given operation, the machine that becomes available the earliest is selected to determine the machine assignment for that operation. * **Particle swarm optimization algorithm based on selective decode string (PSO-SDS)**: This algorithm is developed within the framework of particle swarm optimization (PSO) but does not consider the feasibility of the encoding during the initialization and update processes. Instead, it addresses illegal solutions through a repair mechanism based on selective decode strings. All algorithms were implemented using MATLAB R2016a and executed on an Intel(R) Core(TM) i7-9750H processor, with a frequency of 2.60 GHz and 8.00 GB of RAM. The comparative algorithms also employed range analysis for parameter selection. The processing times are also assumed to follow the normal distribution introduced in Section 4.3, and the confidence interval at a 70% confidence level is used as the range for the interval numbers. The optimal values of \(C_{max}\) obtained from ten independent runs of each algorithm were recorded, and the experimental results are presented in Table 8.  
In Table 8, the size denotes the number of operations and machines utilized in various cases, thereby providing an indication of the problem's scale and the complexity involved in its resolution. As shown in Table 8, the MAE proposed in this paper obtains better solutions for all cases under the same conditions. The commonly used EAM rule for FJSP performs poorly when applied to FAJSP, which includes operation assembly relationships. Figure 12: Gantt chart of the optimal solution for Mk05A obtained by CPLEX. (Confidence level = 90%). The PSO-SDS algorithm with a repair mechanism performs well on problems with a small number of operations and machines (such as cases Mk01A, Mk02A, Mk04A, and Mk07A) but struggles with large-scale problems. The GA-EAM and the MAE designed in this paper utilize the same approach for operation sequencing encoding, with the difference lying in the machine selection component. Specifically, GA-EAM employs the EAM rule to allocate corresponding machines to each operation, while MAE optimizes the machine selection scheme through tabu search and path relinking operators. Figures 13 and 14 illustrate the Gantt charts for the scheduling solutions obtained from the GA-EAM and MAE for the Mk05A test case, respectively. Comparing Figures 13 and 14, it is evident that the schedules generated by GA-EAM exhibit low utilization rates across the machines, with significant idle time remaining. In contrast, the MAE demonstrates a stronger global search capability and better adaptability to changes in the problem. As a result, when addressing machine selection, MAE is able to identify solutions that outperform those derived from heuristic rules. Moreover, the proposed encoding scheme based on the operation sequence constraint matrix effectively ensures the completeness of the solution space. The Gantt charts indicate that the schedule generated by this scheme has tighter sequencing of operations on each machine, thereby significantly reducing the maximum completion time of the products. Figure 13: Gantt chart of the optimal solution for Mk05A obtained by GA-EAM.