## 6 Experiments and Results
### Performance Evaluation of the DMOGWO Algorithm
#### 6.4.2 Performance Comparisons for the Dynamic Problem  
This section presents a comparative analysis of the performance of different algorithms for the dynamic process of the DDFJSP-VPT for all instances. As each instance includes a varying number of rescheduling points, this study adopts the methodology outlined in [45]. For each rescheduling point, each algorithm is executed independently 20 times. The non-dominated solutions from these runs are combined, and the resulting non-dominated set is used as the reference Pareto front to calculate the performance metrics of each algorithm at each rescheduling point. The performance metrics across all rescheduling points in each run are then aggregated and averaged, yielding 20 average results that represent the algorithm's performance over the entire dynamic process of the instance. Finally, these 20 averages are further averaged to compute the overall performance values for each algorithm in the corresponding instance. The results are presented in Tables 5 and 6. The experimental results demonstrate that DMOGWO exhibits excellent optimization performance in addressing the DDFJSP-VPT. As shown in Table 5, DMOGWO achieves the highest HV values in 6 out of 10 instances and the lowest IGD values in 8 out of 10 instances, significantly outperforming other algorithms in most cases, with no instances of values that are significantly inferior to others. Furthermore, in terms of the C metric, DMOGWO consistently surpasses SPEA2, MOEA/D, and MOGWO across the majority of instances. Additionally, when compared with MOEA/D and MOGWO, the C values approximate 1 in some instances, indicating that its solution set almost completely dominates these two algorithms. These findings validate DMOGWO's ability to generate Pareto fronts with better convergence and distribution in dynamic scheduling problems. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{4}{c}{**HV**} & \multicolumn{4}{c}{**IGD**} \\ \cline{2-11}  & **DMOGWO** & **NSGA-II** & **SPEA2** & **MOEA/D** & **MOGWO** & **DMOGWO** & **NSGA-II** & **SPEA2** & **MOEA/D** & **MOGWO** \\ \hline MK01 & 0.7738 & 0.7603 (=) & 0.7069 (+) & 0.6521 (+) & **0.7827** (=) & **0.0199** & 0.0281 (=) & 0.0810 (+) & 0.1016 (+) & 0.0294 (=) \\ MK02 & 0.7362 & 0.6903 (+) & 0.6421 (+) & 0.5926 (+) & **0.7433** (=) & 0.0536 & 0.0699 (=) & 0.0767 (+) & 0.1154 (+) & **0.0502** (=) \\ MK03 & 1.2031 & **1.2032** (=) & 0.8313 (+) & 1.0488 (+) & 1.1829 (+) & **0.0114** & 0.0138 (=) & 0.1450 (+) & 0.0713 (+) & 0.0254 (+) \\ MK04 & **0.7616** & 0.7588 (=) & 0.5943 (+) & 0.6655 (+) & 0.7495 (+) & **0.0141** & 0.0153 (=) & 0.0848 (+) & 0.0661 (+) & 0.0310 (+) \\ MK05 & **1.2028** & 1.1762 (+) & 0.9385 (+) & 0.9083 (+) & 1.0734 (+) & **0.0487** & 0.0622 (+) & 0.1579 (+) & 0.1776 (+) & 0.1188 (+) \\ MK06 & 0.8062 & **0.8098** (=) & 0.7011 (+) & 0.6305 (+) & 0.6290 (+) & 0.0449 & **0.0402** (=) & 0.1082 (+) & 0.2068 (+) & 0.2008 (+) \\ MK07 & **1.0272** & 0.9310 (+) & 0.6847 (+) & 0.7477 (+) & 0.9003 (+) & **0.0282** & 0.0399 (+) & 0.1691 (+) & 0.1240 (+) & 0.1036 (+) \\ MK08 & **0.7434** & 0.6699 (+) & 0.5093 (+) & 0.5493 (+) & 0.6314 (+) & **0.0391** & 0.0523 (+) & 0.1373 (+) & 0.1116 (+) & 0.1193 (+) \\ MK09 & **1.1828** & 1.1618 (+) & 0.8553 (+) & 1.2061 (+) & 1.0592 (+) & **0.0249** & 0.0362 (+) & 0.2166 (+) & 0.1188 (+) & 0.1223 (+) \\ MK10 & **0.9470** & 0.9171 (+) & 0.7176 (+) & 0.7122 (+) & 0.8573 (+) & **0.0966** & 0.1132 (+) & 0.1949 (+) & 0.2087 (+) & 0.1508 (+) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison between DMOGWO and other algorithms using HV and IGD values. To provide a more intuitive comparison of the algorithm's performance, we introduce the performance score metric [46]. The performance score for each algorithm is determined by aggregating the results from all rescheduling points across the instances (33 rescheduling points in total) and counting the number of instances at each rescheduling point where the algorithm is significantly worse than the others.