## 4 Simulation Testing and Analysis
### Algorithm Performance Evaluation  
The parameter settings of the MOWPA algorithm are shown in Table 5. To verify the performance of the algorithm, this article aims to establish a multi-objective mathematical model that considers transportation time and energy consumption to minimize the maximum completion time and minimum energy consumption. MOWPA and NSGA-II are used for solving. The Pareto optimal solution set obtained through simulation is shown in Table 8. The data format in the Pareto solution set column in the table is \((x;y)\), where \(x\) represents the maximum completion time and \(y\) represents the total energy consumption of the workshop. From the table, It is evident that the solved range distribution exhibits greater width and uniformity. In order to compare the convergence performance of the two algorithms, this article uses the widely used Coverage (C) [6] and Inverted Generational Distance (_IGD_) [27] in multi-objective optimization problems as evaluation algorithm indicators. Their meanings and formulas are as follows. \[C(F_{1},F_{2})=\frac{|\{sol_{2}\in F_{2}|\exists sol_{1}\in F_{1}:sol_{1} \succ sol_{2}\}|}{|F_{2}|} \tag{16}\]  
where \(F_{1}\) and \(F_{2}\) are the Pareto fronts by the two algorithms, respectively, and \(|F_{2}|\) is the size of \(F_{2}\). The larger \(C(F_{1},F_{2})\) is, the better the surface \(F_{1}\) is. For example, \(C(F_{1},F_{2})=1\) means that all solutions in \(F_{2}\) are dominated by \(F_{1}\), and \(C(F_{1},F_{2})=0\) means that there is no solution in \(F_{1}\) that can dominate \(F_{2}\). \[IGD(F_{1},F*)=\frac{1}{|F*|}\sum_{sol_{1}\in F_{*}}\min_{sol_{2}\in F_{1}}d(sol _{1},sol_{2}) \tag{17}\]  
where, \(F^{*}\) is the non-dominated solution set of the first frontier, \(|F^{*}|\) is the size of \(F^{*}\), \(d(sol_{1},sol_{2})\) representing the Euclidean distance between \(sol_{1}\) and \(sol_{2}\). The smaller the  
\begin{table}
\begin{tabular}{c c c} \hline
**Test Data** & **The Set of Pareto Solutions Obtained by MOWPA** & **The Set of Pareto Solutions Obtained by NSGA-II** \\ \hline MK01 & (46;525.53), (52;516.62), (58;513.43) & (49;546.97), (50;539.37), (51;532.77) \\  & (47;522.11), (51;518), (45;526.91) & (52,524.79), (58;501.11) \\  & (37;526.91), (38;519.49), (39;514.91) & (38;539.11), (39;53  
\(IGD(F_{1},F^{\star})\), the better the \(F_{1}\). In this paper, the \(F^{\star}\) of each example is formed by averaging the non-dominated solution set obtained after each algorithm runs 20 times, respectively. The results are shown in Table 9.  
Table 9 reveals that in the Brandimarte case, the solutions obtained by the MOWPA algorithm dominate the solutions obtained by the NSGA-II algorithm in the majority of cases. Only in the case of MK03 did there exist individual solutions in NSGA-II that are not dominated. However, considering the overall results, it is evident that the MOWPA algorithm outperforms the NSGA-II algorithm. This implies that the Pareto frontier generated by MOWPA is superior to that of NSGA-II. Additionally, based on the IGD index, it can be observed that the IGD value of MOWPA is consistently smaller than that of NSGA-II in most cases. This indicates that the proposed MOWPA algorithm exhibits better convergence performance compared to the NSGA-II algorithm. Figure 9 illustrates the population iteration diagram of the maximum completion time and total energy consumption values obtained using the MOWPA algorithm and the NSGA-II algorithm with MK04 data. In Figure 9a, which displays the population iterations for the maximum completion time, the red solid and dashed lines represent the optimal and average values, respectively, obtained by the MOWPA algorithm. Similarly, the blue solid and dashed lines represent the optimal and average values, respectively, obtained by the NSGA-II algorithm. The MOWPA algorithm achieves a stable optimal maximum completion time after approximately 35 generations, while the NSGA-II algorithm achieves this after around 25 generations.