## 4 Simulation Test
### Algorithm Performance Testing  
This paper investigated a multi-objective, no-idle HFSP while considering setup time and handling time, and the order type was a multi-variety, small-lot MTO order. Since there was no benchmark related to the research problem of this paper, the benchmark proposed by [PERSON] was adapted. The adaptations were to add setup time and handling time in the ranges of 1-10 and 10-30, to classify the workpiece types, and to change the original single-equipment processing to multiple-equipment processing. The adapted examples are \(j10k3i3\), \(j20k4i4\), \(j30k5i6\), \(40k6i5\), and \(j50k7i5\), where \(j\) is the total number of workpieces, \(k\) is the number of workpieces, and \(i\) is the number of work centers. To evaluate the performance of each strategy proposed in this paper further, the NSGA-II, NSGA-II-V, and NSGA-II-1 were used to compare the performance improvement. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Example**} & \multicolumn{2}{c}{**NSGA-II-V**} & \multicolumn{2}{c}{**DSOA**} & \multicolumn{2}{c}{**DFWA-LS**} & \multicolumn{2}{c}{**IWO**} \\  & **ARPD** & **SD** & **ARPD** & **SD** & **ARPD** & **SD** & **ARPD** & **SD** \\ \hline T01 & 0.05 & 0 & 0.05 & 0.008 & 0.15 & 0.013 & 0.92 & 0.083 \\ T11 & 0.21 & 0 & 0.21 & 0.034 & 0.62 & 0.06 & 2.37 & 0.156 \\ T21 & 0.22 & 0.065 & 0.21 & 0.057 & 0.51 & 0.093 & 2.33 & 0.272 \\ T31 & 0 & 0 & 0 & 0 & 0.01 & 0.004 & 0.49 & 0.103 \\ T41 & 0.3 & 0.073 & 0.29 & 0.048 & 0.52 & 0.085 & 2.76 & 0.311 \\ Average & 0.158 & 0.028 & 0.152 & 0.029 & 0.362 & 0.051 & 1.774 & 0.185 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the results of the algorithms. Among them, the NSGA-II-V was the algorithm designed in this paper. The NSGA-II-1 has the same genetic part as the traditional NSGA-II, and the rest conditions are consistent with the NSGA-II-V. Each case was simulated 10 times, and the best results were compared. Since there is no standard algorithm for the problem studied in this paper, it was not possible to evaluate the performance of the multi-objective optimization algorithm using, for example, generational distance, inverse generational distance, or the space metric. Therefore, this section uses the Pareto front surface for comparison, as shown in Figure 7.  
Due to the small number of j10k3i3 artifacts, the NSGA-II-V had the same Pareto front surface as that obtained by the NSGA-II-1, while the NSGA found only two non-dominated solutions. Therefore, no graphical analysis was performed for j10k3i3. In Figure 7, the dots represent the non-dominated solutions, and the curves are used to connect the dots for easy viewing. In the figure, the Pareto frontier formed by the non-dominated solutions found by the conventional NSGA-II was inferior to that of the NSGA-II-V and NSGA-II-1, which illustrates that the crossover, variation, selection, and variable neighborhood search strategies proposed in this paper can effectively improve the superiority-seeking accuracy of the NSGA-II. The comparison between the NSGA-II-V and the NSGA-II-1 showed that the overall Pareto front surface of the NSGA-II-1 was inferior to that of the NSGA-II-V. This is because there may be better solutions than the current one in the neighborhood of a sure non-dominated solution, and the variable neighborhood search operation can find these solutions. This also proved that the variable neighborhood search strategy designed in this paper was effective. Figure 8 is a Gantt chart of one of the non-dominated solutions of j40k6i5, from which it can be seen that the number of devices in the five work centers was three, two, four, five, and two in order, and each device was unrelated. The book numbers represent the workpiece numbers, where workpieces 1-8 were type 1, 9-13 were type 2, 14-20 were type 3, 21-30 were type 4, 30-34 were type 5, and 35-40 were type 6.