## 4 Comprehensive Experiments
### Benchmark Test  
In this paper, 10 FJSP instances in the standard test set of Brandimarte [44] were selected for the simulation experiments. Different examples were independently run 20 times. Compared with [PERSON]'s IWOA(1) [41], [PERSON]'s IWOA(2) [42], [PERSON]'s WOA-LEDE [43], and the non-dominant sorting genetic algorithm (NSGA-II) commonly used for shop floor scheduling solutions, this paper carried out a comparative analysis. Among them, the first three algorithms were applied to shop scheduling problems and showed excellent performance. These algorithms are highly similar to the algorithm proposed in this paper, and they are all improved algorithms based on the whale optimization algorithm. Therefore, compared with the above algorithms, it can fully prove whether the improvement strategy proposed by the algorithm in this paper is effective. On the other hand, the corresponding literature has compared the above three optimization whale algorithms with some excellent algorithms such as KBACO, TSPCB, HGWO, etc. [39; 40; 41]. Experimental results have shown that the above three improved algorithms can outperform their comparison algorithms in most optimization problems. Therefore, the performance of the proposed algorithm can be effectively verified by comparing it with the above three algorithms of the same type. In addition, NSGA-II is a classical algorithm based on genetic algorithm, which is also a popular algorithm at present. It has been applied to solve various optimization problems [45; 46], so this algorithm was also used as the comparison algorithm in this paper. The parameters of these algorithms are shown in Table 2. The iteration number \(t_{\text{max}}\) of the above algorithm was 200, and the population size \(N\) was 100. The comparison results of the benchmark examples are shown in Table 3. In the table, Best represents the best result in 20 tests, SD refers to the standard deviation of the results of 20 tests, and LB is the best value found so far. Figure 5 shows the \(APRD\) value obtained by IWOA(1), IWOA(2), WOA-LEDE, NSGA-II, and IWOA by solving 10 benchmark instances. \(APRD\) represents the average percentage deviation between 20 experimental results and LB, and the calculation formula is as follows:  
\[APRD=\frac{1}{n}\sum_{i=1}^{n}\frac{C_{i}-LB}{LB}\times 100 \tag{29}\]  
where \(C_{i}\) is the makespan obtained by the algorithm. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \multirow{2}{*}{**Instance**} & \multirow{2}{*}{\(\mathbf{O}_{ij}\times\mathbf{m}_{k}\)} & \multirow{2}{*}{**LB**} & \multicolumn{2}{c}{**IWOA(1)**} & \multicolumn{2}{c}{**IWOA(2)**} & \multicolumn{2}{c}{**WOA-LEDE**} & \multicolumn{2}{c}{**NSGA-II**} & \multicolumn{2}{c}{**IWOA**} \\ \cline{3-13}  & & & **Best** & **SD** & **Best** & **SD** & **Best** & **SD** & **Best** & **SD** & **Best** & **SD** \\ \hline MK01 & 10 \(\times\) 6 & 39 & 40 & 0.60 & 41 & 0.85 & 39 & 0.51 & 41 & 0.7 & 39 & 0.50 \\ MK02 & 10 \(\times\) 6 & 26 & 27 & 0.65 & 26 & 0.79 & 27 & 0.75 & 29 & 1.17 & 26 & 0.74 \\ MK03 & 15 \(\times\) 8 & 204 & 204 & 3.59 & 207 & 2.53 & 204 & 3.36 & 207 & 3.39 & 207 & 3.18 \\ MK04 & 15 \(\times\) 8 & 60 & 65 & 3.25 & 60 & 4.81 & 62 & 3.02 & 67 & 4.34 & 60 & 3.59 \\ MK05 & 15 \(\times\) 4 & 172 & 175 & 4.01 & 179 & 4.54 & 172 & 3.87 & 188 & 5.84 & 175 & 3.55 \\ MK06 & 10 \(\times\) 15 & 58 & 63 & 3.16 & 63 & 4.87 & 61 & 2.01 & 67 & 5.59 & 61 & 1.62 \\ MK07 & 20 \(\times\) 5 & 139 & 139 & 4.03 & 140 & 4.3 & 139 & 3.86 & 142 & 4.58 & 139 & 4.16 \\ MK08 & 20 \(\times\) 10 & 523 & 523 & 3.56 & 523 & 4.46 & 523 & 3.11 & 523 & 4.12 & 523 & 3.46 \\ MK09 & 20 \(\times\) 10 & 307 & 307 & 3.68 & 307 & 4.01 & 312 & 3.55 & 317 & 5.86 & 309 & 3.27 \\ MK10 & 20 \(\times\) 15 & 198 & 214 & 4.37 & 206 & 5.17 & 206 & 4.26 & 208 & 6.63 & 205 & 3.64 \\ \hline \end{tabular}
\end{table} 
Table 3: Comparison results of the Brandimarte benchmark examples. Figure 5: Histogram of the \(APRD\) values obtained by five algorithms.