## 4 Comprehensive Experiments
### Effectiveness of Improvement Strategy  
As shown in Section 3.3, three strategies were adopted for the WOA (Strategy I: Nonlinear convergence factor; Strategy II: Nonlinear adaptive inertia weight; Strategy III: Improve the screw position update model). To analyze the contribution of different strategies to the model, the optimization effects of the IWOA (adopting three strategies), IWOA-I (adopting strategy I), IWOA-II (adopting strategy II), and IWOA-III (adopting strategy III) were compared in five instances. These five instances were the best cases solved by the IWOA in Table 3. Each algorithm was separately run 10 times. The comparison results of the above four algorithms are shown in Table 4.  
According to Equation (29), the larger the value of _APRD_, the more the target value of the algorithm deviates from the optimal value of the objective function, and the worse the performance of the algorithm. Computer results showed that the IWOA with the three strategies had the best performance on the five instances. The _APRD_ of the proposed algorithm was significantly better than that of IWOA-I, IWOA-II and IWOA-III. Meanwhile, its time on MK07 and MK08 was better than that of the comparison algorithm, which showed the effectiveness of the combined application of the three strategies. 1. Effectiveness of nonlinear convergence factor: For comparison, the convergence speed of IWOA-I in the three small-scale instances MK01, MK02, and MK03 was greater than that of IWOA. Furthermore, it can also be calculated that the average running time difference between IWOA-I and IWOA in the five instances was \(-0.05\) (\(22.71-22.76=-0.05\)), and the average convergence speed of IWOA-I was \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Instance**} & \multicolumn{2}{c}{**IWOA**} & \multicolumn{2}{c}{**IWOA-I**} & \multicolumn{2}{c}{**IWOA-II**} & \multicolumn{2}{c}{**IWOA-III**} \\ \cline{2-9}  & _APRD_ & **Time [s]** & _APRD_ & **Time [s]** & _APRD_ & **Time [s]** & _APRD_ & **Time [s]** \\ \hline MK01 & 3.54 & 7.41 & 8.72 & 6.56 & 4.32 & 7.85 & 3.77 & 8.55 \\ MK02 & 6.53 & 13.84 & 7.45 & 8.42 & 6.94 & 15.68 & 7.44 & 17.24 \\ MK04 & 6.27 & 21.45 & 12.62 & 17.28 & 7.81 & 28.74 & 8.15 & 24.31 \\ MK07 & 3.01 & 25.31 & 5.33 & 29.69 & 4.84 & 29.43 & 3.92 & 32.58 \\ MK08 & 2.15 & 45.80 & 14.47 & 51.61 & 6.63 & 51.84 & 4.97 & 55.67 \\
**Average** & 4.30 & 22.76 & 9.72 & 22.71 & 6.12 & 26.708 & 5.65 & 27.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the optimization results of different strategies. ahead of the IWOA. The results show that the nonlinear convergence factor has a very important and positive effect on the convergence speed of the algorithm. This result from the convergence factors is a decrease nonlinearly with the increase in the number of evolutionary iterations, which ensures that whales conduct a global search with a large stride in the early stage of search, and conduct local optimization with a small stride in the later stage to improve the speed of the whole search process while ensuring the quality. 2. Effectiveness of nonlinear adaptive inertia weight: For the \(APRD\) value, the average difference between IWOA-II and IWOA (\(6.12-4.30=1.82\)) was significantly smaller than that between IWOA-I and IWOA (\(9.72-4.30=5.42\)), indicating that the nonlinear adaptive inertia weight has a greater impact on the optimization ability of the IWOA than the nonlinear convergence factor. In addition, for instance MK07 in Table 4, the optimization time of IWOA-II is better than that of IWOA-I, which shows that the nonlinear adaptive weight strategy can also accelerate the convergence speed of the algorithm. 3.