## 4 Experiment and Results
### Evaluation Indicators Analysis  
To evaluate the convergence, diversity, and uniformity of the algorithm, IGD, HV, and Spacing metrics are employed. Recognizing the stochastic nature inherent in optimization algorithms, each algorithm was executed independently 30 times for the dephosphorylation process parameter optimization problem. Subsequently, the mean and standard deviation of each algorithm's solution set indices were calculated, and the results of these evaluation metrics are presented in Table 7.  
Figure 5: Each algorithm’s Pareto feasible solution. (**a**) MOPSO; (**b**) MOGWO; (**c**) MOBWO; (**d**) MOCOA; (**e**) MOWOA. Following extensive empirical evaluations, the mean IGD of the IMOWOA is approximately 0.009, which is close to zero, The mean HV attained is around 7.80, while the mean Spacing value stands at about 0.033. In terms of the IDG metric, although the IMOWOA slightly lags behind the 0.007 of the MOGWO algorithm, it is noteworthy that the IGD produced by the IMOWOA is relatively low, with a standard deviation of 0.0015. The statistically consistent convergence patterns across multiple independent runs demonstrate that the algorithm maintains stable convergence rates without succumbing to local optima entrapment caused by stochastic factors. Comparative analysis of the HV metric reveals that IMOWOA achieves broader coverage of the Pareto optimal front compared to benchmark algorithms, indicating superior solution diversity. Although IMOWOA exhibits a slightly less uniform distribution (Spacing metric) relative to MOPSO, its near-zero Spacing values (approximating 0.033) still confirm advantageous solution spread characteristics. Experimental results collectively demonstrate that IMOWOA's lower IGD values, higher HV values, and near-zero Spacing metrics reflect dual strengths in computational efficiency and optimization accuracy. These findings validate the algorithm's robust performance across convergence stability, solution quality, and distribution uniformity, thereby confirming the efficacy of its architectural design. Consequently, the IMOWOA shows significant potential and effectiveness when applied to the optimization of dephosphorylation process parameters. The experimental results indicate that the solutions derived from the IMOWOA are more effective in addressing the optimization problem associated with dephosphorylation process parameters in converter steelmaking. This algorithm facilitates a more favorable tradeoff between the endpoint phosphorus content and production cost per tonne of steel, leading to improved steel quality with only a marginal increase in the production cost. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **MOPSO** & **MOGWO** & **MOBWO** & **MOCOA** & **MOWOA** \\ \hline Ferric C (\%) & 4.53 & 4.86 & 4.52 & 4.67 & 4.73 \\ Ferric Si (\%) & 0.44 & 0.18 & 0.24 & 0.25 & 0.30 \\ Ferric P (\%) & 0.112 & 0.115 & 0.072 & 0.101 & 0.131 \\ Temperature of iron (T) & 1371 & 1276 & 1265 & 1370 & 1316 \\ Amount of iron (t) & 116.2 & 107.2 & 114.0 & 111.0 & 107.0 \\ Scrap quantity (t) & 12.86 & 39.51 & 13.85 & 19.61 & 39.52 \\ Lime (kg) & 2944 & 1271 & 916 & 2328 & 1323 \\ Limestone (kg) & 4546 & 1864 & 5549 & 1625 & 4710 \\ Raw dolomite (kg) & 2216 & 943 & 3292 & 2787 & 3737 \\ Ore (kg) & 892 & 2664 & 3228 & 2005 & 759 \\ Oxygen blowing quantity (m\({}^{3}\)) & 6694 & 6679 & 6271 & 6465 & 6799 \\ First pour temperature (\({}^{\circ}\)C) & 1624 & 1618 & 1615 & 1637 & 1634 \\ Endpoint P (\%) & 0.0137 & 0.0128 & 0.0136 & 0.0132 & 0.0134 \\ Production cost (CNY/t) & 3510.56 & 3213.36 & 3505.53 & 3435.19 & 3205.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Each algorithm’s Pareto optimal solutions.