## 4 Experiment and Results
### Pareto Solution Set Analysis  
In the context of the dephosphorylation optimization problem, it is imperative to concurrently optimize two objectives to attain outcomes. Typically, the optimal solution is represented by a set, known as the Pareto optimal solution set, which characterizes the trade-offs between the competing objectives. Table 2 delineates the specific parameter configurations within the IMOWOA.All Pareto optimal solutions (Table 3) were validated against historical production data, confirming that the proposed parameter adjustments (e.g., increased limestone substitution) adhere to known physicochemical limits (e.g., slag basicity requirements) while achieving measurable quality-cost improvements. The IMOWOA was employed to address the multi-objective optimization model concerning the production process parameters of Q235B-Z steel grades, obtaining Pareto feasible solutions, as depicted in Figure 4. In this multi-objective optimization framework, the dual objectives are articulated as minimizing the endpoint phosphorus content and minimizing the production cost per tonne of steel. For a comprehensive analysis, the three most optimal solutions from further examination. \begin{table}
\begin{tabular}{c c} \hline
**Type** & **Parameter** \\ \hline Population size & 50 \\ Data dimension & 12 \\ Number of iterations & 800 \\ Solenoidal morphology constant & 1 \\ Alpha & 0.2 \\ Beta & 0.8 \\ \hline \end{tabular}
\end{table} 
Table 2: IMOWOA algorithm parameter settings. \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter** & **Practicality** & **1** & **2** & **3** \\ \hline Ferric C (\%) & 4.67 & 4.47 & 4.48 & 4.48 \\ Ferric Si (\%) & 0.54 & 0.57 & 0.57 & 0.57 \\ Ferric P (\%) & 0.101 & 0.103 & 0.103 & 0.103 \\ Temperature of iron (T) & 1340 & 1302 & 1300 & 1301 \\ Amount of iron (t) & 118.7 & 111.7 & 111.3 & 111.4 \\ Scrap quantity (t) & 21.90 & 38.91 & 38.90 & 38.93 \\ Lime (kg) & 2000 & 2246 & 2248 & 2243 \\ Limestone (kg) & 4000 & 4551 & 4609 & 4594 \\ Raw dolomite (kg) & 1500 & 1812 & 1807 & 1809 \\ Ore (kg) & 1000 & 806 & 792 & 791 \\ Oxygen blowing quantity (m\({}^{3}\)) & 6397 & 6394 & 6405 & 6399 \\ First pour temperature (\({}^{\circ}\)C) & 1641 & 1636 & 1634 & 1634 \\ Endpoint P (\%) & 0.0120 & 0.0107 & 0.0108 & 0.0109 \\ Production cost (CNY/t) & 3414.68 & 3217.03 & 3216.65 & 3216.52 \\ \hline \end{tabular}
\end{table} 
Table 3: Pareto optimal solution set. The Pareto feasible solution graph presented in Figure 4 illustrates that the IMOWOA effectively achieves a favorable trade-off between the dual objectives of minimizing endpoint phosphorous content and production cost per tonne of steel. The presence of multiple solutions in the feasible solution set makes the objectives of \(\mathrm{f_{1}}\) and \(\mathrm{f_{2}}\) research a better solution, reflecting the algorithm's efficacy in navigating the iterative optimization process. In the optimal solution set displayed in Table 3, the endpoint phosphorus content of the first optimized solutions is recorded as 0.0107%, alongside the consumed production cost of 3217.03 CNY per tonne of steel. Compared with the actual production data with analogous processing parameters, the endpoint phosphorus content and the production cost of a tonne of steel have been reduced by 10.8% and 5.79%, respectively. The third optimized solution yields an endpoint phosphorous content of 0.0109%, and a steel production cost of 3216.52 CNY per tonne of steel. Compared with the actual production data, two objectives are reduced by 9.17% and 5.80%, respectively. To evaluate the stability of the IMOWOA algorithm, the Pareto optimal solution set delineated in Table 3 underwent further verification. The first optimal objective function was selected as the initial solution, and the same parameter configurations were applied to conduct multiple experimental runs. The optimal endpoint phosphorus content and production cost per tonne of steel attained in each trial were systematically recorded to assess the robustness and stability of the algorithm.