## 4 Experiment and Results
### Modeling Effects Analysis  
The complex nonlinear relationship between process parameters and endpoint phosphorus content originates from fundamental metallurgical mechanisms: the temperature-dependent competitive reaction mechanism where lower temperatures promote dephosphorylation through CaO-P\({}_{2}\)O\({}_{5}\) formation while simultaneously reducing slag fluidity and reaction kinetics; the threshold effect of lime dissolution demonstrating nonlinear saturation characteristics when addition exceeds 80 kg/ton; and the time-dependent reaction kinetics exhibiting distinct inflection points during different blowing stages. These physicochemical complexities manifest mathematically as strong interaction effects and non-Gaussian residual distributions in linear models. The extremely random forest (ERT) [18] algorithm demonstrates superior capability in capturing these nonlinear relationships through three principal mechanisms: the ensemble of randomized decision trees intrinsically models multiplicative parameter interactions; the automatic feature selection weights reveal dominant nonlinear parameters; and the algorithm's inherent robustness to collinear inputs effectively handles the strong correlation between slag-forming materials and furnace temperature. Consequently, the ERT algorithm was employed to construct a regression model for predicting the endpoint phosphorus content. To evaluate the predictive performance of this model, a random selection of 600 sets of sample data was utilized as test samples for experimentation, yielding the prediction results depicted in Figure 3. The results indicate that the fluctuations between the model-predicted values and the actual values exhibit a similar trend.  
The root mean square error (RMSE) of the prediction model for endpoint phosphorus content was measured at 0.09, with an R\({}^{2}\) value of 0.79. Notably, the proportion of furnaces exhibiting a prediction error of endpoint phosphorus content in the range of \(-0.001\%\) to \(0.001\%\) constituted 72.30% of the total furnaces, while those with a prediction error ranging from \(-0.003\%\) to \(0.003\%\) accounted for 85.05% of the total furnaces. The results confirm that the model demonstrates a relatively low average prediction error and high precision, indicating its efficacy and reliability in predicting the endpoint phosphorus content.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline Ferric C (\%) & 4.68 & 4.62 & 4.59 & 4.66 & 4.91 & 4.95 & 4.72 & 4.80 \\ Ferric Si (\%) & 0.32 & 0.25 & 0.30 & 0.31 & 0.39 & 0.32 & 0.41 & 0.39 \\ Ferric P (\%) & 0.112 & 0.120 & 0.122 & 0.125 & 0.114 & 0.113 & 0.123 & 0.122 \\ Temperature of iron (T) & 1344 & 1330 & 1330 & 1280 & 1301 & 1320 & 1320 & 1300 \\ Amount of iron (t) & 116.00 & 116.90 & 113.70 & 110.05 & 109.10 & 109.00 & 112.30 & 108.80 \\ Scrap quantity (t) & 15.90 & 19.95 & 20.40 & 21.00 & 23.00 & 19.00 & 21.25 & 19.29 \\ Lime (kg) & 3200 & 1200 & 1700 & 1500 & 1800 & 1100 & 2000 & 1700 \\ Limestone (kg) & 600 & 3000 & 4100 & 1800 & 2200 & 3900 & 2700 & 3700 \\ Raw dolomite (kg) & 1000 & 1000 & 1500 & 2000 & 1600 & 1000 & 2600 & 1000 \\ Ore (kg) & 800 & 1500 & 1100 & 3400 & 3600 & 1300 & 700 & 1900 \\ Oxygen blowing quantity (m\({}^{3}\)) & 5566 & 5965 & 6242 & 5132 & 5326 & 6197 & 5755 & 5808 \\ First pour temperature (\({}^{\circ}\)C) & 1682 & 1646 & 1674 & 1640 & 1667 & 1680 & 1680 & 1644 \\ Endpoint P (\%) & 0.019 & 0.015 & 0.024 & 0.018 & 0.021 & 0.020 & 0.022 & 0.010 \\ Production cost (CNY/t) & 3740.70 & 3429.21 & 3420.80 & 3420.58 & 3401.90 & 3426.18 & 3407.73 & 3428.74 \\ \hline \end{tabular} Note: The table presents eight representative samples selected from the complete dataset comprising 4000 heats, with endpoint phosphorus content (Endpoint P) and production cost per ton of steel established as the dual optimization objectives.  
\end{table}
Table 1: Partial sample data.