## 2 Optimization Model for Process Parameters of Zinc Smelting Slag Melting Point
### Model Building  
Regression of slag melting point is a complex nonlinear problem. Many machine learning methods can handle nonlinear problems [27], but traditional learning methods require a large amount of data for training to achieve high accuracy [28]. However, for practical factory production, collecting a large amount of qualified data is laborious and time-consuming. During the operation of the smelting furnace, the production parameters of different batches may change, requiring the collection of new data for training to obtain a new regression model. Traditional learning methods entail repetitive and extensive work, making it difficult to adapt to the rapid diagnostic requirements. Therefore, this study selects an improved gradient-boosting algorithm--the CatBoost algorithm. CatBoost is a gradient-boosting algorithm based on symmetric decision trees, offering the advantages of minimal parameter setting, support for categorical feature data, and high classification accuracy [29]. In comparison to similar earlier algorithms such as XGBoost [30] and LightGBM [31], CatBoost incorporates enhanced functionalities. It automatically handles categorical features and adopts fully symmetric trees. It incorporates combination features specifically designed for categorical features and utilizes sorting boosting to counteract noise points in the training set, thus avoiding gradient estimation errors and addressing prediction shift issues to alleviate overfitting. Notably, its method for automatic handling of categorical features involves initially using one-hot encoding. If a maximum number of categorical values for one-hot encoding is set, when this limit is exceeded, CatBoost first conducts a statistical analysis of the categorical features to calculate the frequency of each categorical value, thereby generating new numerical features [32]. Hyperparameter optimization is pivotal in machine learning for enhancing the performance of models. While models come with default hyperparameters, fine-tuning them to a specific dataset can substantially boost their efficacy [33]. In recent years, various methods for hyperparameter optimization have emerged, such as the Tree-structured Parzen Estimator (TPE) for independent parameter sampling [34], grid search for finding the optimal combination through exhaustive search [35], and random search, which reduces computational costs based on grid search [36]. These methods have demonstrated promising performance across different domains. TPE is a Bayesian optimization technique. Unlike grid or random search, which treats hyperparameters as isolated, TPE considers the relationship between hyperparameters and the objective function. The advantage of TPE over other methods lies in its efficiency. Constructing a probabilistic model of the objective function can suggest hyperparameters that are more likely to yield better results, hence reducing the number of trials [37]. In this study, we apply TPE to the popular machine-learning models CatBoost, XGBoost, and LightGBM. TPE adjusts these settings more effectively for specific data, reducing overfitting and enhancing the model's ability to generalize to new data. This ultimately leads to improved overall model performance, as hyperparameters play an important role in how efficiently these algorithms handle the dataset. Tables 1-3 contain the hyperparameters used in the XGBoost, LightGBM, and CatBoost models after applying TPE hyperparameter tuning. Using the relationship model regressed by machine learning algorithms, the iron-silicon ratio, calcium-silicon ratio, coal quantity, oxygen flow, and river sand usage are modeled as independent variables, and the slag melting point is the dependent variable. The mathematical description is as follows:  
\[t_{\rm s}=f(x_{1},x_{2},x_{3},x_{4},x_{5}) \tag{1}\]  
Based on this relationship model, we construct the objective function with the minimum slag melting point as the target.