## 2 Materials and Methods
### Creation of a Proactive System Based on Predictions
At this moment, the mold \(V_{j}\) has the set of variables \(PC_{i},SC_{i},M_{i},I_{i}\) that can now be sent to the machine learning models, which will attempt to predict the state the part will reach when the final inspection is carried out. In the same way that the system manages the events, the predictions will also be triggered at every grouping step illustrated in Figure 5.  
In order to develop the aforementioned predictions and considering that we have extracted all the manufacturing process results, as well as the recorded results in both prediction objectives, the best approach is the employment of supervised learning. Specifically, this is a type of machine learning where models are trained using labeled data. The labeled data include the corresponding correct output. This learning is performed using statistical classifiers to categorize data based on patterns in the labeled examples. These classifiers learn from the labeled data to make predictions on new and unseen data. The process relies on statistical methods to model the relationship between input features and target labels. To make future predictions and to properly evaluate the machine learning models for the prediction of both aforementioned objectives, we applied the following methodology. * Cross validation: In order to obtain a proper representation of the data, we must use as much available information as possible. For this purpose, K-fold cross validation is usually used in machine learning experiments [43]. In our experiments, we performed a K-fold cross validation with \(k=10\). In this approach, our dataset was split into 10 sets of learning (66% of the total dataset) and testing (34% of the total data). * Teaching the model: For each fold, we performed the learning phase of each algorithm using the corresponding training dataset, applying different parameters or learning algorithms depending on the model. Due to the lack of knowledge about the perfor  
Figure 5: Investment casting process division and grouping for predictions and optimizations. First group, with the slurry coating work (\(PC\)). Then, as second group, the combination of both coating stages, slurry, and stucco (\(PC+SC\)). Finally, the complete group including previous coating stages and the pouring (\(PC+SC+M\)). mance of this type of prediction in the investment casting process and considering that this research is in an initial stage, we decided to use classical machine learning models to validate whether the methodology could be useful. More accurately, we used the following models: * Classical statistical classifiers: This type of classifier has been widely used in machine learning due to its simplicity, interpretability, and effectiveness with relatively small datasets [44]. One of the key advantages of these classifiers is that they are computationally efficient and easy to implement. Although newer machine learning models, such as deep learning approaches, have gained popularity, classical statistical classifiers remain valuable for their reliability and ease of use in many practical applications [45; 46; 47; 48; 49]. Specifically, we conducted our research by evaluating the following classical algorithms: * Bayesian networks: For Bayesian networks, we used different structural learning algorithms, including K2 [50], hill climber [51], and Tree Augmented Naive (TAN) [52]. Moreover, we also performed experiments with a naive Bayes classifier. * K-nearest neighbor: For K-nearest neighbor [53], we performed experiments with \(k=1\), \(k=2\), \(k=3\), \(k=4\), \(k=5\), and \(k=6\). * Artificial neural networks: We used a three-layer Multilayer Perceptron (MLP) [54] taught using a back-propagation algorithm. * Support vector machines: We performed our experiments with a polynomial kernel [55], a normalized polynomial kernel [56], a Pearson VII function-based universal kernel [57], and a radial basis function (RBF)-based kernel [58].