# Application of Artificial Neural Network (ANN) in Predicting Box Compression Strength (BCS)
## 3 Result
### Training ANN Model to Evaluate BCS in the Industry  
With the extracted data of 395 data points covering the majority box dimensions used in the industry and optimized hidden neuron configuration, epoch number, modeling cycle number, a generalized ANN model was trained to predict BCS at an industrial applicable level. The average BCS prediction error of the training and test data across 70 modeling cycles is 10.1% and 10.5%, respectively, as shown in Figure 9. \begin{table}
\begin{tabular}{c c c} \hline
**ANN Architecture Parameters** & **Value/Description** & **Rationale for Selection** \\ \hline \multirow{4}{*}{Input} & ECT & Identified as the most influential parameter affecting BCS \\ \cline{2-3}  & Length & \\ \cline{2-3}  & Width & Included to enable prediction across the majority of commercial box sizes \\ \cline{2-3}  & Depth & \\ \hline Hidden layer number & 1 & Determined via optimization using five different methods and exhaustive search to balance complexity and performance \\ \hline Minimum Epoch number & 50 & Chosen based on error convergence to ensure stable and robust training \\ \hline Modeling cycle number & 70 & training \\ \hline Output & BCS & Target variable to be predicted by the ANN model \\ \hline \end{tabular}
\end{table} 
Table 2: ANN Architecture Parameters and Rationale for Selection. Figure 9: Average BCS error in the training and test data across 70 modeling cycles with real data (395 data points). Through the standard regression metrics (RMSE, MAE, R\({}^{2}\)) analysis, it shows the model achieved good predictive performance, with a training MSE of 8908.33, RMSE of 94.38, MAE of 74.91, and coefficient of determination (R\({}^{2}\)) of 0.896. On the test set, the model maintained high accuracy with an MSE of 11,206.82, RMSE of 105.86, MAE of 82.69, and R\({}^{2}\) of 0.870. Figures 10 and 11 show parity plots comparing predicted and actual BCS values for the training and test data, respectively. The strong linear correlation and close clustering around the diagonal line in both plots indicate the model's effectiveness in capturing the underlying relationships between input features and BCS. Figure 11: Parity plot of predicted BCS and tested BCS across 100 modeling cycles (test data). Figure 10: Parity plot of predicted BCS and tested BCS of one modeling cycle of training data (**left**) and test data (**right**). To assess the robustness and generalizability of the ANN model, 100 modeling cycles were performed using different random splits of the dataset. The parity plot of predicted versus tested BCS values across these 100 cycles (Figure 11) demonstrates consistently strong agreement, with data points closely aligned along the identity line. This indicates stable model performance and reliable predictive capability across varying data partitions. The overall linear fit observed in the plot further confirms that the ANN model effectively captures the relationship between input features and BCS under repeated training and testing conditions. To investigate the causes of the BCS error in the ANN model's predictions, the BCS distribution for two randomly selected modeling cycles was analyzed and plotted, as shown in Figure 12. The actual BCS distribution is represented by the blue columns, while the predicted BCS distributions over 70 modeling cycles are shown in orange. A comparison of the actual and predicted BCS distributions reveals that the model struggles to accurately predict data points with BCS values between 347 lbs and 427 lbs, as well as those between 1999 lbs and 2172 lbs. These values consistently exhibit higher prediction errors compared to other data points. This suggests that boundary data points play a significant role in the high prediction error of the ANN model for BCS evaluation. Boundary points often lie near the edges or outside the range of the training data, resulting in a lack of neighboring data points. While ANNs generally interpolate well within the training range, they tend to struggle with extrapolation, leading to higher errors at the boundaries.