# Evaluating Packaging Design Relative Feature Importance Using an Artificial Neural Network (ANN)
## 3 Results
### Case Study 2: Relative Feature Importance of the Real Dataset
#### 3.2.2 Relative Feature Importance Analysis of the Real Dataset  
Given the unreliability of the connection weights method for assessing BCS feature importance, particularly for the depth feature, it was excluded from the analysis of the real dataset. Instead, the BCS feature importance for the real data were evaluated using the remaining three methods. Following the same procedure applied to the synthetic dataset, the BCS feature importances were normalized across these methods to ensure that the sum of the five BCS feature importances equaled 1 for each method, as illustrated in Figure 13.  
Figure 11: Optimal neuron numbers in the hidden layer(s) determined by different methods for the real dataset. Figure 12: Model mean squared error (MSE) versus epoch plot with 34 neurons in the hidden layer. The results from the three methods demonstrate overall consistency and have been averaged to establish a comprehensive ranking of the five BCS feature importances. The average BCS feature importance is ranked as ECT \(>\) perimeter \(>\) EI\({}_{y}\) \(>\) EI\({}_{x}\) \(>\) depth, which aligns well with the theoretical BCS feature importance calculated using the mathematical model, as shown in Figure 14. The average BCS feature importance from these three methods shows that ECT has a weight of 0.480, perimeter has a weight of 0.235, EI\({}_{y}\) has a weight of 0.116, EI\({}_{x}\) 0.099, and depth has a weight of 0.070. The analysis of the real dataset reveals that, although depth is ranked last, it still has a notable influence. In general, depth remains an important factor in determining BCS (buckling compression strength). As the depth value increases, buckling theory suggests that depth can significantly affect compression strength, making it a critical consideration. Despite the McKee equation [20] theoretically assigns a zero-importance value to depth, the real-world data demonstrate that the depth's effect should not be disregarded. Since the real dataset is derived from real-world scenarios, it naturally reflects various real-world influencing factors, including the buckling of the corrugated board under compression. Therefore, it is expected that the importance of depth is not zero. Our results emphasize the influence of depth, validating that the proposed methods for measuring feature importance accurately reflect practical applications. Furthermore, although the theoretical BCS feature importance ranks EI\({}_{x}\) EI\({}_{y}\) equally, the average importance ranking from the three methods shows a 0.017 difference between these two features. This small variation is understandable considering the fluctuations of measurement inaccuracies that mathematical models may not fully account for. In summary, the BCS feature importance ranking for the real dataset is consistent with the findings from the theoretical feature importance. The evaluation of the real dataset provides a real-world context to the findings and further demonstrates the capability of ANN approach in terms of feature importance evaluation for packaging design. Based on the results of this study, there are practical implications for improving the packaging design process. On the one hand, packaging designers seeking to significantly enhance a package's BCS can prioritize modifying key features with the greatest impact, such as the ECT of the corrugated board. A relatively small increase in ECT can enhance BCS more effectively than changes to other features, minimizing design effort. Understanding the relative importance of different factors helps designers avoid modifications with minimal impact, such as increasing the box's depth, which has the lowest effect on BCS  
Figure 13: ANN-evaluated BCS feature importance of the real dataset. compared to features like ECT, perimeter, thickness, and flexural stiffness.