# Evaluating Packaging Design Relative Feature Importance Using an Artificial Neural Network (ANN)
###### Abstract
In recent years, ANNs have gained growing interest in various engineering and multidisciplinary research fields, such as the tourism industry [11], financial sector [12], and complex engineering applications [13], as well as the optimization of sustainable construction material and technique [14,15]. However, in the field of packaging, artificial neural networks (ANNs) have been applied in only a few areas, such as food packaging's influence on food shelf life [16, 17, 18, 19], estimating edge crush resistance and evaluating other packaging properties [20,21], with limited applications beyond these. In principle, training an ANN model involves building a function that recognizes the underlying relationships between input variables and output variables. Several methods have been employed within ANN models to determine the relative importance of input variables. One commonly used method is the weight connections method [22]. In this method, the importance of input variables is determined by the connection strengths (weights) within a trained neural network [23]. In 1991, [PERSON] et al. [23,24] proposed a method to determine the relative impact of each input variable by calculating the percentage of output weight values associated with the contribution of a single input across the entire network. This method was utilized by [PERSON] et al. [22] to evaluate the contribution of inputs to outputs in trained neural networks for both classification and regression problems. Another widely used method is the gradient-based method [25]. [PERSON] et al. [25] proposed a gradient-based method to identify the relative importance of influencing factors for robotic control by obtaining the gradient of the output with respect to any component of the neural network using the chain rule in 2020. [PERSON] al. [26] conducted dimension reduction for both feature extraction and variable selection based on the gradient-based method in supervised learning. In addition, the sensitivity analysis [27] and SHAP (Shapley additive explanations) values [28] are also commonly applied. One of the most commonly used techniques within the sensitivity analysis is the permutation method. [PERSON] et al. [29] used the permutation method to measure the sensitivity of a model with the presence or absence of a feature to determine the importance scores of input features in fluid dynamics using a neural network-based turbulence model. [PERSON] et al. [30] estimated the distribution of measured importance for each variable in a non-informative setting based on the permutation method in random forest (RF) models. Regarding SHAP values, [PERSON] et al. [28] presented a unified framework for interpreting predictions, assigning each feature an importance value for a particular prediction in a deep learning model. [PERSON] [31] applied SHAP values to extract spatial effects for interpreting and visualizing complex geographical phenomena and processes in machine learning models. Given ANNs' effectiveness in evaluating input feature importance across various fields and the limitations of traditional methods in assessing packaging design features, ANNs show strong potential for application in evaluating the importance of input features in packaging design to support cost savings and minimize material usage. Therefore, this study explores the application of ANN models to assess the relative importance of packaging design features, providing valuable insights that can be directly applied to optimize the packaging design process. In this study, box compression strength (BCS) was used as an example packaging property, and the relative importance of up to six packaging design features was assessed using ANN-based methods. Two datasets were utilized: one synthetic dataset generated using a widely used mathematical model (the McKee formula) and one real dataset [32]. These datasets were used to train artificial neural networks (ANNs) to assess packaging design feature importance.