## 2 Materials and Methods
### Development and Cross Validation of a Gaussian Process Model to Predict Unit Load Segment Bending  
From the results of the screening analysis of the unit load bending model, the main significant factors were identified. In order to develop a model that could accurately predict the bending behavior of the boards supporting the payload and the influence of different payload characteristics, each variable was limited to specific ranges, as follows. The number of columns, which relate to the box sizes, was limited to 3, 4, 5, and 6 columns \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multirow{2}{*}{**Name**} & \multicolumn{2}{c}{**Levels**} \\ \cline{3-4}  & & **Low (–)** & **High (+)** \\ \hline \(x_{1}\) & Payload Weight & 109 kg & 218 kg \\ \(x_{2}\) & Unit Load Columns & 3 & 5 \\ \(x_{3}\) & Unit Load Layers & 3 & 5 \\ \(x_{4}\) & Pallet Friction & 0.20 & 0.60 \\ \(x_{5}\) & Box Friction & 0.20 & 0.60 \\ \(x_{6}\) & Pallet Stiffness & 2.76 GPa & 8.27 GPa \\ \(x_{7}\) & Payload Height & 0.762 m & 1.778 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Factor level combinations for the fractional factorial \(2^{7-1}_{IV}\) design. of stacked boxes along the length of the board. Larger boxes were not part of the validated scope of the model, and smaller boxes, having seven or more columns, were considered too small to be commonly handled on unit loads without any additional shippers. The height of the boxes, in the model referenced as the overall height of the payload, was limited from relatively short unit loads of 508 mm to tall unit loads of 2.03 m. Very short unit loads, or abnormally tall ones, were not included in the model scope. It is important to note that the number of layers, or the number of divisions that the unit load has along its height, was not considered a significant factor within these ranges. Even though pallet stiffness was not considered a significant factor, this was validated only within the ranges studied, from 2.76 GPa to 8.27 GPa. These values are equivalent to a multiple use plastic pallet and a wooden pallet with characteristics similar to a standard pool pallet [15]. Stiffer pallets using different materials, such as metal, might present different behavior, as well as very lightweight plastic pallets; therefore, these were not included in the model. The payload weight range for the unit load segment modeled was the same as for the screening design study, from 109 kg to 218 kg. When comparing the payload weight for a full unit load on a standard GMA pallet (1219 mm \(\times\) 1016 mm), this range correlates to 523 kg to 1045 kg, using the same pressure level. Although this model's accuracy is expected to be acceptable, even around the limits, moving further away from this range could drastically reduce the prediction's accuracy. The coefficient of friction (CoF) ranges for the different contact interactions were limited based on the commonly identified ranges for each material. The CoFs of the pallet's top deck's contact with corrugated fiberboard were based on the findings by [PERSON] et al. [28], where multiple pallet decks were evaluated for friction characteristics. Plastic pallets with little to no surface treatment presented coefficients around 0.30, while wooden pallets had values under 0.60. To account for variability, the model input was limited to between 0.20 and 0.70. Higher frictions or the use of lips at the edges of the pallets might affect the movement of the components and therefore were not considered. Contacts between corrugated boxes were shown to be between 0.50 and 0.60 [20]. To account for the possibilities of different packaging materials, the model range for the coefficient of friction between the box contacts was between 0.20 and 0.70. Using material treatments that significantly increase friction, such as tie sheets, could generate different interactions between the components.