# Application of Machine Learning Methods for Pallet Loading Problem
## 3 Methodology
### Number of the Boxes Loaded onto the Pallet
#### 3.1.2 Phase II: Computing the Number of Horizontal Layers on the Basis of the \(H_{max}\)  
In the second phase, the total number of boxes per pallet and the number of the horizontal layers per pallet are determined considering three parameters: (a) the total height of the horizontal layers should not exceed the \(H_{max}\); (b) the total weight on the pallet should not be more than \(M_{max}\); and (c) the average load on the box at the bottom horizontal layer of the pallet should be less than its dynamic compression strength, considering humidity, interlock stacking pattern, dimensions of the box, and storage time. The \(H_{max}\) is determining either on the basis of storage on pallet within the warehouse, i.e., racks, floor, etc., or on the basis of transportation constraints, i.e., trailer internal height, number of horizontal layers loaded, etc. In this case, the height of the total number of the horizontal layers should be less than the \(H_{max}\) since more than one horizontal layer can be used per pallet (Equation (1)). This paper assumes that all the layers have the same pattern, where overlap between layers is not allowed; therefore, all the horizontal layers are identical and have the same height, as depicted in Figure 5.  
\[NHL_{max}=\left\lfloor\frac{H_{max}}{BDPB}\right\rfloor \tag{1}\]  
_NHL:_ The maximum number of horizontal layers  
_BDPB:_ Box's dimension perpendicular to the base  
Furthermore, the total weight on the pallet should not be more than \(M_{max}\). This is important for several reasons. First, if there is no height limit, the number of boxes per pallet will be determined based on the \(M_{max}\) per pallet. Second, the total weight of the boxes per pallet when loading on the shelves or the trailers should be less than the maximum weight capacity to avoid damage to the pallets, the shelves, and the trailers.  
Compression strength is another parameter that is used to define the number of horizontal layers per pallet to maximize the total number of boxes loaded. Compression strength represents the capacity of a material or structure to withstand loads without any deterioration. For a pallet, compression strength is the maximum weight that the boxes in  
Figure 5: Pallet Loading Problem with identical horizontal layers based on \(H_{max}\).  
the bottom layer can support without any damage. In this section, pallet dynamic compression strength is calculated by modifying the McKee formula by adding considerations for humidity, storage time, interlock stacking pattern, and the pallet shape coefficients. The McKee formula is usually used to determine the static compression strength, which is a theoretical value under ideal laboratory conditions, with the temperature and humidity controlled within 73 \(\pm\) 2 \({}^{\circ}\)F and 50% \(\pm\) 2% RH, respectively [36]. On the other hand, under real conditions, dynamic compression strength is the actual compression strength of a box. Equations (2) and (3) show the static compression strength and dynamic compression strength. \(ECT\) is the Carton Edge Crush Test, which measures the ability of a corrugated board to sustain a top-to-bottom load [37]. \(CAL\) is caliper of the corrugated board, \(PER\) is box perimeter from the top view, \(S_{S}\) is the static compression strength, \(F_{O}\) is the orientation coefficient form strength, \(F_{T}\) is the storage time coefficient, \(F_{I}\) is the interlock coefficient, \(F_{H}\) is the humidity coefficient, and \(F_{G}\) is the pallet shape coefficient (Table 1)  
\[\text{Static Compression Strength}\left(S_{S}\right)\ =5.874*ECT*CAL^{0.508}*PER^{0.492} \tag{2}\]  
\[\text{Dynamic Compression Strength}\left(S_{D}\right)\ =S_{S}*\ F_{T}*\ F_{H}*\ F_{G}*F_{G}*F_{I} \tag{3}\]