## 2 The Research Methodology
### Constraints  
The optimization model of the design the stacked pallet load units problem is constructed with nine constraints; see Formulas (3)-(11) below. Constraint (3) indicates that each \(i\)-PLU is assigned exactly to one of the \(j\)-SPLU within the considered order. The next constraint (4) indicates that on a \(k\)-layer of the \(j\)-SPLU, only one \(i\)-PLU can be located. Two other constraints say that the gross weight (5) and gross height (6) of the \(j\)-SPLU result from the requirements of the \(j\)-SPLU user, i.e., \(W_{j}{}^{\text{max}}\) and \(H_{j}{}^{\text{max}}\). Based on the constraint (7), successive layers on a \(j\)-SPLU are planned while preserving the fragility class of individual \(i\)-PLU on a \(k\)-layer. It means that the \(k\)-layer may contain an \(i\)-PLU of the same or higher fragility class than layer \(k-1\). Fragility class\(f=1\) means the lowest fragility, i.e., the product has a significant compression resistance, while \(f=n\) means the highest fragility, i.e., the product has the lowest compression resistance. The constraints (8) and (9) result from the classification of \(i\)-PLUs into the _base_ and _top_ status. Status _base_ means that on \(i\)-PLU, another \(k\)-layer of \(j\)-SPLU can be considered, and status _top_, otherwise. The unit \(i\)-PLU of the status _top_ can be on the top of \(j\)-SPLU, exclusively. Constraint (8) guarantees that, at maximum, one \(i\)-PLU of _top_ status can be allocated to a single \(j\)-SPLU. However, constraint (9) ensuresthat the \(i\)-PLU of the _top_ status will not be located below the \(i\)-PLU of the _base_ status. The last two constraints, i.e., (10) and (11), indicate that both decision variables \(y_{j}\) and \(x_{ijk}\) are of binary nature.  
\[\sum_{j=1}^{J}\sum_{k=1}^{K}x_{ijk}=1,\,\forall i=1,\,2,\,\ldots,\,I \tag{3}\]  
\[\sum_{i=1}^{I}x_{ijk}\,\leq\,1,\,\forall j=1,\,2,\,\ldots,\,J;\,\forall k=1,\, 2,\,\ldots,\,K \tag{4}\]  
\[\sum_{i=1}^{I}\sum_{k=1}^{K}w_{i}\cdot x_{ijk}\,\leq\,W_{j}^{\max},\,\forall j =1,\,2,\,\ldots,\,J \tag{5}\]  
\[\sum_{i=1}^{I}\sum_{k=1}^{K}h_{i}\cdot x_{ijk}\,\leq\,H_{j}^{\max},\,\forall j =1,\,2,\,\ldots,\,J \tag{6}\]  
\[\sum_{i=1}^{I}f_{i}\cdot\left(x_{ijk}-x_{\bar{y}(k-1)}\right)\geq\,0,\,\forall j =1,\,2,\ldots,\,J;\,\forall k=2,\ldots,\,K \tag{7}\]  
\[\sum_{k=1}^{K}\sum_{i=1}^{I}t_{i}\cdot\left(x_{ijk}-x_{\bar{y}(k-1)}\right) \leq\,1,\,\forall j=1,\,2,\,\ldots,\,J \tag{8}\]  
\[\sum_{i=1}^{I}t_{i}\cdot\left(x_{ijk}-x_{ij(k-1)}\right)\geq\,0,\,\forall j=1, \,2,\ldots,\,J;\,\forall k=2,\ldots,\,K \tag{9}\]  
\[x_{ijk}\,=\,\left\{\begin{array}{l}1\text{ if $i$-PLU is located at $k$-layer in the $j$-SPLU,}\\ 0\text{ otherwise}\end{array}\right. \tag{10}\]  
\[y_{j}=\left\{\begin{array}{l}1\text{ if $j$-SPLU is planned to be designed,}\\ 0\text{ otherwise}\end{array}\right. \tag{11}\]  
Fragility is the parameter that defines the compressive strength of a single package of a product exerted by products of the same type and piled on it. Products, usually homogenous, with a certain fragility of a full load unit, i.e., filled up to the maximum weight, are characterized by the height of this PLU, i.e., \(h_{i}{}^{\max}\). It should also be assumed that the weight of a full pallet load unit of products with lower fragility \(f_{i}\) is substantially heavier than the weight of a full pallet load unit of products with higher fragility \(f_{i+1}\), i.e., \(w_{i}\langle f_{i}\rangle>w_{i+1}\langle f_{i+1}\rangle\). Thus, in case of stacking incomplete pallet load units, i.e., \(h_{i}<h_{i}{}^{\max}\), with different fragilities and maintaining the fragility ranking from the lowest to the highest \(f_{i}>f_{i+1}>f_{i+2}\) etc., starting from the background of SPLU, the total height \(H_{j}^{\max}\) of this SPLU may exceed the maximum height of each of the pallet full load unit of products, i.e., \(H_{j}{}^{\max}>h_{i}{}^{\max}\).