## 2 The Research Methodology
### Implementation Procedure  
The optimization of the design of the stacked pallet load units is based on the formal assumptions and the mathematical model presented in Sections 3.1 and 3.2. The practical application of this approach is shown as a BPMN notation procedure, see Figure 2.  
Figure 2: The implementation of stacking pallet load units optimization procedure.  
The starting point is set \(A\) of \(i\)-PLUs, where \(i=1,\,2,\,\ldots,\,I,\,I+1,\,\ldots,\,N\), which constitutes all pallets prepared upon customer order. Initially, two boundary parameters of \(j\)-SPLU that should be designed on \(A\), i.e., \(W_{j}^{\rm max}\) and \(H_{j}^{\rm max}\), are defined (step 1). Then, each \(i\)-PLUs from the set \(A\): \(i=1,\,\ldots,\,N\) is identified in terms of basic parameters (step 2), such as weight \(w_{i}\), height \(h_{i}\), fragility class \(f_{i}\), and the degree of filling of the top layer \(t_{i}\). Based on the identified parameters of each \(i\)-PLU and the boundary parameters of SPLU to be constructed, in the next step of the procedure (step 3), the assessment of the possibility of building stacked \(j\)-SPLUs from available \(i\)-PLUs is carried out. As a result, set \(A\) is divided into two subsets, i.e., \(A1\): \(i=I+1,\,\ldots,\,N\), which includes \(i\)-PLUs that do not meet at least one boundary condition of \(W_{j}^{\rm max}\) or \(H_{j}^{\rm max}\), and subset \(A2\): \(i=1,\,2,\,\ldots,\,I\), which includes those \(i\)-PLUs that do not exceed any of the boundary parameters. In step 4, a set of feasible solutions is built on \(A2\); for this purpose, all the constraints described by Formulas (3)-(11) are applied. If, as a result of this operation, the set of feasible solutions is non-empty, in step 5, the optimal solution is searched; at this stage, the Formulas (1)-(2) of the objective function are applied. The computations within steps 4 and 5 are performed by optimization engines of the evolutionary algorithm. If, as a result of step 5, an optimal solution is found, in step 6.1, an analysis of whether it is necessary or applicable to modify boundary parameters for further improvement should be carried out. If not, then step 7 is performed, which consists of joining the \(j\)-SPLUs constructed in steps 4 and 5 with the \(i\)-PLUs that were separated in step 3 due to unmet boundary conditions; see subset \(A1\) of step 3. The result of the procedure at this stage is a reduced number of pallet spaces that are necessary for the transport or storage of \(i\)-PLUs because some of them can be transformed to SPLUs, upon boundary conditions of \(W_{j}^{\rm max}\) or \(H_{j}^{\rm max}\).  
If, as a result of step 6.1, the change of boundary parameters is applicable, then it is necessary to return to step 1 and repeat the optimization procedure step by step. If, after step 4, it is not possible to define a set of feasible solutions, or in step 5, it is not possible to find an optimal solution, it is recommended to consider changing the values of the boundary parameters (step 6.2) and repeat the procedure from step 1. If such a change of parameters is not possible, the problem is unsolved, and the procedure should be stopped.