## 4 Multi-Criteria Process Analysis
### Defining the Number of Experimental Repetitions  
Since the degree of the multi-factorial plan is directly related to the degree of the mathematical model and given that this is a second-degree mathematical model, a second-degree multi-factorial experimental design will be applied. After defining the process input values, i.e., independent influencing factors (independence assumed in the first approximation), it was necessary to determine the levels of factor variation. Orthogonal plans with two-level factor variation are most commonly used, i.e., \(x_{i,min}\) and \(x_{i,max}\). Plans in which factors are varied at two levels are called \(2^{k}\), which represents the number of factors, or independent variables [56].  
The total number of experimental repetitions for each combination of measurement conditions was calculated using the following formula:  
\[N=2^{k}+2k+n \tag{5}\]  
where: \(N\) is the total number of experimental repetitions, \(k\) is the number of factors, and \(n\) is the number of repetitions at the center of the experiment to ensure statistical stability by minimizing the influence of variability in the experimental conditions. Central point repetitions play a crucial role in estimating inherent process variation and improving the reliability of the derived mathematical model.  
In this study, the value of \(n=10\) was determined based on the factorial experimental design and the structural requirements of the second-degree mathematical model. The selection of ten central repetitions was made to ensure a sufficient number of data points for evaluating process stability while maintaining experimental feasibility. The applied design incorporated \(3^{3}\) combinations of three controlled factors, while two additional fixed factors were introduced to maintain stable experimental conditions. The number of central repetitions was optimized to improve model accuracy and mitigate variability due to uncontrolled influences, in accordance with established experimental design principles.  
The formula for calculating the number of repetitions is based on five factors, where \(X_{1}\) (paper temperature) and \(X_{2}\) (paper humidity) are treated as controlled constant factors. Although they were not varied in the experiment, their inclusion in the formula ensures consistency of experimental conditions and reduces potential variability.  
The three controlled factors (\(W_{1}\), \(W_{2}\), \(W_{3}\)) were defined with three levels, resulting in a total of 27 combinations (\(3^{3}\)) in the experimental model. In addition, the constant factors (\(X_{1}\), \(X_{2}\)) were set to stable initial values to ensure the consistency of the input conditions during all measurements.  
This approach enabled the analysis of interactions between variable factors at the sigma level, while the constant factors contributed to the stability of the experimental process. In addition, the uncontrolled factors (\(Z_{1}\), \(Z_{2}\), \(Z_{3}\), \(Z_{4}\)) were kept constant during the experiments to eliminate their influence on the results.  
Using \(k=5\) and \(n=10\), the total number of repetitions was calculated as:  
\[N=2^{5}+2\cdot 5+10=52 \tag{6}\]  
Therefore, with 3 factors, \(3^{3}\) or 27 combinations were obtained for the controlled measurement conditions, and 52 repeated measurements were carried out for each condition.