## 2 Materials and Methods
### Analysis of the Obtained Data Using Selected Methods  
For determining the weights and the resulting model, the Analytic Hierarchy Process (AHP) method was used. The AHP method to support multi-criteria decision-making was originally developed by [PERSON]. The basis for decision-making is empirical decision-making criteria, and using AHP outlines the whole decision problem as a hierarchical structure [66]. AHP is based on the value of the information obtained and derives ratio scales from paired comparisons of criteria [67] to discover and correct logical contradictions. Questionnaire surveys, in general, contain the subjective opinions of respondents. These opinions were taken as inputs and divided according to alternatives \(A_{1}\dots\ A_{m}\) (competencies) and specific criteria \(K_{1}\dots\ K_{m}\) (abilities of project managers). This process allows translating these opinions into measurable numeric relations.  
Internationally, AHP is used in a wide range of applications, for example, for the evaluation of suppliers, in project management, or for the selection of the best alternative in a decision tree. AHP helps us make decisions in a more rational way and make them more transparent and more easily understood ([PERSON], 2018). Using AHP as a supporting tool for decision-making helps gain a better insight into complex decision problems. It means the selection of the best alternative in the decision tree. As you need to structure the problem as a hierarchy, it forces you to think through the problem, consider possible alternatives (decision criteria), and select the most significant criteria with respect to the decision objective.  
The criteria matrix for evaluation alternatives according to criteria is represented by \(Y=(y_{ij})\), where rows are alternatives and columns are criteria. For every criterion, it was necessary to calculate its weight (number from 0 to 1). The more important criterion, the higher its weight (denoted \(w_{j}\) for criterion \(K_{j}\), \(j=1\dots\ n\)). Some of the methods of criteria weights:  
Method of entropy: no preferences, weights of criteria are equal (\(w_{j}=1/n\));  
Method of order and Fuller's method: ordinal information about criteria is known;  
Figure 7: Equipment used for measurements.  
The Scoring Method and the AHP Method ([PERSON]'s method) [61]: ordinal information and distances between criteria are known. The scale of points for criteria is 1, 3, 5, 7, 9, where 1 = criteria are equal, 3 = the first criterion is more significant than the second one, 9 = the first criterion is absolutely more significant than the second one.  
Mathematically, the AHP method [68] is based on the solution of an Eigenvalue problem. The results of the pair-wise comparisons are arranged in a matrix. [PERSON]'s matrix is squared, and consists of estimated elements \(s_{ij}\approx\frac{w_{i}}{w_{j}}\) which are ratios of weights \(i\)-th and \(j\)-th criterion. Elements on the main diagonal are equal to 1. In an ideal case, for every \(i,j,\,k=1,\,\ldots,\,n\) should apply \(s_{ij}=\frac{w_{i}}{w_{j}}=\frac{w_{i}}{w_{k}}\frac{w_{k}}{w_{j}}=s_{ik}\cdot s _{kj}\) and then this matrix is called consistent and reciprocal. So it could be multiplied from the right side by normalized right Eigenvector \((w_{1},w_{2},\ldots,w_{n})^{T}\) (\(n\) is Eigenvalue) gives the ratio scale (weighting), the Eigenvalue determines the consistency ratio:  
\[\left(\begin{array}{cccc}1&\frac{w_{1}}{w_{2}}&\cdots&\frac{w_{1}}{w_{n}}\\ \frac{w_{2}}{w_{1}}&1&\cdots&\frac{w_{2}}{w_{n}}\\ \cdots&\cdots&\cdots&\cdots\\ \frac{w_{n}}{w_{1}}&\frac{w_{n}}{w_{2}}&\cdots&1\end{array}\right)\cdot\left( \begin{array}{c}w_{1}\\ w_{2}\\ \cdots\\ w_{n}\end{array}\right)=\left(\begin{array}{c}w_{1}\\ w_{2}\\ \cdots\\ w_{n}\end{array}\right)\cdot n \tag{1}\]  
In practice, it is very rare to obtain a fully consistent matrix.