# New Possibilities of Using DEMATEL and ERPN in the New PFMEA Hybrid Model
## 2 Materials and Methods
### Creation of DEMATEL Model  
We used the well-known DEMATEL methodology, primarily following [PERSON] et al. [21]. This methodology [21] was used to develop the model on which we built our research (Section 2.7).  
Establishing a direct-relation matrix (Equation (1)):  
Each team member will determine the intensity of the relations between the selected failure modes as an expert, and thus the basis of the initiation matrix for the DEMATEL model will be created. The coupling between failure modes (FM's) will be evaluated on a scale of 0-6, where 0 means no relations and 6 is the maximum number of relations (Figure 2).  
Subsequently, a direct-relation matrix is created (Equation (1))  
\[x=\begin{bmatrix}0&x_{12}&\cdots&x_{1n}\\ \vdots&0&\cdots&\vdots\\ \vdots&\vdots&\ddots&\vdots\\ x_{n1}&x_{n2}&\cdots&0\end{bmatrix} \tag{1}\]  
Determination of [PERSON]'s alpha and CITC:  
[PERSON]'s alpha (Equation (2)) will be used to assess the reliability of the survey data, and CITC will be calculated. The test results will show how reliable the questionnaires used in the DEMATEL analyses are. The acceptable limit value of [PERSON]'s alpha is 0.7 [33; 39].  
\[\alpha=\frac{k}{k-1}\bigg{(}1-\frac{V_{i}}{V_{t}}\bigg{)} \tag{2}\]  
where \(k\) is the number of experts, \(V_{i}\) is the sum of item variances, and \(V_{t}\) response variance of total scores.  
A corrected total item correlation (CITC) score is determined for each item and each expert. The CITC expresses how well each item contributes to the internal consistency of a particular construct, as measured by [PERSON]'s alpha coefficient [40]. CITC was evaluated by Pearson's product-moment correlation coefficient.  
Determination of the normalized direct-relation matrix \(N\):  
Using column vectors and maximum values (Equation (3)) as the basis for matrix normalization (Equation (4)) [21].  
\[\lambda=1/max_{1\leq i\leq n}\sum_{j=1}^{n}x_{ij} \tag{3}\]  
Figure 2: Relations by DEMATEL.  
\[N=\lambda X \tag{4}\]  
Determination of the direct/indirect relation matrix (\(T\)) or the total-relation matrix (Equation (5)):  
\[T=\lim_{n\rightarrow\infty}\left(N+N^{2}+\ldots+N^{k}\right)=N(I-N) \tag{5}\]  
where \(I\) represents the identity matrix.  
Determination of the values in each row and column:  
This is followed by performing a mathematical summation of the values in each row and column of the total relation matrix (\(T\)). We denote \(D_{i}\) (Equation (6)) as the sum of the \(i\)-th column and \(R_{i}\) (Equation (7)) as the sum of the \(j\)-th row. These values of \(D_{i}\) and \(R_{i}\) consider both indirect and direct influences [21].  
\[D_{i}=\sum_{j=1}^{n}t_{ij}(i=1,2,...,n) \tag{6}\]  
\[R_{i}=\sum_{j=1}^{n}t_{ij}(i=1,2,...,n) \tag{7}\]  
Creating the DEMATEL cause and effect diagram:  
The horizontal axis (D + R) and vertical axis (D - R) are used to represent the cause and effect diagram, respectively. The diagram simplifies complex causal relations into an easy-to-understand visual structure. Decision-makers can determine factor types based on the characteristics of the factors and formulate appropriate solutions based on the extent of influence of each factor [21].  
Attribute k is either a cause or effect attribute when (Dk - Rk) is a positive or negative value, respectively. The size of (Dk + Rk) represents the extent of the attribute's cause or effect [21].  
Based on the coordinates in (Dk + Rk) and (Dk - Rk), \(k\) can be categorized into four categories:  
* Rk) and large (Dk + Rk) values: \(k\) is a cause factor and an actuating factor for solving the problem;
* Rk) and small (Dk + Rk) values: \(k\) is an independent factor and influences only a small number of other factors;
* Rk) and small (Dk + Rk) values: \(k\) is an independent factor and is influenced by only a small number of factors;
* Rk) and large (Dk + Rk) values: \(k\) is a core problem that requires resolution. However, it is an effect attribute; thus, it cannot be directly improved [21; 41; 42].