## 2 Framework of the Proposed ExJ-PSI Model  
The present study takes six criteria into account: severity of the failure (SV), probability of the occurrence (O), degree of detectability (D), degree of maintainability (M), degree of safety (S), and production or quality loss (L). It is difficult to determine the interdependence of these attributes and to assign impo trance when determining the criticality of the components. As a result, the Preference Section Index (PSI) approach is utilized to rank the risk priority of FMs properly. The framework of the developed model is presented in Figure 1. It is also specified step-by-step for the convenience of the readers:  
* **Step 1:** Select and arrange a set of risk factors and failure modes in the decision matrix.
* **Step 2:** Collect the risk factors rank data from the various experts of several industries.
* **Step 3:** Normalize the data collected by assigning overall weights to experts.
* **Step 4:** Construct initial decision matrix 'A' with criteria ranks, \[A=\begin{bmatrix}a_{ij}\end{bmatrix}=\begin{bmatrix}a_{11}&a_{12}&\dots&a_{1 1n}\\ a_{21}&a_{22}&\dots&a_{2n}\\ \dots&\dots&\dots&\dots\\ \dots&\dots&\dots&\dots\\ a_{m1}&a_{m2}&\dots&a_{mm}\end{bmatrix}\] where \(a_{ij}\) is the index value. \(i\) = 1, 2,..., \(m\) along the row represents the failure modes and \(j\) = 1, 2,..., \(n\) represents the risk criteria's along the column.
* **Step 5:** Normalize decision matrix 'A' Normalization of A is achieved by the following equations [30], For maximizing attributes, \[\text{N}_{\bar{\text{\text{\text{\text{\text{\text{ij}}}}}}}}=\frac{a_{ij}}{a _{ijmax}}\] (1) For minimizing attributes, \[\text{N}_{\bar{\text{\text{\text{\text{\text{\text{\text{\text{ij}}}}}}}}}}= \frac{a_{ijmin}}{a_{ij}}\] (2) where; \(a_{ijmax}\) and \(a_{ijmin}\) are the maximum and minimum values of each alternative (Criteria). The normalized decision matrix \(A1\) is shown below; \[A1=\begin{bmatrix}a_{ij}\end{bmatrix}=\begin{bmatrix}N_{11}&N_{12}&\dots&N_{1 n}\\ N_{21}&N_{22}&\dots&N_{2n}\\ \dots&\dots&\dots&\dots\\ \dots&\dots&\dots&\dots\\ N_{m1}&N_{m2}&\dots&N_{mn}\end{bmatrix}\]* **Step 6:** Calculate the mean value of the normalized data [30]. \[\text{N}=\frac{1}{m}\sum_{i=1}^{m}\text{N}_{ij}\] (3)
* **Step 7:** Calculate the values of the variation of preferences between the values of every attribute [30]. \[\varphi_{j}=\frac{1}{m}\sum_{i=1}^{m}\left(\text{N}_{ij}-\text{N}\right)^{2}\] (4)
* **Step 8:** Calculate the deviations in the preference value for all criteria [30]. \[\Omega_{j}=\left(1-\varphi_{j}\right)\] (5)
* **Step 9:** Calculate the overall criteria weights for each criteria [30]. \[\text{w}_{\text{j}}=\frac{\Omega_{\text{j}}}{\sum_{\text{j=1}}^{n}\Omega_{ \text{j}}}\] (6)
* **Step 10:** Prepare a matrix by multiplying \(\text{N}_{\text{ij}}\) and \(\text{w}_{\text{j}}\).
* **Step 11:** Calculate the Criticality Index for all failure modes and all components. The criticality index of all failure modes can be estimated using Equation (7) [30]. \[\text{CI}_{\text{PSI}}=\sum_{\text{j=1}}^{n}(\text{N}_{ij}\ \times\text{w}_{\text{j}})\] (7)  
Finally, the criticality index of the components can be estimated using Equation (8) [30]. \[(\text{CI}_{\text{PSI}})_{\text{-comp}}=\sum_{\text{i=1}}^{\text{k}}(\text{N} _{ij}\times\text{w}_{\text{j}})\] (8)  
where i = 1, 2,..., k, are the failure modes of that component or subsystem.  
* **Step 12:** Criticality rankings are given by increasing order according to the value of \(\text{CI}_{\text{PSI}}\), i.e., the higher value of \(\text{CI}_{\text{PSI}}\) has a higher priority.