## 3 The Proposed FMEA Model
### Combined Compromise Solution  
The Combined Compromise Solution (CoCoSo) method, introduced by [PERSON] et al. in 2019 [36], is a multi-criteria decision-making (MCDM) approach that integrates the Simple Additive Weighting (SAW) and Weighted Product Model (WPM) techniques to provide a compromised ranking of alternatives. The method addresses the limitations of traditional MCDM methods by combining additive and multiplicative strategies, resulting in a more robust and balanced decision-making process. The CoCoSo method starts with constructing a decision matrix, where alternatives are evaluated against multiple criteria, classified as either benefit (where higher values are preferred) or non-benefit (where lower values are preferred). The data are normalized to ensure comparability, scaling benefit criteria upwards and non-benefit criteria downwards. It then aggregates normalized values by combining the SAW and WPM techniques, applying k-values to adjust the influence of additive and multiplicative strategies or to emphasize specific criteria. This hybrid approach results in a comprehensive score for each alternative, enabling their ranking and supporting more effective and objective decision-making. As previously outlined, the CoCoSo method begins with the construction of the initial decision matrix:  
\[X= \begin{bmatrix}x_{11}&x_{12}&\dots&x_{1n}\\ x_{21}&x_{22}&\dots&x_{2n}\\ \dots&\dots&\dots&\dots\\ x_{m1}&x_{m2}&\dots&x_{mn}\end{bmatrix} ;\quad i=1,\,2,\dots,\,m;\quad j=1,\, 2,\,\dots,\,n \tag{3}\]The decision matrix \(X=\big{(}x_{ij}\big{)}_{m\times n}\) establishes a relationship between decision variables, represented by the index \(j\) (with its maximum value denoted as \(n\)), and the alternatives, represented by the index \(i\) (with its maximum value denoted as \(m\)), i.e., \(i=1,2,\ldots m\) and \(j=1,2,\ldots,n\). The next step is normalizing the initial decision matrix, where for benefit criteria we apply the following expression, Equation (4):  
\[r_{ij}=\frac{x_{ij}-\min\limits_{i}(x_{ij})}{\max\limits_{i}(x_{ij})-\min \limits_{i}(x_{ij})}\ ;\quad i=1,\,2,\,\ldots,\,m;\quad j=1,\,2,\,\ldots,\,n \tag{4}\]  
For cost criteria, we apply the following, Equation (5). \[r_{ij}=\frac{\max\limits_{i}(x_{ij})-x_{ij}}{\max\limits_{i}(x_{ij})-\min \limits_{i}(x_{ij})}\ ;\quad i=1,\,2,\,\ldots,\,m;\quad j=1,\,2,\,\ldots,\,n \tag{5}\]  
The next step is to calculate the sum of the weighted comparability matrix, Equation (6):  
\[S_{i}=\sum\limits_{j=i}^{n}(w_{j}\cdot r_{ij}) \tag{6}\]  
and the power of the weighted comparability matrix, Equation (7):  
\[P_{i}=\sum\limits_{j=i}^{n}(r_{ij})^{wj} \tag{7}\]  
The next step is to calculate the relative weights of the alternatives using the following aggregation equations. Equations (8)-(10). \[k_{ia}=\frac{P_{i}+S_{i}}{\sum\limits_{i=1}^{m}(P_{i}+S_{i})} \tag{8}\]  
\[k_{ib}=\frac{S_{i}}{\min\limits_{i}(S_{i})}+\frac{P_{i}}{\min\limits_{i}(P_{i})} \tag{9}\]  
\[k_{ic}=\frac{\lambda S_{i}+(1-\lambda)P_{i}}{\lambda\max\limits_{i}S_{i}+(1- \lambda)\max\limits_{i}P_{i}} \tag{10}\]  
The final step is to rank the alternatives according to their k-values, as indicated in Equation (11), with higher values leading to better rankings. \[k_{i}=(k_{ia}\cdot k_{ib}\cdot k_{ic})^{\frac{1}{3}}+\frac{1}{3}(k_{ia}+k_{ib} +k_{ic}) \tag{11}\]  
Based on the calculated compromise ratios, the alternatives are ranked, providing a more balanced and flexible evaluation compared to traditional methods such as TOPSIS or VIKOR. The CoCoSo method addresses the problem of rank reversals and allows for a more accurate assessment of alternatives, especially in situations where there are conflicting or complex criteria. In the proposed model, we consider the prioritization of failure modes as a multi-criteria decision problem. The decision criteria are the FMEA risk variables: severity, occurrence, and detectability. To ensure a balanced assessment depending on the risk scenario, the weights for these criteria are determined using the Rank Order Centroid (ROC) method.