## 3 Materials and Methods
### Proposal of a Extended RPI Model  
The extension of the RPI model to the risk space with nth variables was done in two steps. First, the \(RI\) function was extended to include more than three variables, and then the equations of the scale variables (Equation (2)) were also extended. The \(RI\) extension is described in Equation (5), which is an extrapolation of Equation (3) to nth risk variables. Equation (5) retains injectivity and surjectivity and has an output range between 1 and 10. For \(n=3\), Equation (5) is equivalent to Equation (3). \[RI(x_{1},x_{2},\ldots,x_{n})_{x_{1}>x_{2}>\ldots>x_{n}}=\sum_{i=1}^{n}(x_{i}- 1)\cdot\alpha^{n-i}+1 \tag{6}\]  
Figure 1: Materials and methods framework followed in this research. The order of importance of the risk variables is determined by \(x_{1}>x_{2}>x_{3}\ldots>x_{n}\), while \(\alpha\) is the scale used to evaluate the risk variables, and \(n\) is the dimension of the risk space. For example, in a four-dimensional risk space, Equation (5) is set as follows:  
\[RI(x_{1},x_{2},x_{3},x_{4})_{x_{1}>x_{2}>x_{3}>x_{4}}=(x_{1}-1)\cdot\alpha^{3}+( x_{2}-1)\cdot\alpha^{2}+(x_{3}-1)\cdot\alpha+(x_{4}-1)+1 \tag{6}\]  
For a five-dimensional risk space, it is determined as shown in Equation (7):  
\[RI(x_{1},x_{2},x_{3},x_{4},x_{5})_{x_{1}>x_{2}>x_{3}>x_{4}>x_{5}}=(x_{1}-1)\cdot \alpha^{4}+(x_{2}-1)\cdot\alpha^{3}+(x_{3}-1)\cdot\alpha^{2}+(x_{4}-1)\cdot \alpha+(x_{5}-1)+1 \tag{7}\]  
and so on. The next step is to extend the formulation of the scale variables (Equation (2)) and reduce the scenario risk variables to the same risk scale to account for their combined effect in an nth risk space. They must be determined by considering the permutations of the risk variables, as shown in Equation (2). Thus, for a space with 4 dimensions, there are 4! Permutations (24 permutations), and for a space with 5 dimensions, there are 5! Permutations (120 permutations) and so on. Equation (8) represents 5! possible permutations for a five-dimensional risk space, which corresponds to 120 possible importance orders with five risk variables. \[ \begin{array}{l}a,P(b,c,d,e)\\ b,P(a,c,d,e)\\ c,P(a,b,d,e)\\ d,P(a,b,c,e)\\ e,P(a,b,c,d)\end{array}  \tag{8}\]  
Here the function \(P()\) stands for permutations without repetition of the risk variables \(a\), \(b\), \(c\), \(d\), and \(e\). Equation (9) represents the extension of Equation (8) to any length of the risk space:  
\[ \begin{array}{l}x_{1},P(x_{2},x_{3},\ldots,x_{n})\\ x_{2},P(x_{1},x_{3},\ldots,x_{n})\\ \ldots\\ x_{n},P(x_{1},x_{2},\ldots,x_{n-1})\end{array}  \tag{9}\]  
where \(n\) is the number of risk variables (of the dimension of the risk space). For simplicity, the first risk variable in each permutation set is called the master risk variable, since its position is maintained for its permutation set. For example, in the set \(x_{1},P(x_{2},x_{3}\ldots x_{n})\) of permutations, there are \((n-1)!\) permutations where the first variable is always \(x_{1}\), so \(x_{1}\) is a master risk variable in this set of permutations. So, for each master risk variable there are \((n-1)!\) orders of importance, and for each of these orders of importance, the \(RI\) function is evaluated in each failure mode, as shown in Equation (10):  
\[R_{ijk}=RI(x_{j},P_{k}(x_{1},x_{2},\ldots,x_{n})) \tag{10}\]  
where \(i\) is the failure mode number, \(k\) is the permutation number from 1 to \((n-1)!\), and \(j\) is the master risk variable number from 1 to \(n\). Therefore, \(R_{ijk}\) is the output value of the extended \(RI\) function for failure mode \(i\) with respect to master risk variable \(j\) and order of importance \(k\). For each master risk variable \(j\) and order of importance \(k\), the failure modes are prioritized according to their respective \(R_{ijk}\) values. After these prioritizations, the \(R_{ijk}\) values are replaced by their respective rank orders, which are represented as \(\delta_{ijk}\) from now on.