## 3 Method  
### Preliminaries for the Computational Framework  
Fermatean Fuzzy Sets  
The conventional approach to computing is performed using real or \"crisp\" numbers. In dealing with systems with epistemic uncertainties, such an approach is deemed inadequate because quantities in such systems lie on a \"fuzzy\" spectrum rather than on a fixed value, which crisp numbers suggest. This problem was addressed by [PERSON] [45] by proposing the fuzzy set theory. Since then, various extensions have been introduced for the past decades. Among the recent advances is the notion of Fermatean fuzzy set (FFS) theory, which corresponds to the fuzzy environment adopted in this work. Some useful definitions of the FFS are presented as follows:  
**Definition 1** ([14]). : _Let \(X\) be a universe of discourse. The general form of an FFS, \(\mathcal{F}\), in \(X\) can be presented as:_  
\[\mathcal{F}=\left\{\left(x,\;\mu_{\mathcal{F}}^{F}(x),\
u_{\mathcal{F}}^{F}(x )\right):x\in X\right\} \tag{1}\]  
_where \(\mu_{\mathcal{F}}^{F}(x):X\rightarrow[0,1]\) and \(\
u_{\mathcal{F}}^{F}(x):X\rightarrow[0,1]\), such that \(0\leq\left(\mu_{\mathcal{F}}^{F}(x)\right)^{3}+\left(\
u_{\mathcal{F}}^{F}(x )\right)^{3}\leq 1\) for all \(x\in X\). Furthermore, \(\mu_{\mathcal{F}}^{F}(x)\) and \(\
u_{\mathcal{F}}^{F}(x)\) refer to the degree of membership and non-membership of the element \(x\) in the set \(\mathcal{F}\), respectively [14]. Meanwhile, the degree of indeterminacy, \(\pi_{\mathcal{F}}^{F}(x)\) is obtained through:_  
\[\pi_{\mathcal{F}}^{F}(x)=\sqrt[3]{1-\left(\mu_{\mathcal{F}}^{F}(x)\right)^{3}- \left(\
u_{\mathcal{F}}^{F}(x)\right)^{3}} \tag{2}\]  
**Definition 2** ([14]). : _Let \(\mathcal{F}_{i}=\left(\mu_{\mathcal{F}_{i}}^{F},\
u_{\mathcal{F}_{i}}^{F} \right)(i=1,2,\ldots,n)\) be a set of FFS, and \(w=\left(w_{1},w_{2},\ldots,w_{n}\right)^{T}\) be the corresponding weight vector such that \(\sum\limits_{i}w_{i}=1\). Then, the Fermatean fuzzy weighted average (FFWA) aggregation operator is defined by:_  
\[FFWA(\mathcal{F}_{1},\mathcal{F}_{2},\ldots\mathcal{F}_{n})=\left(\sum_{i=1}^ {n}w_{i}\mu_{\mathcal{F}_{i}^{F},\sum_{i=1}^{n}w_{i}\
u_{\mathcal{F}_{i}}^{F}}\right) \tag{3}\]  
**Definition 3** ([60]). : _Let \(\mathcal{F}=\left(\mu_{\mathcal{F}}^{F},\
u_{\mathcal{F}}^{F}\right)\) be an FFS. The score function \(S^{F}\) for \(\mathcal{F}\) is defined as:_  
\[S^{F}(\mathcal{F})=\frac{1}{2}\bigg{[}\bigg{(}\left(\mu_{\mathcal{F}}^{F} \right)^{3}-\left(\
u_{\mathcal{F}}^{F}\right)^{3}-ln\bigg{(}1+\left(\pi_{ \mathcal{F}}^{F}\right)^{3}\bigg{)}\bigg{)}+1\bigg{]} \tag{4}\]  
_where \(S^{F}:\;\mathcal{F}\rightarrow\mathbb{R}\)._  
Decision-Making Trial and Evaluation Laboratory  
Let \(i\in\{1,\ldots,n\}\) be an element of the system under study. First, generate the direct-relation matrix. An expert group of \(K\) members performs pairwise comparisons of the causal relationships between \(n\) elements. The response from the expert group generates a direct-relation matrix \(X^{k}=\left(x_{ij}^{k}\right)_{n\times n}\), where \(x_{ij}^{k}\) represents the causal influence of the element \(i\) on element \(j\) as perceived by the \(k^{th}\) member, \(k\in\{1,2,\ldots,K\}\), of the group. An evaluation scale of 0, 1, 2, 3, and 4 is used for the causal influence, representing \"no influence\", \"medium influence\", \"high influence\", and \"very high influence\", respectively. Second, aggregate \(X^{k}\), considering that \(w_{k}>0\) (\(\sum_{k}w_{k}=1\)) is assigned to the importance of the \(k^{th}\) member, as described in Equation (5):  
\[X=\left(x_{ij}\right)_{n\times n}=\left(\sum_{k}w_{k}\;x_{ij}^{k}\right)_{n \times n} \tag{5}\]Third, normalize the aggregate direct-relation matrix \(X\).