# A Hybrid IVFF-AHP and Deep Reinforcement Learning Framework for an ATM Location and Routing Problem
## 3 The Proposed Methodology
### Fermatean Fuzzy Analytic Hierarchy Process
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Linguistic Terms** & \multicolumn{4}{c}{**IVFFN Equivalents**} \\  & \(\mu_{L}\) & \(\mu_{U}\) & \(v_{L}\) & \(v_{U}\) \\ \hline Certainly High Importance (CHI) & 0.95 & 1 & 0 & 0 \\ Very High Importance (VHI) & 0.8 & 0.9 & 0.1 & 0.2 \\ High Importance (HI) & 0.7 & 0.8 & 0.2 & 0.3 \\ Slightly More Importance (SMI) & 0.6 & 0.65 & 0.35 & 0.4 \\ Equally Importance (EI) & 0.5 & 0.5 & 0.5 & 0.5 \\ Slightly Less Importance (SLI) & 0.35 & 0.4 & 0.6 & 0.65 \\ Low Importance (LI) & 0.2 & 0.3 & 0.7 & 0.8 \\ Very Low Importance (VLI) & 0.1 & 0.2 & 0.8 & 0.9 \\ Certainly Low Importance (CLI) & 0 & 0 & 0.95 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Linguistic terms and IVFFN equivalents [20]. comparison, using the IVFFWG aggregation operator. This ensures that the influence weights of individual experts are preserved and that uncertainty is properly propagated in the aggregation process. Let \(\text{Ek}=E_{1}\), \(E_{2}\),...,\(E_{k}\), with \(k=1,2,\ldots,K\), representing the set of experts with influence weights \(w_{k}\); \(\sum_{k=1}^{K}w_{k}=1\). \[IVFFWG(z_{1},z_{2},\ldots,z_{k})=\left(\left[\prod_{k=1}^{K}\left(\mu_{k}^{L} \right)^{m},\prod_{k=1}^{K}\left(\mu_{k}^{L}\right)^{m}\right],x\left[\sqrt[]{ \left(1-\prod_{k=1}^{K}\left(1-\left(v_{k}^{L}\right)^{3,m}\right)\right), \sqrt[]{1-\prod_{k=1}^{K}\left(1-\left(v_{k}^{L}\right)^{3,m}\right)}}\right]\right) \tag{11}\]  
Step 5: The differences matrix, \(D=\left(z_{ij}\right)_{m\times m}\), representing the strength of preference between criteria \(i\) and \(j\), is determined between the lower and upper points of the membership and non-membership functions using Equations (12) and (13). \[d_{ij}^{L}=\left(\mu_{ij}^{L}\right)^{3}-\left(v_{ij}^{L}\right)^{3} \tag{12}\]  
\[d_{ij}^{U}=\left(\mu_{ij}^{U}\right)^{3}-\left(v_{ij}^{L}\right)^{3} \tag{13}\]  
Step 6: The interval multiplicative matrix \(S=\left(s_{ij}\right)_{m\times m}\) is calculated using Equations (14) and (15). As a result, the \(s_{ij}\) matrix, which indicates the relative dominance between criterion pairs, is obtained using the distance matrix. \[s_{ij}^{L}=\sqrt[3]{1000^{d_{ij}^{L}}} \tag{14}\]  
\[s_{ij}^{U}=\sqrt[3]{1000^{d_{ij}^{U}}} \tag{15}\]  
Step 7: Applying Equation (16), the indeterminacy value \(T=\left(t_{ij}\right)_{m\times m}\) of \(z_{ij}\) is calculated. The degree of indeterminacy is represented by \(t_{ij}\), which reflects the uncertainty between the bounds of membership and non-membership values. \[t_{ij}=1-\left(u_{ijU}^{3}-u_{ijL}^{3}\right)-\left(v_{ijL}^{3}-v_{ijL}^{3}\right) \tag{16}\]  
Step 8: Applying Equation (17), the indeterminacy degrees are multiplied by the matrix \(S=\left(s_{ij}\right)_{m\times m}\) to create the matrix of unnormalized weights \(R=\left(r_{ij}\right)_{m\times m}\). It combines the average interval multiplicative matrix (s) and the indeterminacy value (t), reflecting both preference strength and uncertainty. \[r_{ij}=\left(\frac{s_{ij}^{L}+s_{ij}^{U}}{2}\right)t_{ij} \tag{17}\]  
Step 9: \(w_{i}\), which represents the normalized priority weights for criterion \(i\), is calculated by dividing the relative weight of the criterion by the sum of all non-normalized weights, as shown in Equation (18):  
\[w_{i}=\frac{\sum_{j=1}^{m}r_{ij}}{\sum_{i=1}^{m}\sum_{j=1}^{m}r_{ij}} \tag{18}\]  
Step 10: The alternatives are ranked based on the priority weight values calculated in the previous step.