# 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
To better reflect the uncertainty in expert judgments, membership and non-membership degrees are delivered in closed intervals, as in Equations (5)-(7):  
\[\mu_{\widetilde{F}}(x)=\left[\mu_{\widetilde{F}}^{\cdot L}(x),\ \mu_{\widetilde{F}}^{\cdot U}(x)\right]\subseteq\ \left[0,1\right] \tag{5}\]  
\[v_{\widetilde{F}}(x)=\left[v_{\widetilde{F}}^{\cdot L}(x),\ v_{\widetilde{F}} ^{\cdot U}(x)\right]\subseteq\ \left[0,1\right] \tag{6}\]  
\[0\leq\left(\ \mu_{\widetilde{F}}^{\cdot U}(x)\right)^{3}+\left(v_{\widetilde{F}}^{ \cdot U}(x)\right)^{3}\leq 1 \tag{7}\]  
In IVFFSs, the hesitancy degree is defined as \(\pi_{\widetilde{F}}(x)=\left[\pi_{\widetilde{F}}^{\cdot L}(x),\ \pi_{\widetilde{F}}^{\cdot U}(x)\right]\) for every x \(\epsilon\) X, as in Equations (8) and (9), which allows for a more flexible representation of uncertainty compared to traditional fuzzy sets:  
\[\pi_{\widetilde{F}}^{\cdot L}(x)=\sqrt[3]{1-\left(\ \mu_{\widetilde{F}}^{\cdot U}(x) \right)^{3}-\left(v_{\widetilde{F}}^{\cdot L}(x)\right)^{3}} \tag{8}\]\[\pi_{F}^{\omega U}(x)=\sqrt[3]{1-\left(\mu_{F}^{-L}(x)\right)^{3}-\left(v_{F}^{-L}(x )\right)^{3}} \tag{9}\]  
[PERSON] developed AHP, one of the literature's most widely used MCDM approaches. The method has a specific format for weight criteria and decision-making in complicated MCDM problems. However, judgments of decision-makers in unclear situations cannot be represented using the traditional AHP technique. To define the uncertainty in human judgment and preference, traditional AHP has been extended to fuzzy AHP. Fuzzy AHP has been used in many MCDM problems in the literature in different ways, such as intuitionistic fuzzy AHP [34], interval-valued intuitionistic fuzzy AHP [35], hesitant fuzzy AHP [36], neutrosophic AHP [37], and interval-valued neutrosophic AHP [38]. [PERSON] and [PERSON] developed IVFF-AHP by combining their method with the IVFF set and AHP to determine the best digital transformation strategy [20]. The following are the steps of the IVFF-AHP approach suggested by [PERSON] and [PERSON]:  
Step 1: The criteria and alternatives are determined before constructing the hierarchical structure. An objective, decision criteria, and alternatives are decided for the given situation. The set \(A_{i}=A_{1},A_{2},\ldots\ldots\), \(A_{n}\), which has \(I=1,2,\ldots,n\) alternatives, is assessed by \(m\) decision criteria from the set \(C_{j}=\{C_{1},C_{2},\ldots\ldots,C_{m}\}\), which has \(j=1,2,\ldots.m\). Let \(w_{j}=w_{1},w_{2},\ldots,w_{m}\) be the vector set used to define the criterion weights, where \(w_{j}>0\) and \(\sum_{j=1}^{n}w_{j}=1\). Table 2 lists linguistic terms and their associated interval-valued Fermatean fuzzy numbers (IVFFNs). Step 2: The pairwise comparison matrix \(Z=\left(z_{ij}\right)_{m\times m}\) is created using the expert views in Table 2. The pairwise comparison between criterion \(i\) and criterion \(j\) is represented as an interval-valued Fermatean fuzzy judgment \(z_{ij}\), defined by the membership interval \(\left[n_{ij}^{L},n_{ij}^{U}\right]\) and non-membership interval \(\left[v_{ij}^{L},v_{ij}^{U}\right]\), as shown in Equation (10):  
\[\widetilde{F}\cong\left\{x,\ \mu_{\widetilde{F}}(x),\ v_{\widetilde{F}}(x);xeX\right\} \tag{2}\]0  
Step 3: Table 2 determines the consistency in the pairwise comparison matrix created according to expert judgments. After matching the numbers obtained from defuzzification with the IVFFNs in Table 2, [PERSON]'s classical consistency steps are followed. Step 4: Expert opinions are gathered. The pairwise comparison matrices generated for each expert are combined using the interval-valued Fermatean fuzzy weighted geometric (IVFFWG) aggregation operator. This step involves aggregating the judgements of multiple experts into a single fuzzy pairwise