## 3 Methodology
### Fuzzy Analytic Hierarchy (FAHP) Procedure  
In the 1980s, [PERSON] presented an analytical hierarchy process for solving complex problems by breaking the problem down into subgroups that were hierarchically arranged [9]. Even though AHP has merit in evaluating multiple criteria, it has shortcomings that the decision maker came up with, as it substitutes for pairwise comparison. [PERSON] also observed the fallbacks of the AHP method when comparing alternatives [46]. Also, in its comparison procedure, the scale used for judgment in this technique did not extract the meaning of unpredictability and delinquency in the data [46]. Therefore, [PERSON] presented an improved AHP referred to as the fuzzy analytical hierarchy process [11]. The FAHP technique was used in this research in order to avoid fallbacks. Fuzzy offers more versatility to expression. There are several types of fuzzy numbers: triangular, trapezoidal fuzzy numbers, etc. [17]. The triangular fuzzy number contains the group of three real integer numbers starting from the least, most ascertain, and the highest weights for the triangular fuzzy. However, trapezoidal and triangular fuzzy numbers are said to be the most popularly used. Triangular fuzzy numbers were used in this study. The procedures involved in the improved AHP are described as follows, as presented by [47]. Let's declare G as a set of objects, where G = {G1, G2, G3,.......,Gm} and P is a goal set where P = {P1,P2,P3,..........,Pn}. Based on this technique, the individual object is taken, and the extent analysis set for every goal is carried out respectively. Therefore, for an individual object, the respective m extent analysis values are acquired. In this case, the individual value of \(M_{S_{P}}^{j}\) where j = 1, 2, 3, 4,...,m are triangular fuzzy number. The steps for Chang extent analysis are described as follows:  
Step 1: Fuzzy synthetic based on ith objects is stated by an equation as follows;  
\[S_{i}\ =\sum_{j=1}^{m}M_{S_{P}}^{j}\otimes\left[\sum_{i=1}^{n}\sum_{j=1}^{m}M_{ S_{P}}^{j}\right]^{-1} \tag{1}\]  
To acquire the value of \(\sum_{j=1}^{m}M_{S_{P}}^{j}\), m extent analysis is used based on fuzzy summation. \[\sum_{j=1}^{m}M_{S_{P}}^{j}=\left(\sum_{j=1}^{m}l_{j},\ \sum_{j=1}^{m}m_{j}, \sum_{j=1}^{m}u_{j}\right) \tag{2}\]  
\[\sum_{i=1}^{n}\sum_{j=1}^{m}M_{S_{P}}^{j}=\left(\sum_{i=1}^{n}l_{i},\ \sum_{i=1}^{n}m_{i}, \sum_{i=1}^{n}u_{i}\right) \tag{3}\]  
Equation (2) is a vector that is then reversed to acquire Equation (4), shown below;  
\[\left[\sum_{i=1}^{n}\sum_{j=1}^{m}M_{S_{P}}^{j}\right]^{-1}=\left(\frac{1}{ \sum_{i=1}^{n}u_{i}},\frac{1}{\sum_{i=1}^{n}m_{i}},\frac{1}{\sum_{i=1}^{n}l_{ i}}\ \right) \tag{4}\]  
Step 2: The degree of possibility depends on the values of M1 and M2, of which, while M1 remains (l1, m1, u1) and M2 remains (l2, m2, u2) and can be evaluated as stated in Equation (5);  
\[l_{i}=min_{k}\{a_{ik}\};\ m_{i}=\frac{1}{k}\sum_{k=1}b_{ik};u_{ij}=max_{k}\{ \ c_{ik}\} \tag{5}\]  
Therefore, M2 \(\geq\) M1 for the degree of possibility, which is stated as follows:  
\[V(M2\geq M1)=\text{hgt}\Big{(}M2\bigcap M1\Big{)}=\mu_{M_{2}}\ (d)= \begin{cases}1,&m_{2}\geq\ m_{1}\\ 0,&l_{1}\ \geq\ u_{2}\\ \frac{l_{1}-u_{2}}{(m_{2}-u_{2})(m_{1}-l_{1})}&Otherwise\end{cases}  \tag{6}\]  
V(M2 \(\geq\) M1) is equal to d; this indicates that the highest convergence within M1 and M2 is the ordinate of D, as shown in Figure 7.  
In Figure 7, d defines the highest convergence point among \(\mu_{M_{1}}\) and \(\mu_{M_{2}}\). Comparing M\({}_{1}\) and M\({}_{2}\), the values of V(M\({}_{2}\)\(\geq\) M\({}_{1}\)) and V(M\({}_{1}\)\(\geq\) M\({}_{2}\)) becomes relevant.