## 2 Preliminaries
### Fuzzy Analytic Hierarchy Process Approach  
[PERSON] [3] introduced the AHP approach, leveraging mathematics and psychology concepts to solve complex MCDM problems featuring hierarchical structures. However, the typical AHP method cannot process the fuzzy semantic information experts provide. [PERSON] [11] pioneered the fuzzy set approach for processing cognitive information to solve the fuzzy semantics and uncertain information of decision-making problems. The FAHP approach combines the strengths of fuzzy sets and AHP approaches, can handle uncertainty and fuzzy information when evaluating decision-making problems, and uses pairwise comparisons of different criteria to compute the relative weight of each criterion.  
The fuzzy set often uses triangular fuzzy numbers (TFNs) for operations. The operation rules of TFN are as follows.  
**Definition 1** ([38]).: _Considering that two TFNs are \(A_{1}=(l_{1},m_{1},u_{1})\) and \(B_{1}=(l_{2},m_{2},u_{2})\), the basic operational laws are as follows:_  
\[A_{1}\oplus B_{1}=(l_{1},m_{1},u_{1})+(l_{2},m_{2},u_{2})=(l_{1}+l_{2},\ m_{1}+m_{2},u_{1}+u_{2}) \tag{4}\]  
\[A_{1}\odot B_{1}=(l_{1},m_{1},u_{1})\odot(l_{2},m_{2},u_{2})=(l_{1}-u_{2},m_{ 1}-m_{2},\ u_{1}-l_{1}) \tag{5}\]  
\[{A_{1}}^{-1}=(l_{1},\ m_{1},u_{1})^{-1}=(1/u_{1},\ 1/m_{1},\ 1/l_{1}) \tag{6}\]  
where \(l_{1}\) and \(l_{2}\) denote the lower bounds of the TFN, \(m_{1}\) and \(m_{2}\) are the middle values of the TFN, and \(u_{1}\) and \(u_{2}\) are the upper bounds of the TFN.  
The calculation procedure of the FAHP approach is as follows [37].  
Step 1. Build the hierarchy structure of decision problems and determine criteria.  
Build the hierarchical structure based on the assessment questions, and then determine the assessment criteria.  
Step 2. Build the pairwise comparison matrix.  
The relative importance of the assessment criteria is evaluated through pairwise comparison. Table 1 shows a pairwise comparison of the relative importance of the evaluation criteria [39].  
Based on Table 1, experts determine the pairwise comparison matrix of the assessment criteria using their past experience.  
Step 3. Defuzzify the TFN values.  
\begin{table}
\begin{tabular}{c c c} \hline
**Relative Importance Scale** & **Linguistic Variable** & **TFN** \\ \hline
1 & Equally important & \((1,1,3)\) \\
3 & Weakly important & \((1,3,5)\) \\
5 & Strongly important & \((3,5,7)\) \\
7 & Very strongly important & \((5,7,9)\) \\
9 & Extremely important & \((7,9,9)\) \\
2, 4, 6, 8 & Between each semantic & \\ \hline \end{tabular}
\end{table}
Table 1: Relative importance of assessment criteria.  
Let \(A_{1}=(l_{1},m_{1},u_{1})\) be a TFN, then defuzzify the \(A_{1}\) value as follows [38].  
\[\text{DF}(A_{1})=\frac{1}{3}(l_{1}+m_{1}+u_{1}) \tag{7}\]  
Step 4. Perform expert opinion consistency check.  
The FAHP approach applies the consistency ratio (CR) to perform an expert opinion consistency check. The expert opinions are consistent if the CR value is less than \(0.1\). The CR value is calculated using the maximum eigenvalue (\(\lambda_{max}\)) of the pairwise comparison matrix (\(M\)), the random index (RI) value, and the consistency index (CI) value. Here, \(w\) denotes the relative weight of assessment criteria and \(n\) denotes the number of square matrices. Table 2 shows the RI values of different square matrices [3].  
\[Mw=\lambda_{max}w \tag{8}\]  
\[\text{CI}=\frac{\lambda_{max}-n}{n-1} \tag{9}\]  
\[\text{CR}=\frac{CI}{RI} \tag{10}\]