## 3 Research Design and Methodology, and Model Integration
### AHP Analysis
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Intensity of Importance** & **Definition** & **Explanation** \\ \hline
1 & Equal importance & Two activities contribute equally to the objective \\
2 & Weak or slight & Experience and judgement slightly favor one activity over another \\
3 & Moderate importance & Experience and judgement strongly favor one activity over another \\
4 & Moderate plus & Experience and judgement strongly favor one activity over another \\
5 & Strong importance & Experience and judgement strongly favor one activity over another \\
6 & Strong plus & An activity is favored very strongly over another \\
7 & Very strong or & An activity is favored very strongly over another; its dominance demonstrated in practice \\
8 & Very, very strong & The evidence favoring one activity over another is of the highest possible order of affirmation \\
9 & Extreme importance & The evidence favoring one activity over another is of the highest possible order of affirmation \\  & If activity \(i\) has one of the above non-zero numbers & A reasonable assumption \\ Reciprocals of above & compared with activity \(j\), then j has the reciprocal value when compared with \(i\) & May be difficult to assign the best value but when compared with other contrasting activities, the size of the small numbers would not be too noticeable, yet they can still indicate the relative importance of the activities. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: [PERSON]’s 9-point scale: scoring system and definitions for AHP analysis [115]. the characteristic that all elements in its main diagonal are \(1\). This reflects the fundamental principle that each element is equally important to itself. \[A=(a_{ij})= \begin{bmatrix}w_{1}/w_{1}&w_{1}/w_{2}&\cdots&w_{1}/w_{n}\\ w_{2}/w_{1}&w_{2}/w_{2}&\cdots&w_{2}/w_{n}\\ \vdots&\vdots&\cdots&\vdots\\ w_{n}/w_{1}&w_{n}/w_{2}&\cdots&w_{n}/w_{n}\end{bmatrix}  \tag{3}\]  
The column vector w, representing the relative importance of evaluation items in matrix \(A\), can be denoted as Equation (4). When w is multiplied by matrix \(A\), this relationship conforms to Equation (5). \[w= \begin{bmatrix}w_{1}\\ w_{2}\\ \vdots\\ w_{n}\end{bmatrix}  \tag{4}\]  
\[ \begin{bmatrix}w_{1}/w_{1}&w_{1}/w_{2}&\cdots&w_{1}/w_{n}\\ w_{2}/w_{1}&w_{2}/w_{2}&\cdots&w_{2}/w_{n}\\ \vdots&\vdots&\cdots&\vdots\\ w_{n}/w_{1}&w_{n}/w_{2}&\cdots&w_{n}/w_{n}\end{bmatrix} \cdot \begin{bmatrix} w_{1}\\ w_{2}\\ w_{n}\end{bmatrix} = \begin{bmatrix}nw_{1}\\ nw_{2}\\ \vdots\\ nw_{n}\end{bmatrix}  \tag{5}\]  
[PERSON] [116] suggested that by employing the eigenvalue method and using the principal eigenvector of the pairwise comparison matrix \(A\), this concept can be formulated as depicted in Equation (6). \[A\cdot w=\lambda_{\max}\cdot w \tag{6}\]  
where \(\lambda_{\max}\) refers to the maximum eigenvalue (principal eigenvalue) of matrix \(A\). Equation (6) represents an eigenvalue problem that seeks a non-zero solution within a system of n simultaneous linear equations. The solution obtained for \(w\) from Equation (6) is utilized as the weight vector for each evaluation criterion. The process for deriving matrix \(A\) through pairwise comparison involves setting the importance of elements in each column as \(1\) as a baseline, and then determining the relative importance of the elements above the diagonal. If the elements \(a_{ik}\) of matrix \(A\), obtained through pairwise comparison, each correspond to the ratio \(w_{i}/w_{j}\), cardinal consistency must be established, as demonstrated in Equation (7). \[a_{ij}\times a_{jk}=a_{jk} \tag{7}\]  
Equation (7) implies that if element \(i\) is deemed \(x\) times more important than \(j\), and \(j\) is \(y\) times more important than \(k\), then \(i\) should be evaluated as \(x\times y\) times more significant than \(k\). However, maintaining such cardinal consistency in actual survey responses is challenging. As a result, verifying the cardinal consistency of matrix \(A\) becomes essential.