# A Path Recommendation Method Considering Individual Driving Preferences
## 3 A Path Satisfaction Model Considering Individual Preferences
### Hierarchical Analysis Method
The traditional AHP achieves this objective by employing a numerical scale ranging from 1 to 9, as illustrated in Table 1. This scale allows for clear differentiation of value judgments, where the values 1, 3, 5, 7, and 9 represent distinct levels of importance. Intermediate values (e.g., 2, 4, 6, and 8) occupy fuzzy spaces, allowing for more nuanced assessments. The comparison matrix is constructed by conducting pairwise comparisons and assigning numerical scores to facilitate the conversion of qualitative problems into quantitative ones. For example, consider the target layer \(A\) and criterion layer \(B\). We defined the comparison matrix as follows:  
\[A= \begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{42}&\cdots&a_{nn}\end{bmatrix} . \tag{16}\]  
Then, the matrix \(A=\left(a_{ij}\right)_{n\times n}\) satisfies the following characteristics:  
\[\left\{ \begin{aligned} a_{ij}>0\\ a_{ji}=\frac{1}{a_{ij}},\\ a_{ii}=1\end{aligned} \right. \tag{17}\]  
where \(a_{ij}\) denotes the comparison outcome between the \(i\)th and the \(j\)th indicator. Typically, each expert constructs a comparison matrix. Therefore, prior to constructing the comparison matrix, an expert questionnaire should be prepared [27], allowing the experts to score the importance of the indicators using values on a scale of 1-9, based on their comparison. 3. Weighting solution  
Determine the maximum eigenvalue \(\lambda_{max}\) of matrix \(A\) and its corresponding eigenvector \(\mathbf{W}^{\prime}=\left(W_{1}^{\prime},W_{2}^{\prime},\cdots,W_{n}^{\prime}\right)\). Once the eigenvector \(W^{\prime}\) is obtained, its elements can be \begin{table}
\begin{tabular}{c c} \hline
**Scale** & **Meaning** \\ \hline
1 & The \(i^{\text{th}}\) indicator is as important as the \(j\)th indicator. \\
3 & The \(i^{\text{th}}\) indicator is slightly more important than the \(j\)th indicator. \\
5 & The \(i^{\text{th}}\) indicator is strongly more important than the \(j\)th indicator. \\
7 & The \(i^{\text{th}}\) indicator is very strongly more important than the \(j\)th indicator. \\
9 & The \(i^{\text{th}}\) indicator is extremely more important than the \(j\)th indicator. \\
2,4,6,8 & Intermediate values between the two adjacent scales from those listed above. \\ Reciprocals & Considering the \(i^{\text{th}}\) indicator relative to the \(j^{\text{th}}\) indicator, the comparison value of \\  & the first indicator is \(a_{ij}\), while the comparison value of the second indicator is \(1/a_{ij}\). \\ \end{tabular}
\end{table} 
Table 1: The comparative scale values and their meanings. normalized to derive \(W_{AB}=(W_{1},W_{2},\cdots,W_{n})\), representing the weight values of each criterion in the target layer \(A\) relative to the criterion layer \(B\). 4. Consistency identification  
The initial comparison values within each layer's comparison matrix serve as the foundation for subsequent calculations in the AHP. In theory, as the number of criteria to be evaluated increases, the potential for conflicting pairwise comparison outcomes also increases. This arises because the scoring process only considers pairwise relationships between indicators and fails to capture the overall relationships among all criteria. For instance, contradictory results such as \(A>B\), \(B>C\), and \(A<C\) may arise. Hence, it becomes essential to conduct a consistency identification on the eigenvector. The consistency identification ensures that the results obtained through constructing the AHP and its comparison matrix are valid, provided the consistency criteria (\(CR<0.1\)) are met. If the consistency check results do not meet the criteria, it is necessary to reconstruct the comparison matrix. The consistency check is performed using Formulas (18) and (19). Consistency Index \(CI\):  
\[CI=\frac{\lambda_{max}-n}{n-1}, \tag{18}\]  
where \(\lambda_{max}\) is the maximum eigenvalue of \(A\).