# User Satisfaction Survey on Public Transport by a New PAHP Based Model
## 3 Methodology  
### Conventional AHP  
AHP questionnaires are structured by the decision hierarchy created in the initial phase of the research. In the hierarchy, the general decision elements (attributes or criteria) are on the top level and the more specific elements are on lower levels; usually the level of alternatives closes the structure as a flat level. The general and more specific attributes are in set-subset relation, and belonging is demonstrated through branches of the decision tree. Based on the structure of the decision tree, pairwise comparisons have to be made branch wisely, so those attributes should be pair wisely compared which ones are on the same level and belong to the same branch. Thus, in the questionnaire, a special type of matrices (Pairwise Comparison Matrices, PCM-s) is created for the pairwise comparisons. For \(n\) attributes to be compared, the structure of a Pairwise Comparison Matrix (Table 1) is needed, and \(w_{n}\) is the relative importance.  
From our research point of view, it is important to emphasize that due to the reciprocity of all PCM-s (\(a_{ji}=\frac{1}{a_{ji}}\), where \(a_{ii}=1\)), the evaluators merely have to fill the brackets above the main diagonal. Consequently, for evaluating an \(n\) number of attributes matrix, \((\frac{n}{2})\) so \(n(n-1)\)/2, questions have to be answered, since all possible pairs are selected out of the \(n\) attributes neglecting the direction of the relation due to the reciprocal characteristics of the matrices. In the AHP process, participants are asked to estimate the relative importance of an attribute over another; for the \(\frac{w_{1}}{w_{2}}\) elements, the reflective question is: please compare, how much more important is Attribute 1 than Attribute 2 in the decision.  
Having gained all required estimations from the evaluators, the researchers calculate the weight scores of the attributes applying [PERSON]'s eigenvector method (we note that there  
\begin{table}
\begin{tabular}{c c c c c} \hline \(\frac{w_{1}}{w_{1}}\) & \(\frac{w_{1}}{w_{2}}\) & \(\frac{w_{1}}{w_{3}}\) & \(\cdots\) & \(\frac{w_{1}}{w_{n}}\) \\ \(\frac{w_{2}}{w_{1}}\) & \(\frac{w_{2}}{w_{  
exist other techniques for calculation, for instance the Least squares method). It is based on the assumption that the estimated scores are similar to consistent evaluations, and thus,  
\[\mathbf{A}\;w=\;\lambda_{\max}\;w \tag{1}\]  
is fulfilled based on the Perron theorem. Here \(\mathbf{A}\) is the PCM; w is the calculated eigenvector, and \(\lambda_{\max}\) is the maximum eigenvalue of \(\mathbf{A}\). Since we are looking for the eigenvector, it can be gained by solving:  
\[(\mathbf{A}-\;\lambda_{\max}\cdot\mathrm{I})\;w=0 \tag{2}\]  
Obviously, the ratings of the evaluators are most likely not consistent, so their consistency should be measured, and the non-tolerably inconsistent evaluations should be omitted. Consistency is generally checked by the Consistency Ratio (CR) created by [PERSON] [41] (Table 2):  
\[CR=CI/RI \tag{3}\]  
\[CI=(\lambda_{max}-n)/(n-1) \tag{4}\]  
\(RI\) is an empirical number of testing random PCM-s with the values.CR is acceptable when its value is lower than 0.1.  
For multiple evaluators, AHP surveys apply definitely multiple decision makers; first, the individual evaluations have to be aggregated by the geometric mean (arithmetic mean or other aggregation techniques are proven to be less effective by [PERSON] and [PERSON] [41]):  
\[\mathbf{A}=\left[\sum_{i=1}^{h}a_{ijk}\right]\;i,j=1,\ldots,n. \tag{5}\]  
where \(a_{ijk}\) denotes entries, in the same position (\(i,j\)), of pairwise comparison matrices, filled in by the \(k\)-th decision maker; \(h\) is the number of total evaluators, and \(\mathbf{A}\) is the aggregated matrix. Formula (5) indicates that exactly the same positioned brackets filled by individuals are multiplied and rooted, and the gain values will construct an aggregated PCM (\(\mathbf{A}\)), for which the eigenvector calculations can be fulfilled.