# An Integrated Decision Support Model for Evaluating Public Transport Quality
## 2 Materials and Methods
### Overview of the Conventional Analytic Hierarchy Process (AHP)  
The hierarchical decision structure is the foundation of the Analytical Hierarchy Process (AHP), which is constructed from complex decision elements [30; 31]. The hierarchical data structure generally consists of multilevels. The multilevels concept is a general structure for the hierarchy. The application was used to evaluate several transportation complex problems [32; 33; 34; 35]  
The importance of the linkages among the criteria in the different levels are determined by a global score for the criteria in the last level. The main steps of the conventional AHP follow:  
* Set up the hierarchical structure of the decision problem. From the top, with the ultimate goal of the decision, then the objectives, through the intermediate levels (criteria on which subsequent elements depend), to the lowest level (which usually is a set of the alternatives). (Please see Figure 1 as an example for a decision hierarchy. This structure was used in this case study.)
* Construct the pairwise comparison matrices based on the hierarchy.
* For the comparisons, a questionnaire needs to be prepared.
* Check the consistency.
* Aggregate by the geometric mean.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Variables** & **Passengers** & **Public Transport Experts** & **Municipal Officials** \\ \hline N & 200 & 5 & 5 \\ Gender (\%) & & & \\ \hline Male & 51 & 100 & 40 \\ Female & 49 & 0 & 60 \\ \hline \end{tabular}
\end{table}
Table 1: Pattern characteristics of participated evaluators.  
* Weight vectors need to be derived.
* Calculate the global scores.
* Perform a sensitivity analysis.  
The pairwise comparison matrices (PCMs) are a positive square matrix \(A_{x}\), where \(x_{ij}>0\) is the subjective ratio between \(w_{i}\) and \(w_{j}\), and \(w_{x}\) is the weight score from \(A_{x}\). The following equation can calculate the eigenvector method of [PERSON], which defined for the PCMs:  
\[A.w_{x}=\lambda_{max}.w_{x}\iff w_{x}(A-\lambda_{max}.I) \tag{1}\]  
where the maximum eigenvalue of the \(A\) matrix is \(\lambda_{max}\).  
Figure 1: The hierarchical model of the public bus transportation supply quality criteria [21].  
The structure of (\(5\times 5\)) consistent theoretical PCMs is defined in (2):  
\[\left[\begin{array}{cccccc}w_{1}/w_{1}&w_{1}/w_{2}&w_{1}/w_{3}&w_{1}/w_{4}&w_{1 }/w_{5}\\ w_{2}/w_{1}&w_{2}/w_{2}&w_{2}/w_{3}&w_{2}/w_{4}&w_{2}/w_{5}\\ w_{3}/w_{1}&w_{3}/w_{2}&w_{3}/w_{3}&w_{3}/w_{4}&w_{3}/w_{5}\\ w_{4}/w_{1}&w_{4}/w_{2}&w_{4}/w_{3}&w_{4}/w_{4}&w_{4}/w_{5}\\ w_{5}/w_{1}&w_{5}/w_{2}&w_{5}/w_{3}&w_{5}/w_{4}&w_{5}/w_{5}\end{array}\right] \tag{2}\]  
Reciprocity is applied for each pairwise comparison matrix \(x_{ji}=1/xij\), where \(x_{ii}=1\) is ensured. However, consistency is probably not performed for empirical matrices. The consistency criterion is  
\[x_{ik}=x_{ij}.x_{jk} \tag{3}\]  
The questionnaire surveys were filled out by three decision-maker groups, taking into consideration the evaluated system factors related to public transportation established upon a [PERSON]'s scale (Table 2).  
Eigenvector methodology contains inconsistency in the empirical matrices despite filled by the evaluators.  
To have more robust outcomes, a consistency check was conducted by Saaty to examine the consistency of the PCMs:  
\[\text{CI}=\frac{\lambda_{max}-m}{m-1} \tag{4}\]  
where CI is the Consistency Index, the maximum eigenvalue of the PCM is \(\lambda_{max}\), and \(m\) is the number of rows in the matrix. The following equation computes the consistency ratio (CR):  
\[\text{CR}=\text{CI}/\text{RI} \tag{5}\]  
where the average of the consistency index (RI) and its values are presented in Table 3. The agreeable value of CR in the AHP approach is CR \(<0.1\).  
Sensitivity analysis enables in perceiving the effects of alternates in the main element on the subelement ranking, and helps the decision maker to check the stability of results throughout the process.