## 3 Methodology
\[(A-\lambda_{max}\ I)w=0. \tag{3}\]  
While in AHP, decision makers most likely do not evaluate PCM-s consistently (for the evaluation, the Saaty scale is recommended, see Table 1) the eigenvector method can be used provided a consistency check has been conducted for the evaluations. Thus, for AHP applications the eigenvector method can be used for deriving the weight scores of the decision elements, however only in those cases in which the pairwise evaluations meet the inconsistency tolerance, measured by the Consistency Ratio. Note that the criterion of consistency must refer to both quality and cost type of attributes. Consequently, during the AHP process the consistency of answers must be examined by [PERSON]'s Consistency Index (CI) and Consistency Ratio (CR) [23; 24; 25]:  
\[CI=\frac{\lambda_{max}-n}{n-1}, \tag{4}\]  
where \(CI\) is the consistency index, _max_ is the maximum eigenvalue of the PCM, and \(n\) is the number of rows in the matrix. \(CR\) can be determined by:  
\[CR=\frac{CI}{RI}. \tag{5}\]  
RI is the random consistency index as demonstrated in Table 2. The threshold was also determined by [PERSON], if CR is below 0.1 (CR < 0.1), the PCM can be considered as acceptable from inconsistency point of view. Since most of the AHP applications involve several evaluators, the individual scores have to be aggregated, and avoiding rank reversal, the calculation of the geometric mean [31] of the respective evaluator scores is recommended for determining the aggregate results. If \"\(h\)\" evaluators exist in the procedure,  
\[f_{(x_{1},\ x_{2},\ldots,x_{h})=\sqrt[h]{\prod\limits_{k=1}^{h}xk}}, \tag{6}\]  
where \(x_{1},\ x_{2},\ \ldots,x_{h}\) denotes entries, in the same matrix position (i, j), of pairwise comparison matrices, filled in by the \(k-\)th decision maker. Having gained the aggregated matrices, calculating final weight vector scores considering hierarchical linkages within the decision is the next step in the procedure. It should be noted that the quality and also cost attributes must be considered as part of the hierarchy so their position in the hierarchy is a key point. As consistency has been acceptable, the final score determination can be applied as:  
\[w_{A_{i}}=\frac{w_{i}}{w}\frac{w_{ij}}{\sum\limits_{k=1}^{n}w_{ik}}=\bigg{(} \frac{w_{i}}{w}\frac{w_{i}}{\sum\limits_{k=1}^{n}w_{ik}}\bigg{)}w_{ij}, \tag{7}\]  
where \(j=1,\ldots\ldots,m\) and \(w=\sum\limits_{i=1}^{m}w_{i},\ w_{j}\ >\ 0\ (j=1,\ldots,m)\) represent the related weight coordinate from the previous level; \(w_{ij}>0\ (i=1,\ldots,n)\) is the eigenvector computed from the matrix in the current level, \(w_{A_{i}}\ (i=1,\ldots,n)\) is the calculated weight score of current level's elements. Sensitivity analysis enables understanding of the effects of changes in the main criteria on the sub criteria ranking and help the decision maker to check the robustness throughout the process. The steps of conducting AHP are clearly presented in Figure 1.  
In the current research, a new hierarchal structure model (Figure 2) was constructed and implemented in order to support the public bus transportation in Amman, Jordan. This is the first model in public transportation multi-criteria decision making where quality and cost issues are pairwise compared. As shown in Figure 2, the first level is more general, the second level is detailed, and the third one is more detailed and specific so the data can be increased essentially on a wide range of components in a complete manner, keeping the hierarchy. The supply quality elements were chosen to cover the qualitative factor such as perspicuity of timetable or safety of travel, as well as quantitative factors such as fare, awaiting time, and frequency of lines. The transport quality, tractability, fare, and service quality are the four main criteria to evaluate the supply quality service in the network.