# User Satisfaction Survey on Public Transport by a New PAHP Based Model
## 3 Methodology
### The Proposed Multi-Level Parsimonious AHP Technique  
Let us have \(n\) criteria structured in a decision problem into \(m\) levels. Select the \(k\)-th level of the structure to be the Parsimonious level of the model. We suggest selecting the level which has more than nine criteria. Let us denote \(j\) the criteria on the selected level of the decision so \(c_{kj}\) denotes a criterion on the Parsimonious level in which if we have \(h\) criteria, \(j=1\),..., \(h\).  
The reason for selecting that level(s) which has \(h\geq 9\) is that it can be considered as a sufficient number of criteria worth for unburdening the evaluators from numerous pairwise comparisons. Moreover, we propose (following [PERSON]'s 7 \(\pm\) 2 rule ([PERSON] and [PERSON], 1983 [41]) for a PCM) to select level(s) for which larger or equal to \(5\times 5\) pairwise comparison matrices should be evaluated. As recommended protocol, the pair wise comparisons for a \(5\times 5\) matrix might be demanding for layman evaluators.  
In the second step, direct evaluations have to be made for the chosen level(s) for the \(c_{kj}\) criteria with respect to the goal of the decision problem on a scale from 0 to 1 or equivalent, e.g., 0-100. By their normalization, we gain the \(\gamma\left(c_{kj}\right)\) values. Please, see the next, case study section. Having gained the normalized values of the criteria, we set up an increasing order and denote the criteria in this order by \(c_{kp}\), where \(p\) denotes the new position of the criterion, \(p=1\),..., \(h\).  
Then, \(t_{p}\) reference elements have to be selected on the chosen level(s) \(k\) based on their new order; we recommend \(t_{p}=3\) and to select \(c_{k1}\), \(c_{kl/2}\) and \(c_{kl}\). Thus, \(s=\left(c_{k1},\ c_{kl/2},\ c_{kl}\right)\), and more generally, \(s=\left(s_{1},\ s_{2},\ \ldots,\ s_{tp}\right)\). Note that in case of a significantly larger \(h\), the number of selected elements can be larger up to 4 or even 5.  
Afterwards, the original AHP pairwise comparisons are conducted for the chosen \(t_{p}\) number of criteria, obtaining the normalized AHP scores for the s criteria: \(u(c_{ks})\), for all \(s=\left(c_{k1},\ c_{kl/2},\ c_{kl}\right)\). Please see the next, case study session. The normalized results of the AHP calculation are denoted by \(u\).  
Following the PAHP procedure by [PERSON] et al. [39], consistency and monotonicity are being checked, and the required modifications are made.  
Finally, the following formula is applied for all criteria (\(c_{kp}\)) existing on the Parsimonious level(s) using the following:  
\[u\left(c_{kp}\ \right)=u(c_{kr})+\frac{u\ (c_{kr+1})-u\ (c_{kr})}{\gamma_{ kc+1}-\gamma_{kc}}\ \left(\tau_{kp}-\gamma_{kc}\right). \tag{6}\]  
with respect to \(e=1\),..., \(t_{p}\), and \(c_{kp}\) has the importance between the two reference criteria \(c_{kr}\) and \(c_{kr+1}\); thus, \(\tau_{kc}<\tau_{kp}<\tau_{kc+1}\ (e=p-1)\). In addition, \(\gamma_{kc+1}\) and \(\gamma_{kc}\) are the normalized direct evaluations of the reference criteria and \(\tau_{kp}\) the normalized direct evaluation of the criterion of interest. Note that the symbol \(u\) denotes the normalized AHP value in all cases. Please, see the next, case study section.  
Having finished with the Parsimonious level(s), the decision structure should be reconstructed in order to gain the final weight and alternative scores and ranking. Consequently, all \(u\left(c_{kp}\right)\)-s have to be multiplied by the weight score of their respective element from the previous level \(k\)-1. Moreover, due to the characteristics of AHP, for the lower levels, the new \(u\left(c_{kp}\right)\) weight scores have to be applied for multiplying the scores of the respective lower elements.  
For better understanding, we provide the proposed general flow chart of the multi-level PAHP surveys (Figure 1).