# An Integrated Decision Support Model for Evaluating Public Transport Quality
## 2 Materials and Methods
### Overview on Best Worst Method (BWM)  
The general model of BWM was customized to the decision problem of improving public bus transport. In our research, no alternatives were applied; only one matrix (\(5\times 5\)) in the second level of the hierarchy structure was evaluated since we relied on three different group considerations and did not strive to influence their judgments by any alternative selection. \begin{table}
\begin{tabular}{c l} \hline \hline
**Numerical Values** & **Explanation** \\ \hline
1 & Elements are equally importance \\
3 & Element is more important over another. \\
5 & One element is strongly important than the other one \\
7 & One element is very strongly important than the other one \\
9 & One element is favored by at least an order of magnitude \\
2, 4, 6, 8 & Used to compromise between two judgments \\ \hline \hline \end{tabular}
\end{table} 
Table 2: [PERSON]’s determination scale for relative weight score of elements [36]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(m\) & **1** & **2** & **3** & **4** & **5** \\ \hline
**RI** & 0 & 0 & 0.58 & 0.9 & 1.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average of the consistency indices (RI) from randomly generated matrices. To provide an overview of all stages for the recent survey, we defined the phases in the following order. **Step 1**: Identify the main and submain criteria for public bus transport supply quality. **Step 2**: Define the most important (MI) and the less important (LI) criteria by simple scoring of the participants. **Step 3**: Evaluate the pairwise comparisons between MI criteria by using a 1-9 scale, where 1 is \"equal importance\" and 9 means \"extremely more important.\" The result of this step is represented by the following vector:  
\[V_{B}=(v_{B1},\ v_{B2},\ \ldots,\ v_{Bn}) \tag{6}\]  
where \(v_{Bj}\) is the preference of criterion \(B\) (the most important or the best) over the criterion \(j\) and \(v_{BB}=1\). In our model \(n=5\), since we have five criteria to compare, \(j=(1,\ 2,\ldots,\ 5)\). **Step 4**: Make the pairwise comparisons between LI criteria by using the 1-9 scale. The result of this step is represented by the following vector:  
\[V_{W}=(v_{1W},\ v_{2W},\ \ldots,\ v_{nW})^{T} \tag{7}\]  
where \(v_{jW}\) is the preference of criteria \(j\) (the most important or the best) over the criteria \(W\), and \(v_{WW}=1\). Since we have five criteria to compare in our model, \(j=(1,\ 2,\ldots\ 5)\), then \(n=5\). **Step 5**: Calculate the final optimal weight scores (\(s_{1}^{*},\ s_{2}^{*},\ \ldots,\ s_{n}^{*}\)) of the criteria, and the indicator of the optimal consistency of comparisons \(\xi^{*}\). The maximum absolute difference must be minimized:  
\[ \begin{array}{l}\min\ max_{j}\left\{\left|\frac{s_{B}}{s_{j}}-v_{Bj}\right| \left|\frac{s_{j}}{s_{W}}-v_{jW}\right|\right\}\\ \sum_{j}^{s.t.}W_{j}=1,\ W_{j}\ \geq 0,\ \text{for all}\ j\end{array}  \tag{8}\]  
After that, the solution can be obtained by solving the linear programming (LP) formula:  
\[ \begin{array}{l}\min\ \xi^{*}\\ \text{s.t.}\\ \left|\frac{s_{j}}{s_{W}}-v_{jW}\right|\leq\ \xi^{*},\ \text{for all}\ j\\ \sum\limits_{j}s_{j}=1\\ s_{j}\geq 0,\ \ \text{for all}\ j\end{array}  \tag{9}\]  
The following formula computes the consistency ratio to check the consistency of the comparisons [37]:  
\[\text{Consistency Ratio}=\frac{\xi^{*}}{\text{Consistency Index}} \tag{10}\]  
The consistency index (CI), presented in Table 4, is gained by random experiments for a different number of comparisons [38]. Consistency Ratio (CR) is acceptable in BWM methodology for the case when its value is between 0 and 1. This ratio can be calculated for individual evaluators, or for groups when the scores of the group are first aggregated by creating the geometric mean of the scores and then conducting the consistency check. In our case, the CR was checked individually.