## 3 Decision Case Modeling and the Comparisons: Methods and the Results
### Verifying the Similarity between the Two ROVs Using the Proposed Comparative Research Flow  
From Equations (6) and (22), the preferential orders obtained individually using AHP-GTMA and AHP-TOPSIS are, respectively:  
Using AHP-GTMA: A7 \(\succ\) A8 \(\succ\) A6 \(\succ\) A5 \(\succ\) A3 \(\succ\) A2 \(\succ\) A4 \(\succ\) A1. Using AHP-TOPSIS: A6 \(\succ\) A7 \(\succ\) A5 \(\succ\) A3 \(\succ\) A8 \(\succ\) A2 \(\succ\) A1 \(\succ\) A4. At a glance, these two preferential orders are not so consistent, but further scrutiny can be made to give more insights to such an observation. Therefore, the systematic yet scientific comparative research flow proposed in Section 3.1 takes place. Phase (1) of the flow is to calculate the absolute distance in rank for each alternative directly based on the ordinal rank values, while the aggregated absolute distance is a similarity measure. In Tables 3 and 5, the two ROVs that are individually solved by the two models are:  
\[ROV_{\text{AHP-GTMA}}\text{[A1 A2 A3 A4 A5 A6 A7 A8]}=\text{[8 6 5 7 4 3 1 2]}; \tag{23}\]  
\[ROV_{\text{AHP-TOPSIS}}\text{[A1 A2 A3 A4 A5 A6 A7 A8]}=\text{[7 6 4 8 3 1 2 5]}. \tag{24}\]  
According to Equation (1), the difference between these two ROVs is calculated as:  
\[Diff=\text{[A1 A2 A3 A4 A5 A6 A7 A8]}=\text{[1 0 1 - 1 1 2 - 1 - 3]}. \tag{25}\]  
Furthermore, the absolute distance in rank for each alternative is assessed in terms of the TV vector as:  
\[TV=\text{[A1 A2 A3 A4 A5 A6 A7 A8]}=\text{[1 0 1 1 1 2 1 3]}. \tag{26}\]  
According to Equation (2), the aggregate absolute distance for these two ROVs, which is the result of applying the pure arithmetic similarity measure, is the summation of the elements in TV:  
\[AD=1+0+1+1+1+2+1+3=10. \tag{27}\]  
Phase (2) is to measure the geometrical distance between the two ROVs in the vector space. As in the Euclidean methods, the absolute distances in rank, \(TV\), can be treated directly as a Euclidean vector, so the Euclidean distance (GD) value is determined, as:  
\[\text{GD}=(1+0+1+1+1+2^{2}+1+3^{2})^{1/2}=(18)^{1/2}. \tag{28}\] \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{A_{\_}i}\) & **RCI Score Vector** & **Rank** \\ \hline A1 & 0.2605186 & 7 \\ A2 & 0.4355761 & 6 \\ A3 & 0.4389373 & 4 \\ A4 & 0.2475097 & 8 \\ A5 & 0.4559113 & 3 \\ A6 & 0.6990469 & 1 \\ A7 & 0.5601782 & 2 \\ A8 & 0.4367003 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The RCI scores assessed by TOPSIS for the alternatives and the rank orders. This is the result of applying the plausible geometrical-based similarity measure. Let us briefly discuss these observations before going on. It is initially seen that, in Equations (23) and (24), one alternative (A2) has the same rank by using both methods. However, in Equation (26), five alternatives (i.e., A1, A3, A4, A5, A7) differ in rank by only one place. Only two of them differ in rank by more than one place: the rankings for A6 differ by two places and those for A8 differ by three. It is seen that the rank orders for both methods are quite similar. _Six alternatives_ (i.e., A1, A2, A3, A4, A5, and A7) of the eight _differ in rank by no more than one place_, so _75% of the alternatives obtained very similar ranks by both methods_. For the two _outliers_ (A6 and A8), the _\"maximum deviation\"_ in the ranking is three places (for A8), which is still not large. There is also supporting evidence in that _the Euclidean distance between these two seven-dimensional ROVs is only 18_ from Phase (2). These are the results of using tangible measures for the first two phases of the proposed flow. However, as using these tangible measures there could be similarity, the intangible statistics measure of Phase (3) is used to confirm the similarity found between these two ROVs. This study views the two ROVs as statistical samples and uses the non-parametric Wilcoxon signed-rank test function (as also offered by R) to test whether they are (from) non-identical populations, as they are in fact \"ordinal variables\" from the perspective of data analytics.