## 3 Decision Case Modeling and the Comparisons: Methods and the Results
### Comparative Research Flow
The _absolute distance in rank_ can be calculated directly based on the ordinal rank values individually solve by the two models, while the _aggregated absolute distance_ is a similarity measure in the arithmetic sense. Even though an ordinal rank is not ratio-scaled, the different values of rankpreserve a meaningful order. Suppose that X is the MADM model which is to be benchmarked and Y is the MADM model used to benchmark X; these processes are expressed as: \[TV_{m\times 1}=\left[ \begin{array}{c}tv_{1}\\tv_{2}\\ \vdots\\tv_{m}\end{array} \right]=\left|ROV_{\text{X}}=\left[ \begin{array}{c}rovx_{1 }\\ rovx_{2}\\ \vdots\\ rovx_{m}\end{array} \right]_{m\times 1}-ROV_{\text{Y}}\left[ \begin{array}{c}rovy_{1 }\\ rovy_{2}\\ \vdots\\ rovy_{m}\end{array} \right]_{m\times 1}\right|_{m\times 1}\] (1) \[\forall i=1,2,..,m,\ rovz_{i}=1,2,..,m\text{ and }\forall i,j\in\{1,2,..,m\},i \
eq j\Rightarrow rovz_{i}\
eq rovz_{j}\left|z\in\{\text{X},\text{Y}\}\right.\] \[AD_{ROVs}=\sum_{i=1}^{m}tv_{i}\] (2) where \(i\) is the index for the alternatives; \(z\) is the index for the methods; \(TV\) is a temporary vector which constrains the element-wise absolute distances between the rank order values assessed by the two methods; and \(AD\) is the aggregated absolute distance, which is the scalar column sum of \(TV\). 2. The _geometrical distance_ can be computed by treating the two valued ROVs as vectors in the space and using the Euclidean method. Thus, the vector \(TV\) in Equation (1) is treated as the Euclidean vector, based on which the Euclidean distance is computed according to the following formula to have another similarity measure in geometrical sense: \[GD=\left(\sum_{i=1}^{m}tv_{i}\right)^{1/2}\] (3) Another _statistical similarity measure_ can be established to confirm and cross-validate the results. This will be especially useful when the two ROVs are found to be very similar or at least similar to some extent in terms of the two previous arithmetic and geometric measures. With this purpose mind, we propose to treat the two ROVs and apply the non-parametric sign rank test in the statistics field to establish this similarity measure. The sign rank test determines whether the two ROVs (not as vectors but as statistical variables) are (from) non-identical populations. Theoretically, using this test here is suitable because the input to such a non-parametric test, i.e., the ROVs, is not ratio-scaled but in the meantime, the ordinal relationship is kept. Thus, if after the test it is shown that these two ROVs are not from non-identical populations and it is far from the evidence to support that they are so (from non-identical populations), this statistical-based observation can be used to confirm the other two similarities which have been observed arithmetically or geometrically. Regardless, setting up such an additional intangible similarity measure is a novelty of this study. To utilize the existing initial real case for paper shredder selection decision in [4] as the exemplar problem of this study (i.e., \(P\)), the following subsections begin with the collection of the required datasets (i.e., the union set \(E\cup F=\{D^{\prime},\ B,\ CWV\}\)). Problem \(P\) is then solved using AHP-GTMA and AHP-TOPSIS, respectively, while two ROVs are obtained. This is followed by using the above systematic comparative research flow to determine the similarity between them in terms of the different measures. As a result, \(ROV_{\text{AHP-GTMA}}\) is shown to be very similar to \(ROV_{\text{AHP-GTMA}}\). Using these processes for illustration, the involved methods for modeling and comparisons are self-explanatory.