## 3 Decision Case Modeling and the Comparisons: Methods and the Results
### Alternative Ranking/Selection Using AHP-TOPSIS
\begin{array}{c}0.15568452\\ 0.02337814\\ 0.11160558\\ 0.05413003\\ 0.42072456\\ 0.06787833\\ 0.16659885\end{array} \right]\text{and }A^{-}=\left[ \begin{array}{c}0.05189484\\ 0.01753360\\ 0.04783096\\ 0.02397804\\ 0.24748504\\ 0.04525222\\ 0.07080747\end{array} \right]. \tag{13}\]  
It is seen that \(A^{+}\) is identical to \(C\!V\!V^{*}\) (while \(A^{-}\) is not). This finding is discussed extensively on a theoretical basis in Section 4.1. The third phase of TOPSIS measures the distance from each alternative to the ideal (i.e., the \"separation measures\") and the distance to the anti-ideal. The process uses the following logic:  
\[S^{\text{to}\_A^{+}}\leftarrow\text{vector(initialize with dimension }\mathit{N\!alts}\times 1); \tag{14}\]  
\[S^{\text{to}\_A^{-}}\leftarrow\text{vector(initialize with dimension }\mathit{N\!alts}\times 1); \tag{15}\]  
\[S^{\text{to}\_A^{+}}[i,1]\ \leftarrow\ \sqrt{\sum_{j}^{\mathit{Ncrit}} \left(WiNDMa[i,j]-A^{+}[j,1]\right)^{2}},\ \forall\ \text{integer}\ i\ \in\ \{1\dots\mathit{N\!alts}\}; \tag{16}\]  
\[S^{\text{to}\_A^{-}}[i,1]\ \leftarrow\ \sqrt{\sum_{j}^{\mathit{Ncrit}} \left(WiNDMa[i,j]-A^{-}[j,1]\right)^{2}},\ \forall\ \text{integer}\ i\ \in\ \{1\dots\mathit{N\!alts}\}; \tag{17}\]  
Applying this logic to the dataset, two separation measure vectors are obtained as follows:  
\[S^{\text{to}\_A^{+}}=\left[ \begin{array}{c}0.18931705\\ 0.14815937\\ 0.14437979\\ 0.20072201\\ 0.13475571\\ 0.08288226\\ 0.12485745\\ 0.15618098\end{array} \right]\text{and }S^{\text{to}\_A^{-}}=\left[ \begin{array}{c}0.06669622 \\ 0.11433725\\ 0.11295291\\ 0.06602163\\ 0.11291660\\ 0.19251697\\ 0.15902443\\ 0.12107992\end{array} \right]. \tag{18}\]  
The relative closeness index (RCI) for each alternative is then calculated to allow the final ranking to proceed. This is the fourth phase of TOPSIS. An alternative has a higher score for this index if it is relatively close to the ideal solution and far from the anti-ideal. This is equivalent to determining the separation from the anti-ideal solution over the entire distance for which it is separated from both the ideal and the anti-ideal solutions:  
\[RCI\leftarrow\text{as vector(initialize with dimension }\mathit{N\!alts}\times 1); \tag{19}\]  
\[RCI\ [i,1]\gets S^{\text{to}\_A^{-}}/(S^{\text{to}\_A^{+}}+S^{\text{to} \_A^{-}}); \tag{20}\]  
Substituting Equation (20) using the information into Equation (18), the \(8\times 1\)_RCI_ vector (_RCIV_), which carries information about the alternatives' relative proximity to the ideal, is obtained. This is another index score vector obtained by AHP-TOPSIS, as shown in the second column in Table 5.  
This _RCIV_ is further used to determine the ranks of the alternatives as a vector, which is \(R\!V\! This ROV also gives another preferential order over the alternatives as suggested by AHP-TOPSIS:  
\[\text{A6}\succ\text{A7}\succ\text{A5}\succ\text{A3}\approx\text{A8}\approx \text{A2}\succ\text{A1}\succ\text{A4}. \tag{21}\]  
If these alternatives are strictly ordered without considering the durable indifference, the preferential order is:  
\[\text{A6}\succ\text{A7}\succ\text{A5}\succ\text{A3}\succ\text{A8}\succ\text{ A2}\succ\text{A1}\succ\text{A4}. \tag{22}\]