## 3 Decision Case Modeling and the Comparisons: Methods and the Results
### Alternative Ranking/Selection Using AHP-TOPSIS  
The decision problem \(P\) is solved using AHP-TOPSIS. Usually, the first step in applying an AHP-TOPSIS involves loading the source decision matrix (i.e., the non-normalized \(D\)), identifying a suitable normalization method according to the problem context and then obtaining the normalized decision matrix, \(D^{\prime}\). However, the dataset \(D^{\prime}\) has been recomputed in Section 3.2 and is thus available. According to the data specification as defined in Section 3.1 (i.e., dataset \(F\)), another solution algorithm for AHP-TOPSIS (which is also designed and implemented in R) initially loads the \(D^{\prime}\) matrix and the \(C\!W\!V^{\prime}\) vector as the data input and uses them to calculate the \"weighted normalized decision matrix\" (named as the _WiNDMa_). This involves the following calculations semantically:  
\[C\!W\!V\!_\!M\leftarrow\mathrm{columnwise\_replicate}(C\!W\!V^{\prime}\!,\!N \!crit); \tag{7}\]  
\[WiNDMa\gets D^{\prime}\cdot C\!W\!V\!_\!M; \tag{8}\]  
where columnwise_replicate(\(m\),\(n\)) is a function replicating vector \(m\) for \(n\) columns, as offered by any software utility and \"\(\cdot\)\" is the standard matrix multiplication operator for linear algebra. For the datasets in Section 3.2, \(C\!W\!V^{\prime}\) is a 7 \(\times\) 1 vector (matrix). Compiling and binding seven of this type of vector using columnwise_replicate() gives a 7 \(\times\) 7 matrix, \(C\!W\!V\). Multiplying the 8 \(\times\) 7 \(D^{\prime}\) by the 7 \(\times\) 7 CWW gives the 8 \(\times\) 7 _WiNDMa_ matrix, as shown in Table 4.  
The most critical element of TOPSIS is to identify the ideal and anti-(negative-)ideal solutions from _WiNDMa_, which are denoted by \(A^{+}\) and \(A^{-}\). This involves the following algorithmic logic:  
\[A^{+}\leftarrow\text{as vector(initialize with dimension }N\!crit\!\times\!1); \tag{9}\]  
\[A^{-}\leftarrow\text{as vector(initialize with dimension }N\!crit\!\times\!1); \tag{10}\]  
\[A^{+}\left[j,1\right]\leftarrow\max(WiNDMa\left[1\,\dots\,\mathit{Nalts }\!,j\right]),\,\forall\,j\in\{1\,\dots\,Ncrit\}; \tag{11}\]  
\[A^{-}\left[j,1\right]\leftarrow\min(WiNDMa[1\,\dots\,\mathit{Nalts }\!,j]),\,\forall\,j\in\{1\,\dots\,Ncrit\}; \tag{12}\]  
Applying these equations to _WiNDMa_, the ideal and anti-ideal alternatives for the decision problem, both being virtual, are identified as: \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Alts.\(\times\)Attrib.** & **c1** & **c2** & **c3** & **c4** & **c5** & **c6** & **c7** \\ \hline A1 & 0.05189484 & 0.01753360 & 0.07971827 & 0.03433167 & 0.30051750 & 0.06787833 & 0.07080747 \\ A2 & 0.10378968 & 0.01753360 & 0.11160558 & 0.02438196 & 0.32363430 & 0.06787833 & 0.07219283 \\ A3 & 0.10378968 & 0.01753360 & 0.10863593 & 0.02411780 & 0.32363430 & 0.06787833 & 0.07838198 \\ A4 & 0.10378968 & 0.01753360 & 0.07242396 & 0.03519999 & 0.24748500 & 0.06787833 & 0.09139847 \\ A5 & 0.10378968 & 0.02337814 & 0.04783096 & 0.05413003 & 0.32363430 & 0.04525222 & 0.12838416 \\ A6 & 0.10378968 & 0.01753360 & 0.06377462 & 0.04659033 & 0.42072460 & 0.04525222 & 0.13074293 \\ A7 & 0.15568452 & 0.2337814 & 0.09656527 & 0.02397804 & 0.30051750 & 0.06787833 & 0.16452456 \\ A8 & 0.10378968 & 0.01753360 & 0.04783096 & 0.04896363 & 0.28816750 & 0.06787833 & 0.16659885 \\ \hline \end{tabular}
\end{table} 
Table 4: The weighted normalized decision matrix (matrix _WiNDMa_). \begin{table}
\begin{tabular}{c c c} \hline
**A\({}_{i}\)** & **Index Score (PermRecur(\(C_{i}\)))** & **Rank** & **Reference** \\ \hline A1 & 9663.529 & 8 & 9663.533 \\ A2 & 10,430.966 & 6 & 10,430.86 \\ A3 & 10,431.094 & 5 & 10,431.03 \\ A4 & 10,409.697 & 7 & 10,049.56 \\ A5 & 10,534.207 & 4 & 10,534.19 \\ A6 & 10,571.709 & 3 & 10,571.69 \\ A7 & 11,654.630 & 1 & 11,654.58 \\ A8 & 10,576.246 & 2 & 10,576.28 \\ \hline \end{tabular}
\end{table} 
Table 3: The index scores and rank orders assessed by GTMA for the alternatives. \[A^{+}=\left[