## 3 Methodology
### Multi-Criteria Analysis
#### 3.2.3 Step 3  
FTOPSIS [70] is then used to weigh and rank the enabling elements of the digital infrastructure pillars acting as alternatives to the MCA. The fuzzy decision matrix for the alternatives is shown in Equation (4), where \(\widetilde{D}=\left[\widetilde{X}_{ij}\right]\). \begin{table}
\begin{tabular}{c c c c} \hline
**Linguistic Judgement for AHP/FAHP** & **Comparison** & & \\
**Pairwise Comparisons** & **Number** & & \\ \hline Equally Important & 1 & (1,1,1) & (1,1,1) \\ Intermediate Value of Relative Importance & 2 & (1,2,3) & (1/3,1/2,1) \\ Between 1 and 3 & (2,3,4) & (1/4,1/3,1/2) \\ Intermediate Value of Relative Importance & 4 & (3,4,5) & (1/5,1/4,1/3) \\ Between 3 and 5 & (4,5,6) & (1/6,1/5,1/4) \\ Intermediate Value of Relative Importance & 6 & (5,6,7) & (1/7,1/6,1/5) \\ Between 5 and 7 & (6,7,8) & (1/8,1/7,1/6) \\ Intermediate Value of Relative Importance & 8 & (7,8,9) & (1/9,1/8,1/7) \\ Between 7 and 9 & (9,9,9) & (1/9,1/9,1/9) \\ \hline \end{tabular}
\end{table} 
Table 5: Linguistic terms and their FAHP numbers. \[\widetilde{D}= \begin{array}{cccc}\begin{array}{ccc}\text{C}_{1}&\text{C}_{j}& \text{C}_{n}\\ A_{1}\\ A_{m}\end{array}&\begin{array}{ccc}X_{11}&\cdots&X_{1n}\\ \vdots&\ddots&\vdots\\ X_{m1}&\cdots&X_{mn}\end{array}\end{array}  \tag{4}\]  
The FTOPSIS decision matrix \(\widetilde{(D)}\) is filled with triangular fuzzy numbers [84] based on the linguistic judgment, as shown in Table 6. The linguistic terms for alternative ratings in this research have been adjusted to suit the context of the topic. The fuzzy decision matrix \(\widetilde{(D)}\) has to be normalized as per Equation (5) into a normalized decision matrix \(\widetilde{R}\). The normalized value for the beneficial criteria is calculated using Equation (6), and the non-beneficial criteria is calculated using Equation (7). \[\widetilde{R}= \begin{bmatrix}\widetilde{r}_{ij}\end{bmatrix} _{m\times n},\;i=1,2,\;\ldots,m\;\text{and}\;j=1,2,\ldots,m \tag{5}\]  
\[\widetilde{r}_{ij}= \begin{pmatrix}\frac{l_{ij}}{u_{j}^{+}},\frac{m_{ij}}{u_{j} ^{+}},\frac{u_{ij}}{u_{j}^{+}}\end{pmatrix} ,\;u_{j}^{+}=max_{i}\;(beneficial\; criteria) \tag{6}\]  
\[\widetilde{r}_{ij}= \begin{pmatrix}\frac{l_{j}^{-}}{u_{ij}^{-}},\frac{l_{j}^{-} }{m_{ij}},\frac{l_{j}^{-}}{l_{ij}}\end{pmatrix} ,\;l_{j}^{-}=min_{i}\;(non-beneficial\; criteria) \tag{7}\]  
The next step is to weigh the normalized decision matrix \(\widetilde{R}\) by multiplying it by the weights of the criteria to obtain a normalized decision matrix \(\widetilde{V}\) as shown in Equation (8). \[\widetilde{V}= \begin{bmatrix}\widetilde{V}_{ij}\end{bmatrix} _{m\times n}= \widetilde{r}_{ij}\times w_{j} \tag{8}\]  
The fuzzy ideal positive solution (\(A^{*}\)) and the fuzzy ideal negative solution (\(A^{-}\)) are calculated using Equation (9) and Equation (10), respectively, for the distance of each alternative from each one of these solutions to be calculated. The distance of an alternative from the fuzzy ideal positive solution (\(A^{*}\)) is denoted as \(d_{i}^{*}\); it is calculated using Equation (11), while the distance from the fuzzy negative ideal solution is denoted as \(d_{i}^{-}\); it is calculated using Equation (12). \[A^{*}= \begin{cases}\widetilde{v}_{1}^{*},\widetilde{v}_{j}^{*},\ldots, \widetilde{v}_{m}^{*}\end{cases} ,\;\text{where}\;\widetilde{v}_{j}^{*}=max_{i }\Big{\{}\widetilde{v}_{ij3}\Big{\}} \tag{9}\]  
\[A^{-}= \begin{cases}\widetilde{v}_{1}^{-},\widetilde{v}_{j}^{-},\ldots, \widetilde{v}_{m}^{-}\end{cases} ,\;\text{where}\;\widetilde{v}_{j}^{-}=max_{i }\Big{\{}\widetilde{v}_{ij3}\Big{\}} \tag{10}\]  
\[d_{i}^{*}=\sum_{j=1}^{n}d\Big{(}\widetilde{v}_{ij,\;\widetilde{v}_{j}^{*}} \Big{)},\;i=1,2,\ldots,m\;\text{and}\;j=1,2,\ldots,n \tag{11}\]