# Analysis and Ranking of IT Risk Factors Using Fuzzy TOPSIS-Based Approach
## 3 Methods and Results
### Fuzzy TOPSIS Method
A normalized decision matrix could be computed using the following relevance predicated on the positive as well as negative ideal options:  
\[\widetilde{r}_{ij}=\left(\frac{a_{ij}}{c_{j}^{+}},\frac{b_{ij}}{c_{j}^{+}}, \frac{c_{ij}}{c_{j}^{+}}\right)\quad;\quad c_{j}^{+}=max_{i}\;c_{ij};\;\text{Positive ideal solution} \tag{2}\] \begin{table}
\begin{tabular}{l l l l} \hline \hline  & **Name** & **Type** & **Weight** \\ \hline
1 & C1 & + & (0.167,0.167,0.167) \\ \hline
2 & C2 & + & (0.167,0.167,0.167) \\ \hline
3 & C3 & + & (0.167,0.167,0.167) \\ \hline
4 & C4 & + & (0.167,0.167,0.167) \\ \hline
5 & C5 & + & (0.167,0.167,0.167) \\ \hline
6 & C6 & + & (0.167,0.167,0.167) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of different criteria. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Code** & **Linguistic Terms** & **L** & **M** & **U** \\ \hline
1 & Very low & 1 & 1 & 3 \\ \hline
2 & Low & 1 & 3 & 5 \\ \hline
3 & Medium & 3 & 5 & 7 \\ \hline
4 & High & 5 & 7 & 9 \\ \hline
5 & Very high & 7 & 9 & 9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fuzzy Scale. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline  & **C1** & **C2** & **C3** & **C4** & **C5** & **C6** \\ \hline A1 & 5.640,7.640,8.760 & 5.240,7.240,8.680 & 4.680,6.680,8.360 & 4.920,6.920,8.520 & 4.920,6.920,8.440 & 4.760,6.760,8.360 \\ \hline A2 & 5.080,7.080,8.520 & 4.840,6.840,8.680 & 5.080,7.080,8.440 & 4.920,6.920,8.440 & 4.680,6.680,8.520 & 5.080,7.080,8.440 \\ \hline A3 & 5.240,7.240,8.840 & 4.920,6.920,8.680 & 4.600,6.600,8.040 & 4.440,6.440,8.280 & 4.120,6.120,7.800 & 4.520,6.520,8.200 \\ \hline A4 & 4.680,6.680,8.280 & 5.000,7.000,8.600 & 5.080,7.080,8.600 & 4.840,6.840,8.440 & 4.680,6.680,8.200 & 4.280,6.280,8.200 \\ \hline A5 & 4.760,6.760,8.360 & 5.080,7.080,8.520 & 5.480,7.480,8.760 & 5.000,7.000,8.440 & 4.280,6.280,8.120 & 4.680,6.680,8.280 \\ \hline A6 & 4.680,6.680,8.360 & 5.240,7.240,8.680 & 5.080,7.080,8.600 & 4.920,6.920,8.440 & 4.840,6.840,8.440\[\widetilde{r}_{ij}=\left(\frac{a_{j}^{-}}{c_{ij}},\frac{a_{j}^{-}}{b_{ij}},\frac{a_ {j}^{-}}{a_{ij}}\right)\quad;\quad a_{j}^{-}=min_{i}\,a_{ij};\,\,\text{Negative ideal solution} \tag{3}\]  
The normalized decision matrix is presented in the following Table 4.  
The weighted normalized decision matrix could be computed by multiplying the criteria weights in the normalized fuzzy decision problem by the following equations, taking into account the distinct weights of every criterion. \[\widetilde{v}_{ij}=\widetilde{r}_{ij}.\widetilde{w}_{ij} \tag{4}\]  
where \(\widetilde{w}_{ij}\) signifies weight of criterion \(c_{j}\). The weighted normalized decision matrix can be seen in Table 5 below. Step 4: Control the fuzzy positive ideal solution (_FPIS_, _A*_) as well as the fuzzy negative ideal solution (_FNIS_, _A-_). The FPIS and the FNIS of the alternatives can be defined as follows:  
\[A^{*}=\left\{\widetilde{v}_{1}^{*},\widetilde{v}_{2}^{*},\ldots,\widetilde{v} _{n}^{*}\right\}=\left\{\left(\underset{j}{max}\,v_{ij}|i\in B\right),\left( \underset{j}{min}\,v_{ij}|i\in C\right)\right\} \tag{5}\]  
\[A^{-}=\left\{\widetilde{v}_{1}^{-},\widetilde{v}_{2}^{-},\ldots,\widetilde{v }_{n}^{-}\right\}=\left\{\left(\underset{j}{min}\,v_{ij}|i\in B\right),\left( \underset{j}{max}\,v_{ij}|i\in C\right)\right\} \tag{6}\]  
The alternative solutions' FPIS and FNIS could be demarcated as presented below: Where \(\widetilde{v}_{i}^{*}\) is the highest amount of \(i\) for all the alternatives, and \(\widetilde{v}_{i}^{-}\) is the lowest amount of \(i\) for all the alternatives options. \(B\) and \(C\) signify the positive as well as negative ideal solutions, correspondingly.