# The Evaluation of Software Security through Quantum Computing Techniques: A Durability Perspective
## 2 Materials and Methods
### Unified Technique of FAHP and FTOPSIS
#### 2.4.2 Fuzzy TOPSIS Method  
The 'M' option in the mathematical geometrical mean plan, with the 'M' point and 'N' dimensional region TOPSIS MCDM approach, is consumed in multi-measure choices for positioning. The TOPSIS strategies is principally found on the possibility of the absolute and furthest separation from the positive ideal planning, the adverse perfect answer for ideal and least ideal arrangements, individually [32]. The TOPSIS approach is valuable for apportioning the ideal situation of the other option and factor, regarding the rules. To accomplish consistency with the fuzzy climate, TOPSIS relegates the fuzzy number, as indicated by the inclination and addresses the meaning of models. We selected the hybrid approach of FAHP-FTOPSIS, in order to facilitate the collective choice decision-making approach in a fuzzy climate. FTOPSIS techniques have the following steps. Step 1: Further estimation of the ranks of the factors, by FTOPSIS and FAHP approaches, is used to evaluate the weight of the factors with the selected alternatives, as mentioned above in Figure 1.  
Step 2: In FTOPSIS, firstly, derive the table for the linguistic terms, used in the affecting factors and alternatives from Table 2, as mentioned below. Further, we used the fuzzy decision matrix, with the assistance of Equation (11) and evaluated the matrix. \[ \begin{array}{cccc}C_{1}&\ldots\ldots&C_{n}\\ &&\widetilde{K}=\begin{array}{c}A_{1}\\ \ldots\\ A_{m}\end{array}&\left[\begin{array}{cccc}\widetilde{x}_{11}&\cdots& \widetilde{x}_{1n}\\ \ldots&\ddots&\ldots\\ \widetilde{x}_{m1}&\ldots&\widetilde{x}_{mm}\end{array}\right]\end{array}  \tag{11}\]  
Here, \(\widetilde{x}_{ij}=\frac{1}{D}\left(\widetilde{x}_{ij}^{1}\cdots\oplus \widetilde{x}_{ij}^{d}\oplus\cdots\widetilde{x}_{ij}^{D}\right)\), \(\widetilde{x}_{ij}^{d}\), is used to calculate the ranking of the alternative \(A_{i}\), the factor \(C_{j}\)_is_ evaluated by the \(d^{th}\) practitioner \(\widetilde{x}_{ij}^{d}=(l_{ij}^{d},m_{ij}^{d},\ u_{ij}^{d})\). Step 3: The normalized fuzzy decision matrices, evaluated by Equation (12) and represented by \(\widetilde{P}\). The normalization is calculated by the Equation (13). \[\widetilde{P}=\left[\widetilde{p}_{ij}\right]_{m\times m} \tag{12}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **TFN** \\ \hline Very scanty & (0, 1, 3) \\ Scanty & (1, 3, 5) \\ Light & (3, 5, 7) \\ Satisfactory & (5, 7, 9) \\ Very satisfactory & (7, 9,10) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Linguistic terms and its TFN. \[\widetilde{p}_{ij}=\left(\frac{l_{ij}}{u_{j}^{+}},\frac{mi_{ij}}{u_{j}^{+}},\frac{ u_{ij}}{u_{j}^{+}}\right),u_{j}^{+}=max\big{\{}u_{ij},\ i=1,2,3\ldots n\big{\}} \tag{13}\]  
The most expected level (\(u_{j}^{+}\)) is 1, and the worst is 0. The normalization process TFNs are calculated by a similar step. Step 4: Further, the weighted normalized fuzzy decision matrix (\(\widetilde{Q}\)) is quantified by Equation (14). \[\widetilde{Q}=\big{[}\widetilde{q}_{ij}\big{]}_{m\times n}i=1,2,\ldots m;\ j=1,2,3 \ldots n \tag{14}\]  
where, \(\widetilde{q}_{ij}=\widetilde{p}_{ij}\otimes\widetilde{w}_{ij}\). Step 5: The fuzzy positive ideal clarification '\(A^{+}\)' and fuzzy negative ideal clarification '\(A^{-}\)' are calculated; the best and worst solutions, respectively, were Equations (15) and (16). This can be done by avoiding the irregular complication of calculation. \[A^{+}=\left(\widetilde{q}_{1,\ldots}^{*}\widetilde{q}_{j,\ldots}^{*} \widetilde{q}_{n_{*}}\right) \tag{15}\]  
\[A^{-}=\left(\widetilde{q}_{1,\ldots}^{*}\widetilde{q}_{j,\ldots}^{*} \widetilde{q}_{n_{*}}\right) \tag{16}\]  
The detachments of alternative were calculating with Equations (17) and (18), respectively.