# The Evaluation of Software Security through Quantum Computing Techniques: A Durability Perspective
## 2 Materials and Methods
### Unified Technique of FAHP and FTOPSIS
#### 2.4.1 Fuzzy AHP Method  
FAHP is the procedure of evaluating the efficient determination of the explicit issues in the security durability of software in a quantum computer. It depends on the characteristics and weights of the substitutes, essentially associated with those characteristics. The FAHP have philological standings, and their corresponding fuzzy numbers, signify the assessment processes. The philological standings have the following corresponding fuzzy numbers, as shown in Table 1. \begin{table}
\begin{tabular}{l l} \hline \hline
**Linguistic Terms** & **TFN** \\ \hline Equal & (1, 1, 1) \\ Not Bad & (2, 3, 4) \\ Good & (4, 5, 6) \\ Very Good & (6, 7, 8) \\ Perfect & (9, 9, 9) \\ Weak Advantage & (1, 2, 3) \\ Preferable & (3, 4, 5) \\ Fairly Good & (5, 6, 7) \\ Absolute & (7, 8, 9) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fuzzy comparison measures or TFN. Afterward, FAHP procedure assesses every individual substance given by the analyst. The subsequent steps are determined using the triangular fuzzy number (TFN) from the hierarchal arrangement. The effect of the feature, and its option, is one measurement to determine various elective rules that have a pair-wise correlation of discrete factors, which assumes a crucial part in the order. The resulting step of FAHP changes the mathematical value from the linguistic terms by utilizing the fuzzy comparison measures [31]. FAHP procedure depends on deciding the weight of the components. The steps are given below. Step 1: To drive the membership function from the triangular fuzzy number, which distributes the yes or no logic in many sub-values in Table 1, and \(t\mu t\), as shown in Equation (2). \[\mu_{\text{a}}(\text{x})=\text{ a}\rightarrow[0,1] \tag{2}\]  
Let us select '\(I\)' (lowest value),'mi' (middle value), and 'u' (uppermost values), as shown in Figure 2.  
Step 2: Thereafter, we have evaluated the environment and transmuted the linguistic terms into TFN values. TFN estimation evaluates by the mathematical geometric mean comparison. The geometric mean was used to evaluate the significant result between factors. Step 3: Further, we evaluated the two-dimensional analysis procedure of a fuzzy pair-wise comparison matrix (Equations (3) and (4)). \[\widetilde{A^{d}}=\left[\widetilde{\mathbb{K}}_{11}^{d}\widetilde{\mathbb{K} }_{12}^{d}\ldots\widetilde{\mathbb{K}}_{1n}^{d},\ \widetilde{\mathbb{K}}_{21}^{d}\widetilde{\mathbb{K}}_{22}^{d}\ldots \widetilde{\mathbb{K}}_{2n}^{d},\ \ldots\ldots\widetilde{\mathbb{K}}_{n1}^{d}\widetilde{\mathbb{K}}_{n2}^{d} \ldots\widetilde{\mathbb{K}}_{nm}^{d}\right] \tag{3}\]  
\[\widetilde{\mathbb{K}}_{ij}=\sum_{i=1}^{d}\widetilde{\mathbb{K}}_{ij}^{d} \tag{4}\]  
where, \(\widetilde{\mathbb{K}}_{ij}^{d}\) mentions the \({}^{t}d^{\prime}\) (decision-maker) on the condition \(\ i^{th}\) over \(j^{th}\) in Equations (2) and (3). If the preference is more than one, the average values are selected. Step 4: The average preferences are evaluated, further creating the hierarchy of affecting factors. From Equation (5), the pair-wise appraisal matrixes are formed for all the affecting aspects in the ladder, basis of preferences. \[\widetilde{A}=\lfloor\widetilde{k_{11}}\ldots\widetilde{k_{1n}}\ldots\ddots \ldots\widetilde{k_{n1}}\ldots\widetilde{k_{nn}}\rfloor \tag{5}\]  
Step 5: The geometric mean and fuzzy weight of factors, derived by the Equation (6), which shows geometric mean technique; Equation (7), derives the fuzzy weight of factors. \[\widetilde{p}_{i}=\left(\prod_{j=1}^{n}\widetilde{\mathbb{K}}_{ij}\right)^{ \frac{1}{n}},\ i=1,2,3\ldots\ldots n \tag{6}\]  
Figure 2: Triangular fuzzy number. \[\widetilde{w}_{i}=\widetilde{p}_{i}\otimes(\widetilde{p}_{1}\oplus\widetilde{p}_{2 }\oplus\widetilde{p}_{3}\ldots\oplus\widetilde{p}_{n})^{-1} \tag{7}\]  
Step 6: Further, we derived and evaluated the normalized weight criteria from Equations (8) and (9).