## 3 Proposed Model
### Obtain the Ranking of the Operation System Alternatives  
In phase 3, the ranking of the operation system alternatives is calculated. A questionnaire was prepared asking experts to evaluate the expected performance of each operating system according to each sub-criteria. Four operation systems were evaluated, and five experts (k1-k5) were invited to complete the questionnaire based on the seven linguistic levels of performance shown in Table 3. The results are shown in Table 9. Through the arithmetic mean method, a fuzzy decision matrix is developed, as shown in Table 10. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sub-Criterion** & **Operation System A1** & **Operation System A2** & **Operation System A3** & **Operation System A4** \\ \hline Processing power (SC1) & (0.44, 0.59, 0.74) & (0.62, 0.74, 0.86) & (0.78, 0.96, 0.98) & (0.72, 0.84, 0.92) \\ Graphic performance (SC2) & (0.38, 0.53, 0.68) & (0.68, 0.81, 0.90) & (0.70, 0.85, 0.92) & (0.74, 0.88, 0.94) \\ Memory storage (SC3) & (0.72, 0.84, 0.92) & (0.70, 0.85, 0.92) & (0.64, 0.78, 0.88) & (0.72, 0.89, 0.94) \\ Connectivity (SC4) & (0.48, 0.62, 0.76) & (0.50, 0.65, 0.80) & (0.52, 0.65, 0.78) & (0.50, 0.65, 0.80) \\ User interface (SC6) & (0.76, 0.92, 0.96) & (0.54, 0.68, 0.82) & (0.62, 0.74, 0.86) & (0.54, 0.68, 0.82) \\ Compatibility (SC7) & (0.76, 0.92, 0.96) & (0.72, 0.84, 0.92) & (0.62, 0.74, 0.86) & (0.74, 0.88, 0.94) \\ Voice assistant (SC8) & (0.04, 0.16, 0.32) & (0.31, 0.44, 0.57) & (0.47, 0.62, 0.77) & (0.33, 0.48, 0.59) \\ APP (SC10) & (0.16, 0.28, 0.44) & (0.10, 0.22, 0.38) & (0.14, 0.29, 0.44) & (0.10, 0.22, 0.38) \\ Cyber security (SC11) & (0.11, 0.26, 0.41) & (0.38, 0.53, 0.68) & (0.14, 0.29, 0.44) & (0.38, 0.53, 0.68) \\ Brand preference (SC12) & (0.32, 0.47, 0.62) & (0.41, 0.56, 0.71) & (0.17, 0.31, 0.47) & (0.41, 0.56, 0.71) \\ Market share (SC13) & (0.29, 0.44, 0.59) & (0.72, 0.84, 0.92) & (0.44, 0.59, 0.74) & (0.74, 0.88, 0.94) \\ Company sustainability (SC14) & (0.14, 0.29, 0.44) & (0.03, 0.12, 0.29) & (0.10, 0.22, 0.38) & (0.03, 0.12, 0.29) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Fuzzy decision matrix \(\mathbb{F}^{\prime}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sub-Criteria** & **\(s_{j}\)** & **Priorities** & **Rank** \\ \hline Processing power (SC1) & \(s_{1}\) & 0.13146 & 1 \\ Graphic performance (SC2) & \(s_{2}\) & 0.12842 & 2 \\ Memory storage (SC3) & \(s_{3}\) & 0.07652 & 5 \\ Connectivity (SC4) & \(s_{4}\) & 0.05687 & 10 \\ Power consumption (SC5) & \(s_{5}\) & 0 \\ User interface (SC6) & \(s_{6}\) & 0.07457 & 6 \\ Compatibility (SC7) & \(s_{7}\) & 0.05621 & 11 \\ Voice assistant (SC8) & \(s_{8}\) & 0.10103 & 3 \\ Update and support (SC9) & \(s_{9}\) & 0 \\ APP (SC10) & \(s_{10}\) & 0.06037 & 8 \\ Cyber security (SC11) & \(s_{11}\) & 0.05972 & 9 \\ Brand preference (SC12) & \(s_{12}\) & 0.06250 & 7 \\ Market share (SC13) & \(s_{13}\) & 0.08679 & 4 \\ Company sustainability (SC14) & \(s_{14}\) & 0.03750 & 13 \\ Maintenance cost (SC15) & \(s_{15}\) & 0.03049 & 14 \\ Upgrade costs (SC16) & \(s_{16}\) & 0.03756 & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Priorities of the sub-criteria. \begin{table}
\begin{tabular}{c cApply Equation (15) to normalize the fuzzy decision matrix \(\widetilde{\mathbf{P}}^{\prime}\), and the result \(\widetilde{\mathbf{P}}\) is listed in Table 11. Next, apply Equation (16) to compute the weighted normalized fuzzy decision matrix, as shown in Table 12. Next, use Equations (17) and (18) to calculate the fuzzy positive-ideal solution (FPIS) and fuzzy negative-ideal solution (FNIS), respectively, and the results are shown in Table 13. Finally, the distance of each operation system from FPIS, \(d_{\tau}^{*}\), and the distance of each operation system from FNIS, \(d_{\tau}^{-}\