## 3 Proposed Model
### Phase 3: Fuzzy TOPSIS  
* Prepare a smart TV operation system evaluation questionnaire. Form a questionnaire asking about the expected performance of each smart TV operation system relative to each sub-criterion [52]. The seven linguistic levels of performance are shown in Table 3[53]. Figure 3: Unweighted supermatrix. * [12] Construct a fuzzy decision matrix. Using the questionnaires collected from the experts, a fuzzy decision matrix for the group can be formed via the arithmetic mean method by using Equation (14) [54; 55; 56]: \[\widetilde{\mathrm{P}}^{\prime}= \begin{bmatrix}\widetilde{x}_{11}^{\prime}& \widetilde{x}_{12}^{\prime}&\cdots&\widetilde{x}_{1j}^{\prime}&&\cdots& \widetilde{x}_{1n}^{\prime}\\ \vdots&\ddots&&&&\vdots\\ \widetilde{x}_{r1}^{\prime}&&\ddots&\widetilde{x}_{rj}^{\prime}&&\widetilde{ x}_{rn}^{\prime}\\ \vdots&&\ddots&&&\vdots\\ \widetilde{x}_{p1}^{\prime}&\cdots&&&\ddots&\cdots&\widetilde{x}_{pn}^{\prime }\\ \end{bmatrix} \] (14) where \(\widetilde{\mathrm{P}}^{\prime}\) is a fuzzy decision matrix, and \(\widetilde{x}_{rj}^{\prime}=(l_{\tau j},m_{\tau j},u_{\tau j})\) is the synthesized performance of system \(\tau\) with respect to sub-criterion \(j\). * Normalize the fuzzy decision matrix. \[\widetilde{\mathrm{P}}= \begin{bmatrix}\widetilde{x}_{11}&\widetilde{x}_{12}& \cdots&\widetilde{x}_{1j}&&\cdots&\widetilde{x}_{1n}\\ \vdots&\ddots&&&&\vdots\\ \widetilde{x}_{r1}&&\ddots&\widetilde{x}_{rj}&&\widetilde{x}_{rn}\\ \vdots&&\ddots&&&\vdots\\ \widetilde{x}_{p1}&\cdots&&&\ddots&\cdots&\widetilde{x}_{pn}\\ \end{bmatrix} \] (15) where \(\widetilde{x}_{\tau j}=(\frac{l_{\tau j}}{u_{j}^{\prime}},\frac{m_{\tau j}}{u_ {j}^{\prime}},\frac{u_{\tau j}}{u_{j}^{\prime}})\) and \(u_{j}^{+}=\max_{\tau}u_{\tau j}\). * Obtain the weighted normalized fuzzy decision matrix. \[\widetilde{\mathrm{P}}\times\boldsymbol{s}_{j}= \begin{bmatrix}\widetilde{x}_{ 11}&\widetilde{x}_{12}&\cdots&\widetilde{x}_{1j}&&\cdots&\widetilde{x}_{1n}\\ \vdots&\ddots&&&&\vdots\\ \widetilde{x}_{r1}&&\ddots&\widetilde{x}_{\tau j}&&\widetilde{x}_{rn}\\ \vdots&&\ddots&&&\vdots\\ \widetilde{x}_{p1}&\cdots&&&\ddots&\cdots&\widetilde{x}_{pn}\\ \end{bmatrix} \times \begin{bmatrix}\boldsymbol{s}_{1}\\ \boldsymbol{s}_{j}\\ \boldsymbol{s}_{n}\\ \end{bmatrix} \] (16) \[= \begin{bmatrix}\widetilde{v}_{11}&\widetilde{v}_{12}&\cdots& \widetilde{v}_{1j}&&\cdots&\widetilde{v}_{1n}\\ \vdots&\ddots&&&&\vdots\\ \widetilde{v}_{r1}&&\ddots&\widetilde{v}_{rj}&&\widetilde{v}_{rn}\\ \vdots&&\ddots&&&\vdots\\ \widetilde{v}_{p1}&\cdots&&&\ddots&\cdots&\widetilde{v}_{pn}\\ \end{bmatrix} \] where \(\boldsymbol{s}_{j}\) is calculated from Step 10. \begin{table}
\begin{tabular}{c c} \hline \hline
**Linguistic Variables** & **Positive Triangular Fuzzy Numbers** \\ \hline Very Weak (VW) & (0, 0, 0.2) \\ Weak (W) & (0.05, 0.2, 0.35) \\ Medium Weak (MW) & (0.2, 0.35, 0.5) \\ Fair (F) & (0.35, 0.5, 0.65) \\ Medium Good (MG) & (0.5, 0.65, 0.8) \\ Good (G) & (0.7, 0.8, 0.9) \\ Very Good (VG) & (0.8, 1, 1) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Membership function of fuzzy numbers for operation system evaluation. Step 15. Compute the Fuzzy Positive-Ideal Solution (FPIS) and the Fuzzy Negative-Ideal Solution (FNIS) [53; 54; 55]. \[F^{*}=\left\{\widetilde{v}_{1}^{*},\widetilde{v}_{2}^{*}\ldots,\widetilde{v}_{j }^{*},\ldots,\widetilde{v}_{n}^{*}\right\}=\left\{\left(\max_{\tau}\widetilde{v }_{\tau j}|\tau=1,\ldots,p\right),j=1,\ldots,n\right\} \tag{17}\]  
\[F^{-}=\left\{\widetilde{v}_{1}^{-},\widetilde{v}_{2}^{-}\ldots,\widetilde{v}_{ j}^{-},\ldots,\widetilde{v}_{n}^{-}\right\}=\left\{\left(\min_{\tau} \widetilde{v}_{\tau j}|\tau=1,\ldots,p\right),j=1,\ldots,n\right\} \tag{18}\]  
where \(F^{*}\) is the FPIS for sub-criterion \(j\), and \(F^{-}\) is the FNIS for sub-criterion \(j\). Step 16. Compute the distance of each smart TV operation system from FPIS and FNIS [54; 55; 56].