## 3 Proposed Model
### Calculate the Importance Weights of the Sub-Criteria  
In phase 2, the importance weights of the sub-criteria are calculated by applying the FANP. In accordance with Figures 4-6, a FANP questionnaire was prepared and distributed to five experts. Based on the questionnaire feedback, fuzzy pairwise comparison matrices  
Figure 5: Interrelationships among the criteria.  
Figure 6: Interrelationships among the sub-criteria under system on chip (C1).  
for each expert are formed. For instance, the fuzzy pairwise comparison matrix of criteria for expert 1 is as follows:  
\[\widetilde{\mathbf{W}}_{\text{C1}}=\begin{matrix}\text{C1}&\text{C2}&\text{C3}& \text{C4}\\ \text{C2}&\begin{bmatrix}(1,1,1)&(0.2,0.33,1)&(1,3,5)&(1,3,5)\\ (1,3,5)&(1,1,1)&(1,3,5)&(3,5,7)\\ \text{C3}&(0.2,0.33,1)&(0.2,0.33,1)&(1,1,1)&(3,5,7)\\ \text{C4}&\begin{bmatrix}(0.2,0.33,1)&(0.14,0.2,0.33)&(0.14,0.2,0.33)&(1,1,1) \end{bmatrix}\end{matrix}\]  
Fuzzy aggregated pairwise comparison matrices are prepared by integrating the experts' opinions using the geometric mean approach. The fuzzy aggregated pairwise comparison matrix for the criteria is as follows:  
\[\widetilde{\mathbf{W}}_{\text{C}}=\begin{matrix}\text{C1}&\text{C2}&\text{C3}& \text{C4}\\ \text{C2}&\begin{bmatrix}(1,1,1)&(0.38,0.64,1.38)&(1.25,3.32,5.35)&(1.93,4.08,6.12 )\\ (0.72,1.55,2.63)&(1,1,1)&(0.72,1.93,3.62)&(3.32,5.35,7.36)\\ \text{C3}&(0.19,0.30,0.80)&(0.28,0.52,1.38)&(1,1,1)&(1.93,4.08,6.12)\\ \text{C4}&(0.16,0.25,0.52)&(0.14,0.19,0.30)&(0.16,0.25,0.52)&(1,1,1)\end{bmatrix}\end{matrix}\]  
Next, the center of gravity method is used to calculate the defuzzified aggregated pairwise comparison matrices, and then the importance vectors of the criteria, the importance vectors of the sub-criteria, the interdependence among criteria, and the interdependence among sub-criteria are calculated. Then, compute the defuzzified aggregated pairwise comparison matrix and the importance vector of the criteria, and run a consistency test:  
\[\begin{matrix}\text{C1}&\text{C2}&\text{C3}&\text{C4}\\ \text{C1}&\begin{bmatrix}1&0.80&3.31&4.04\\ 1.25&1&2.09&5.34\\ \text{C3}&0.30&0.48&1&4.04\\ \text{C4}&0.25&0.19&0.25&1\end{bmatrix}\end{matrix}\]  
\[\begin{matrix}\text{\boldmath$w$}_{\text{C}}=[0.373&0.381&0.179&0.066]^{T} \end{matrix}\]  
\[\begin{matrix}\text{C}I_{\text{C}}=\frac{\lambda_{\text{max}}-N}{N-1}=\frac{4.133-4}{4-1}=0.044\end{matrix}\]  
\[\begin{matrix}\text{C}R_{\text{C}}=\frac{CI_{\text{C}}}{\text{RI}}=\frac{0.04 4}{0.9}=0.049\end{matrix}\]  
Since \(CR_{\text{C}}\) is less than 0.1, the comparison matrix is deemed to have passed the consistency test. After calculating the importance vector of the criteria, the importance vectors of the sub-criteria, the interdependence among the criteria, and the interdependence among the sub-criteria, an unweighted supermatrix is formed, as shown in Table 5. The weighted supermatrix and limit supermatrix are also obtained, as shown in Tables 6 and 7, respectively. The importance weights of the sub-criteria can be found in the sub-criteria-to-goal column of the limit supermatrix in Table 7 and in Table 8. They are as follows:  
\[\begin{matrix}\text{\boldmath$\mathsf{s}$}_{j}=[0.131&0.128&0.077&0.057&0&0.075&0.056&0.101&0&0.06&0.059&0.063&0.087&0.038&0.030&0.038]^{T}\end{matrix}\]  
In this case study, the experts stressed that the most important criterion in the criteria is the operation system requirement (C2) criterion, with an importance weight of 0.381, followed by the system on chip (C1) criterion, with an importance weight of 0.373. Regarding the sub-criteria, processing power (SC1) is the most important criterion, with an importance weight of 0.131, followed by graphic performance (SC2), with an importance weight of 0.128. The third and fourth most important sub-criteria are, respectively, voice assistant (SC8), with an importance weight of 0.101, and market share (SC13), with an importance weight of 0.087.