## 4 Case Study
### Solution of the Proposed Novel Fuzzy Analytic Hierarchy Process-Based Approach  
In the context of construction industry Internet platform selection, traditional MCDM methods exhibit limitations in simultaneously addressing fuzzy semantic information, neglecting relative weights between evaluation criteria, and handling incomplete expert-provided information. To effectively address these shortcomings, this study integrated the FAHP and the WASPAS approaches to handle the challenges of selecting Internet platforms for the construction industry within an incomplete information environment. In the selection process of the construction industry Internet platforms, the experts initially conferred to determine the possible alternatives and evaluation criteria. Subsequently, they assigned rating values to the potential alternatives under different evaluation criteria (Table 3) and established the pairwise comparison matrix for various evaluation criteria (Table 4). This sequence of steps constitutes the proposed approach's initial stages (Steps 1 to 4). Step 5. Data filling of incomplete information. In Table 3, it was noted that expert E4 provided information containing partial data for alternatives P21 and P22; thus, the average of the information provided by the other experts was used to fill in the incomplete information. Step 6. Determine the relative weight of the evaluation criteria calculation and perform consistency checks. Based on the data in Table 4, which aggregated the evaluation opinions of the four experts, the results of the pairwise comparison matrix for evaluation criteria were determined, as expressed in Table 9. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Platform** & **TM** & **UN** & **PF** & **CS** & \(\mathbf{Q_{1}^{1}}\)**Value** & \(\mathbf{Q_{2}^{2}}\)**Value** & \(\mathbf{Q_{i}Value}\) & **Rank** \\ \hline P8 & 2.000 & 2.333 & 3.000 & 3.667 & 0.451 & 0.431 & 0.441 & 21 \\ P9 & 2.667 & 3.667 & 1.667 & 1.333 & 0.367 & 0.347 & 0.357 & 24 \\ P10 & 4.333 & 4.000 & 2.000 & 3.333 & 0.555 & 0.527 & 0.541 & 15 \\ P11 & 4.000 & 5.333 & 6.000 & 5.000 & 0.817 & 0.809 & 0.813 & 4 \\ P12 & 3.333 & 4.333 & 4.333 & 4.333 & 0.659 & 0.653 & 0.656 & 8 \\ P13 & 2.667 & 4.333 & 2.667 & 1.667 & 0.444 & 0.431 & 0.437 & 22 \\ P14 & 3.667 & 5.000 & 5.000 & 2.333 & 0.628 & 0.615 & 0.622 & 11 \\ P15 & 2.667 & 2.667 & 4.000 & 2.000 & 0.450 & 0.442 & 0.446 & 19 \\ P16 & 1.667 & 2.333 & 5.333 & 3.667 & 0.525 & 0.475 & 0.500 & 16 \\ P17 & 4.667 & 3.333 & 5.000 & 3.667 & 0.673 & 0.661 & 0.667 & 7 \\ P18 & 2.000 & 2.000 & 2.000 & 2.000 & 0.324 & 0.322 & 0.323 & 25 \\ P19 & 1.667 & 2.000 & 5.000 & 2.333 & 0.438 & 0.402 & 0.420 & 23 \\ P20 & 6.000 & 4.000 & 4.667 & 5.333 & 0.818 & 0.795 & 0.807 & 5 \\ P21 & 4.333 & 4.333 & 3.000 & 2.667 & 0.573 & 0.563 & 0.568 & 14 \\ P22 & 4.667 & 6.000 & 5.000 & 1.667 & 0.674 & 0.629 & 0.652 & 9 \\ P23 & 4.667 & 7.000 & 4.000 & 1.667 & 0.673 & 0.618 & 0.645 & 10 \\ P24 & 6.000 & 6.000 & 4.667 & 5.000 & 0.874 & 0.866 & 0.870 & 3 \\ P25 & 1.333 & 2.333 & 2.000 & 1.667 & 0.292 & 0.289 & 0.290 & 26 \\ P26 & 2.667 & 2.667 & 2.333 & 4.333 & 0.497 & 0.468 & 0.483 & 17 \\ \hline \end{tabular}
\end{table} 
Table 8: _Cont._ \begin{table}
\begin{tabular}{c c c c c} \hline
**Criteria** & **TM** & **UN** & **PF** & **CS** \\ \hline TM & (1.000, 1.000, 1.000) & (0.667, 0.750, 0.750) & (2.000, 2.750, 3.000) & (3.750, 4.750, 5.000) \\ UN & (1.500, 1.500, 2.000) & (1.000, 1.000, 1.000) & (3.250, 4.000, 4.500) & (3.750, 4.500, 5.250) \\ PF & (0.354, 0.396, 0.583) & (0.244, 0.292, 0.342) & (1.000, 1.000, 1.000) & (1.500, 1.500, 2.000) \\ CS & (0.208, 0.217, 0.279) & (0.196, 0.229, 0.279) & (0.583, 0.750, 0.750) & (1.000, 1.000, 1.000) \\ \hline \end{tabular}
\end{table} 
Table 9: Pairwise comparison matrix for evaluation criteria of the proposed approach. According to the information in Table 9, Equation (7) was used to defuzzify the TFN and calculate the value of \(\lambda_{max}\) as 4.202. Equations (9) and (10) were used to calculate the values of CI and CR. The calculation process is as follows.