## 4 Case Study
### Solution of the Weighted Sum Model and the Weighted Product Model  
Both the WSM approach [37] and the WPM approach [35] are widely applied in decision analysis known for its simplicity and accessibility, requiring no complex mathematical calculations and skills. It involves calculating a weighted sum and a weighted product of the evaluation results based on various criteria. However, the WSM approach [37] and WPM approach [35] are limited to handling complete information and cannot process the expert-provided incomplete information. Due to expert E4 providing partially incomplete information for alternatives P21 and P22 (see Table 3), only the information provided by experts E1, E2, and E3 were utilized for calculation.  
For example, experts E1, E2, and E3 determined the evaluation criteria TM score in platform P1 to be 4, 3, and 5, respectively; the aggregated expert-provided information is shown as follows.  
\[(1/3)\times 4+(1/3)\times 3+(1/3)\times 5=4.000\]  
Based on the evaluation scores presented in Table 3, the experts employed the arithmetic average to calculate the evaluation scores for each construction industry Internet platform. The calculation results of the WSM and WPM approach are shown in Table 5.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **Expert** & **TM** & **UN** & **PF** & **CS** \\ \hline \multirow{4}{*}{TM} & E1 & (1, 1, 1) & (1/3, 1/2, 1/2) & (2, 2, 3) & (3, 4, 4) \\  & E2 & (1, 1, 1) & (1, 1, 1) & (3, 3, 3) & (4, 6, 6) \\  & E3 & (1, 1, 1) & (1, 1, 1) & (1, 2, 2) & (5, 5, 6) \\  & E4 & (1, 1, 1) & (1/3, 1/2, 1/2) & (2, 4, 4) & (3, 4, 4) \\ \hline \multirow{4}{*}{UN} & E1 & (2, 2, 3) & (1, 1, 1) & (3, 4, 4) & (3, 4, 6) \\  & E2 & (1, 1, 1) & (1, 1, 1) & (2, 2, 3) & (5, 6, 6) \\  & E3 & (1, 1, 1) & (1, 1, 1) & (5, 6, 7) & (4, 4, 5) \\  & E4 & (2, 2, 3) & (1, 1, 1) & (3, 4, 4) & (3, 4, 4) \\ \hline \multirow{4}{*}{PF} & E1 & (1/3, 1/2, 1/2) & (1/4, 1/4, 1/3) & (1, 1, 1) & (2, 2, 2) \\  & E2 & (1/3, 1/3, 1/3) & (1/3, 1/2, 1/2) & (1, 1, 1) & (1, 1, 2) \\  & E3 & (1/2, 1/2, 1) & (1/7, 1/6, 1/5) & (1, 1, 1) & (2, 2, 3) \\  & E4 & (1/4, 1/4, 1/2) & (1/4, 1/4, 1/3) & (1, 1, 1) & (1, 1, 1) \\ \hline \multirow{4}{*}{CS} & E1 & (1/4, 1/4, 1/3) & (1/6, 1/4, 1/3) & (1/2, 1/2, 1/2) & (1, 1, 1) \\  & E2 & (1/6, 1/6, 1/4) & (1/6, 1/6, 1/5) & (1/2, 1, 1) & (1, 1, 1) \\ \cline{1-1}  & E3 & (1/6, 1/5, 1/5) & (1/5, 1/4, 1/4) & (1/3, 1/2, 1/2) & (1, 1, 1) \\ \cline{1-1}  & E4 & (1/4, 1/4, 1/3) & (1/4, 1/4, 1/3) & (1, 1, 1) & (1, 1, 1) \\ \hline \end{tabular}
\end{table}
Table 4: Pairwise comparison matrix of four evaluation criteria.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Platform**} & **TM** & **UN** & **PF** & **CS** & \multicolumn{2}{c}{**WSM Approach**} & \multicolumn{2}{c}{**WPM Approach**} \\ \cline{6-9}  & & & & & \(\mathbf{P_{i}^{1}}\)**Value** & **Rank** & \(\mathbf{P_{i}^{2}}\)**Value** & **Rank** \\ \hline P1 & 4.000 & 3.333 & 2.000 & 2.000 & 2.833 & 19 & 2.702 & 20 \\ P2 & 4.333 & 5.000 & 4.667 & 3.000 & 4.250 & 8 & 4.173 & 6 \\ P3 & 5.667 & 6.000 & 6.000 & 5.000 & 5.667 & 1 & 5.651 & 1 \\ P4 & 2.333 & 3.667 & 4.000 & 1.667 & 2.917 & 18 & 2.748 & 18 \\ P5 & 5.333 & 6.000 & 6.667 & 4.000 & 5.500 & 2 & 5.405 & 2 \\ P6 & 5.333 & 5.333 & 3.000 & 1.667 & 3.833 & 12 & 3.453 & 14 \\ P7 & 3.667 & 4.000 & 4.000 & 3.000 & 3.667 & 13 & 3.642 & 12 \\ P8 & 2.000 & 2.333 & 3.000 & 3.667 & 2.750 & 22 & 2.677 & 22 \\ P9 & 2.667 & 3.667 & 1.667 & 1.333 & 2.333 & 24 & 2.159 & 24 \\ P10 & 4.333 & 4.000 & 2.000 & 3.333 & 3.417 & 15 & 3.279 & 15 \\ P11 & 4.000 & 5.333 & 6.000 & 5.000 & 5.083 & 4 & 5.030 & 4 \\ P12 & 3.333 & 4.333 & 4.333 & 4.083 & 10 & 4.058 & 8 \\ \hline \end{tabular}
\end{table}
Table 5: Calculation results of the WSM and WPM approach.