## 4 The Application of the Proposed Model
### Case Study
#### 4.1.2 Application of the Fuzzy VIKOR Method  
Based on the results of the criteria weights obtained from the ANP method, the fuzzy VIKOR approach was applied to determine the best plan that was the most suitable for the construction site. The specific calculation process was conducted in the following steps:  
(1) Performance assessments. The experienced experts were required to assess the performance of each alternative based on the identified criteria. The verbal expressions were adopted in the expert assessment process. As shown in Table 9, the expert assessment results for Alternative 1 were obtained. After the assessment results for all alternatives were determined, the assessment responses obtained from the experts were then transformed into fuzzy numbers using the triangular fuzzy scale in Table 3 to model the uncertainty and ambiguity from the subjective information. (2) Aggregation. Then the fuzzy numbers were aggregated using Equation (11) to calculate the aggregated fuzzy number \(P(\theta)\) for each criterion. \[P(\theta)\cong\sum_{i=0}^{n}\left[w_{i}\otimes P_{i}(\theta)\right] \tag{11}\]  
where \(w_{i}\) is the weight of criterion \(i\), \(P_{i}(\theta)\) is the fuzzy assessment result for criterion \(i\), and \((\theta)\cong(a,b,c)\). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline
**Criteria** & **C1** & **C2** & **C3** & **C4** & **C5** & **C6** & **C7** & **C8** & **C9** & **C10** & **C11** & **C12** & **C13** & **C14** & **C15** \\ \hline C1 & 0.0369 & 0.0343 & 0.0443 & 0.0411 & 0.0806 & 0.1010 & 0.0649 & 0.1805 & 0.0898 & 0.0802 & 0.0851 & 0.0727 & 0.0527 & 0.0385 & 0.0304 \\ C2 & 0.1385 & 0.1371 & 0.1286 & 0.0919 & 0.1227 & 0.0671 & 0.1145 & 0.0902 & 0.1271 & 0.1222 & 0.1346 & 0.0969 & 0.1470 & 0.1474 & 0.1475 \\ C3 & 0.0903 & 0.0969 & 0.0845 & 0.1411 & 0.0634 & 0.0473 & 0.0435 & 0.0638 & 0.0635 & 0.0631 & 0.0492 & 0.0517 & 0.0842 & 0.0895 & 0.0927 \\ C4 & 0.1964 & 0.1939 & 0.2048 & 0.1881 & 0.1954 & 0.2468 & 0.2393 & 0.1276 & 0.1797 & 0.1945 & 0.1912 & 0.2389 & 0.1782 & 0.1867 & 0.1915 \\ C5 & 0.1242 & 0.0935 & 0.1296 & 0.1458 & 0.0831 & 0.1242 & 0.1225 & 0.1142 & 0.0845 & 0.0991 & 0.0904 & 0.0899 & 0.0795 & 0.1346 & 0.1233 \\ C6 & 0.0512 & 0.0335 & 0.0447 & 0.0441 & 0.0311 & 0.0780 & 0.0795 & 0.0880 & 0.0422 & 0.0400 & 0.0586 & 0.0400 & 0.0354 & 0.0342 & 0.0860 \\ C7 & 0.0403 & 0.0535 & 0.0344 & 0.0310 & 0.0481 & 0.0403 & 0.0354 & 0.0363 & 0.0299 & 0.0252 & 0.0261 & 0.0231 & 0.0560 & 0.0544 & 0.0362 \\ C8 & 0.0780 & 0.1133 & 0.0851 & 0.0729 & 0.1315 & 0.0512 & 0.0560 & 0.0553 & 0.0597 & 0.0520 & 0.0412 & 0.0633 & 0.1228 & 0.0706 & 0.0493 \\ C9 & 0.0153 & 0.0166 & 0.2098 & 0.0135 & 0.0150 & 0.0127 & 0.0099 & 0.0143 & 0.0225 & 0.0280 & 0.0364 & 0.0225 & 0.0132 & 0.0163 & 0.0099 \\ C10 & 0.0693 & 0.0700 & 0.0654 & 0.0793 & 0.0693 & 0.0747 & 0.0727 & 0.0830 & 0.1099 & 0.1061 & 0.1120 & 0.1154 & 0.0824 & 0.0798 & 0.0750 \\ C11 & 0.0470 & 0.0443 & 0.0423 & 0.0413 & 0.0286 & 0.0450 & 0.0458 & 0.0371 & 0.0652 & 0.0614 & 0.0594 & 0.0397 & 0.0395 & 0.0354 & 0.0281 \\ C12 & 0.0249 & 0.0256 & 0.0189 & 0.0223 & 0.0435 & 0.0241 & 0.0280 & 0.0220 & 0.0377 & 0.0397 & 0.0275 & 0.0577 & 0.0214 & 0.0249 & 0.0434 \\ C13 & 0.0292 & 0.0237 & 0.0209 & 0.0251 & 0.0201 & 0.0280 & 0.0191 & 0.0219 & 0.0211 & 0.0253 & 0.0177 & 0.0220 & 0.0273 & 0.0284 & 0.0292 \\ C14 & 0.0463 & 0.0565 & 0.0548 & 0.0501 & 0.0569 & 0.0490 & 0.0606 & 0.0574 & 0.0553 & 0.0505 & 0.0604 & 0.0525 & 0.0433 & 0.0516 & 0.0463 \\ C15 & 0.0123 & 0.0075 & 0.0120 & 0.0125 & 0.0107 & 0.0107 & 0.0080 & 0.084 & 0.0121 & 0.0126 & 0.0103 & 0.0139 & 0.0172 & 0.0078 & 0.0123 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Weighted super matrix for criteria weighting.