# Dynamic BIM Adoption Impact on Contract Cost Variance Factors Using PLS-SEM Techniques
## 3 Methodology
### Evaluating the Impact of BIM Adoption on CCV Factors over Time (Component 3)  
After knowing the BIM adoption with time, the change in the weight of group factors with time was determined based on the relative weight of the CCV factor in 2022 (\(w_{jk}^{2022}\)). The values of \(w_{jk}^{2022}\) (as shown in the second column in Table 9) were obtained using the developed causal model and mentioned in the third stage. Based on the BIM adoption in 2022 (\(BIM_{2022}\)), BIM adoption in the required year (\(BIM_{Y_{i}}\)), and the values of \(w_{jk}^{2022}\), the following steps were performed to determine the relative weight of a latent group with their factors.  
Step 1: At a given year (\(Y_{i}\)), the \(w_{jk}^{2022}\) of all significant CCV factors was utilized. The ratio of BIM adoption (\(\delta_{i}\)) was computed based on the BIM adoption in 2022 (\(BIM_{2022}\)) and BIM adoption at (\(Y_{i}\)) using Equation (2).  
\[\delta_{i}=\frac{BIM_{2022}}{BIM_{Y_{i}}} \tag{2}\]  
Step 2: The BIM strength influence (\(\lambda_{l}\)) related to BIM group factors (\(l\)) was utilized. The expert construction industry determined the \(\lambda_{l}\). It should be noted that the summation of strength influence is equal to one (\(\sum_{l=1}^{n}\lambda_{l}=1\)), where \(n\) is the number related to BIM group factors (IQ, EP, and CP), which was equal to 3 (n = 3).  
Step 3: The \(w_{jk}^{2022}\) related to BIM group factors was modified by multiplying it with \(\delta_{i}\) and \(\lambda_{l}\). At the same time, there is no change in the \(w_{jk}^{2022}\) for non-related BIM group factors. In other words, the \(\lambda_{l}\) and \(\delta_{i}\) set as units for non-related BIM group factors. The \(w_{jk-mod}^{Y_{i}}\) for related and non-related BIM group factors can be computed in Equation (3).  
\[w_{jk-mod}^{Y_{i}}=\left\{\begin{array}{ll}\delta_{i}\lambda_{l}w_{jk}^{2022 }\text{ for k }=\text{ IQ, EP, or CP}\\ w_{jk}^{2022}\text{ for k }=\text{ EF, or PC}\end{array}\right. \tag{3}\]  
Step 4: The \(w_{jk-mod}^{Y_{i}}\) of the BIM-affected group and non-BIM-affected factors (as shown in the third column in Table 9) were normalized, and the relative weight factor in the year of \(Y_{i}\) (\(w_{jk}^{Y_{i}}\)) was computed such as \(w_{jk}^{2023}\), as shown in the fourth column in Table 9.  
Step 5: The relative weight of the group was determined by summating the relative weight of its components (\(w_{ik}^{Y_{i}}\)), as shown in the fifth column in Table 9.  
The above steps were repeated to obtain the relative weight of the groups that were computed for several years. Figure 6 shows a sketch of the group's relative weight computation. The illustrative example of the computation of weight factors for 2023 is shown in Table 8.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Factors** & \(w_{jk}^{2022}\) & \(w_{jk-mod}^{2023}\) = \(\delta_{i}A_{k}w_{jk}^{2022}\) & \(w_{jk}^{2023}\) & **The Relative Weight of the Group** \\ \hline CP1 & 2.53 & 3.97 & 2.87 & \\ CP3 & 2.70 & 4.24 & 3.06 & \\ CP5 & 3.05 & 4.78 & 3.46 & \\ CP6 & 2.98 & 4.67 & 3.38 & \\ CP7 & 2.51 & 3.93 & 2.84 & \\ CP8 & 2.99 & 4.68 & 3.38 & \\ CP9 & 3.44 & 5.39 & 3.90 & \\ EF1 & 1.80 & 1.80 & 1.30 & \\ EF10 & 4.11 & 4.11 & 2.97 & \\ EF12 & 1.82 & 1.82 & 1.31 & \\ EF13 & 3.11 & 3.11 & 2.25 & 15.30 \\ EF7 & 1.82 & 1.82 & 1.32 & \\ EF8 & 3.45 & 3.45 & 2.50 & \\ EF9 & 5.04 & 5.04 & 3.65 & \\ EP4 & 5.98 & 10.22 & 7.39 & \\ EP5 & 4.16 & 7.11 & 5.14 & \\ EP6 & 4.50 & 7.68 & 5.55 & 25.16 \\ EP7 & 5.73 & 9.78 & 7.08 & \\ IQ1 & 7.59 & 12.95 & 9.37 & \\ IQ2 & 9.95 & 16.98 & 12.28 & 21.65 \\ PC1 & 3.21 & 3.21 & 2.32 & \\ PC2 & 1.98 & 1.98 & 1.43 & \\ PC3 & 2.04 & 2.04 & 1.47 & \\ PC4 & 2.94 & 2.94 & 2.13 & \\ PC5 & 3.20 & 3.20 & 2.32 & 15.00 \\ PC6 & 2.78 & 2.78 & 2.01 & \\ PC8 & 2.63 & 2.63 & 1.91 & \\ PC9 & 1.95 & 1.95 & 1.41 & \\ \multicolumn{4}{c}{\(\sum\) = 100\%} \\ \hline \hline \end{tabular}
\end{table}
Table 9: Illustrative example of computation of weight factors for 2023.