# Hybrid Methods for Selecting Precast Concrete Suppliers Based on Factory Capacity
## 3 Methodology
### Estimate the CQW of the Alternative Based on Its Capacity
To ensure the suitability of the data, the Kaiser-Meyer-Olkin (KMO) coefficient, which should exceed 0.5, was examined. The KMO value was 0.708, greater than 0.5, as shown in Table 7. The KMO test is intended to measure the adequacy of the sample size. In this study, the KMO value is 0.708, which exceeds the recommended threshold of 0.5. This result implies a sufficient sample size ([PERSON], 2021) [46]. The findings in Table 8 display the initial eigenvalue of the nine components. Eigenvalues represent the variance explained by each principal component. The initial eigenvalue of the first component is 49.117, indicating that it explains a significant amount of variance in the data. The eigenvalues of the subsequent components decrease, with the second component having an initial eigenvalue of 9.345. A rotated component matrix in PCA represents the relationship between the original variables and the principal components that have been rotated. It is a transformed representation of the loadings of each variable on the principal components. A rotation technique is often applied to enhance the interpretability of the components. The rotation aims to achieve a more straightforward and meaningful structure by maximizing the loadings of some variables on specific components while minimizing loadings on others. The rotated matrix loadings (RMLs) for the first and second components are shown in Table 9. The values in the matrix indicate the strength and direction of the relationship between the variables. Each cell in the matrix represents the correlation between a criterion and a principal component that has been rotated. The first component was chosen based on its high eigenvalue and the comprehensiveness of the first component over the nine criteria. Therefore, the RML of the first component was used to compute the normalized coefficients (\(\overline{C}\)) of the criteria, as shown in Equation (6). The values of \(\overline{C}\) for the nine criteria are presented in Table 10.  
\[\overline{C}_{i}=\frac{RML_{i}}{\sum RML} \tag{6}\]  
The research team visited their respective manufacturing facilities in person to gain a deeper understanding of each of the supplier's capacities. The framework was applied to suppliers in Riyadh, Saudi Arabia, where the precast concrete market has few reliable providers. Therefore, five suppliers were selected using the developed framework to select the best among them. Table 11 summarizes each supplier's annual production capacity, measured in cubic meters. For context and a better understanding of capacity values, the research team noted that a precast house characteristically needs an average capacity of 200 to 300 cubic meters. The normalized capacity supplier (\(NCS\)) entails dividing \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(RML\) & \(\overline{C}\) \\ \hline C1 & 1.865 & 0.128 \\ \hline C2 & 1.827 & 0.125 \\ \hline C3 & 1.760 & 0.121 \\ \hline C4 & 2.383 & 0.164 \\ \hline C5 & 1.371 & 0.094 \\ \hline C6 & 2.484 & 0.171 \\ \hline C7 & 1.130 & 0.078 \\ \hline C8 & 0.568 & 0.039 \\ \hline C9 & 1.170 & 0.080 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: RML and \(\overline{C}\) of the nine criteria. each supplier's capacity (\(CS_{i}\)) by the maximum capacity among all suppliers (\(CS_{max}\)), as computed in Equation (7), facilitating the easier comparison and evaluation of each supplier's capacity. \[NCS_{i}=\frac{CS_{i}}{CS_{max}} \tag{7}\]  
The final column in Table 11 presents the \(NCS\) values. These capacity assessments and site visits yielded valuable insights into each supplier's production capacity. The selection of these five specific suppliers for the case study was deliberate, aiming to encompass a representative range of production capacities within Riyadh's limited pool of reliable precast concrete providers. The observed variation in supplier productivity, as detailed in Table 11, is not an arbitrary aspect but a fundamental characteristic that the proposed framework is designed to explicitly address and leverage.