# Hybrid Methods for Selecting Precast Concrete Suppliers Based on Factory Capacity
## 3 Methodology
### Estimate Each Supplier's Qi Score Using the WASPS Method  
The WASPS method is one of the most commonly used Multicriteria Decision-Making (MCDM) techniques. Before detailing its application, it is essential to clearly state the underlying assumptions of this method and their implications for this study. The WASPS method, like other multi-attribute utility theory (MAUT) models, operates on several key prerequisites. First, it assumes that the evaluation criteria are mutually independent, meaning the performance score of a supplier on one criterion (e.g., safety standards) does not influence its score on another (e.g., on-time delivery). While this assumption simplifies the model, it may not accurately reflect real-world scenarios, where factors such as cost and quality are often interrelated. Second, the method assumes that the criteria weights, derived here from SWARA, accurately reflect the decision-maker's valid preferences. Finally, it assumes that a linear trade-off between criteria is acceptable, as established by the WSM component. In this study, these assumptions are deemed reasonable for a strategic-level assessment, as the criteria are selected to be as distinct as possible, and the weights are derived from a panel of industry experts. However, for projects requiring a deeper analysis of criterion interdependencies, an extension of this model using methods like the Analytic Network Process (ANP) could be considered as a future research direction. WASPS combines the Weighted Sum Model (WSM) and the Weighted Product Model (WPM) to enhance ranking accuracy. According to [PERSON] and [PERSON] (2019) [39], the WASPS method is a compensatory technique with independent attributes. The qualitative features are changed into quantitative characteristics. Besides generating accurate outputs, most users also prefer using WASPS because its computing process is simple. There are four significant steps associated with analyzing data using WASPS. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & **C1** & **C2** & **C3** & **C4** & **C5** & **C6** & **C7** & **C8** & **C9** \\ \hline A1 & 0.092 & 0.09 & 0.087 & 0.118 & 0.118 & 0.068 & 0.123 & 0.056 & 0.028 \\ \hline A2 & 0.077 & 0.075 & 0.073 & 0.098 & 0.098 & 0.056 & 0.103 & 0.047 & 0.023 \\ \hline A3 & 0.128 & 0.125 & 0.121 & 0.164 & 0.164 & 0.094 & 0.171 & 0.078 & 0.039 \\ \hline A4 & 0.077 & 0.075 & 0.073 & 0.098 & 0.098 & 0.056 & 0.103 & 0.047 & 0.023 \\ \hline A5 & 0.064 & 0.063 & 0.061 & 0.082 & 0.082 & 0.047 & 0.086 & 0.039 & 0.02 \\ \hline \end{tabular}
\end{table} 
Table 12: CQWs of the alternative supplier using PCA. After computing the CQW matrix and the weights of the nine criteria (Table 5), the WASPAS method was applied to calculate the Q\({}_{i}\) of the precast suppliers based on the WSM and WPM. The WSM and WPM methods are used to calculate the total relative importance of the \(j\)th alternative using Equations (9) and (10). For these equations, \(w_{i}\) represents the weight (relative importance) of the \(i\)th criterion. \[Q_{i}^{(1)}=\sum_{i=1}^{n}{CQW_{ij}W_{i}} \tag{9}\]  
\[Q_{i}^{(2)}=\prod_{i=1}^{n}{\left({CQW_{ij}}\right)^{w_{i}}} \tag{10}\]  
In the last step, a generalized equation (Equation (11)) is used to calculate the total relative significance of each alternative. Establishing this generalized equation increases the ranking accuracy and improves the effectiveness of the associated decision-making processes. \[Q_{i}=\lambda Q_{i}^{(1)}+(1-\lambda)Q_{i}^{(2)}=\lambda{\sum_{i=1}^{n}{CQW_{ ij}w_{i}}}+(1-\lambda)\prod_{i=1}^{n}{\left({CQW_{ij}}\right)^{w_{i}}} \tag{11}\]  
The values of \(\lambda\) range between 0 and 1. The WASPAS method becomes the WPM or WSM method if the value of \(\lambda\) is 0 and 1, respectively [40]. When using WASPAS, the most suitable alternative is the one with the highest Q value. The results of the WSM and WPM are shown in Tables 13 and 14, respectively.