## 2 Materials and Methods
### Construction of Quality Risk Evaluation Indicator System for Assembled Building
#### 2.1.2 Determination of Weights of Indicators for Quality Risk Evaluation of Assembled Building  
The ordinal priority approach method is a multi-attribute decision-making (MADM) method [17], which establishes a mathematical model to solve the multi-attribute decision-making problem by allowing the decision maker to rank the indicators to determine the weights. The method is simple and intuitive to operate, only needs to be ranked without complex numerical scoring, it can fully reflect the subjective preference and experience of the decision maker, and it simplifies the process of determining the weights by converting the ranking results into weights. The weights are determined by direct comparison of the experts' rankings, avoiding the inconsistency problem that may be caused by the pairwise comparison matrix in the traditional AHP method, so there is no need for consistency testing. In addition, the core strength of the ordinal priority approach method is its high sensitivity to changes in decision preferences. Compared with the traditional weight calculation methods, the OPA method accurately captures the subtle differences in the ranking through a linear optimization model. For example, in the quality risk evaluation of this study, for indicators U\({}_{13}\) (number of prefabricated component design changes) and U (construction network rate) to be ranked under U\({}_{1}\) in the decision-making design phase the differences in the experts' rankings must reflect their different emphasis on the rationality of the design scheme and the importance of construction management. The OPA method is able to directly quantify these subtle changes in ranking into weight differences, thus improving the science and rationality of weight allocation. The use of the OPA method first requires policymakers to identify the basic criteria and sub-criteria and to select expert groups. To ensure the accuracy of the weighting calculation, the panel members come from the fields of assembly building design, production and construction, with diverse professional backgrounds and rich practical experience. The experts ranked the indicators according to their experience, education and other factors to prioritize them, and then prioritized each criterion or sub-criterion separately and prioritized the alternatives within each criterion. Finally, a linear model (1) was constructed based on the collected data and programmed to be solved using MATLAB R2020a to ensure the accuracy of the sorted data and the reliability of the results. Max Z  
S.t:  
\[ \begin{array}{c}Z\leq i\left(j\left(k\left(W_{ijk}^{k}-W_{ijk}^{k+1}\right) \right)\right)\quad\forall i,j\text{ and }k\\ Z\leq jimW_{ijk}^{m}\quad\forall i,j\text{ and }k\\ \sum\limits_{i=1}^{p}\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{m}W_{ijk}=1\\ W_{ijk}\geq 0\quad\forall i,j\text{ and }k\end{array}  \tag{1}\]  
where \(i\) represents experts, \(j\) represents standardized preferences, and \(k\) represents alternatives. After solving, the model subjective weights are determined through Equation (2):  
\[W^{(1)}=\sum\limits_{i=1}^{p}\sum\limits_{k=1}^{m}W_{ijk},\forall j \tag{2}\]  
(2)Anti-entropy weighting method  
Compared with the traditional entropy weighting method, the anti-entropy weighting method [18] can more effectively reflect the differences between indicators when calculating objective weights. By defining a kind of anti-entropy value, the anti-entropy weight method attenuates the sensitivity to the differences of the indicators, which makes it possible to consider the importance of each indicator more fairly in the weight calculation, trying to avoid the situation of extreme weights in the process of assigning weights, and especially when dealing with some key indicators, it can better reflect their actual impact. The steps for determining the objective weights of the indicators by the anti-entropy weighting method are as follows:  
1. Determine the original risk judgment matrix.