## 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
There are m experts to evaluate n risk evaluation indicators and so the evaluation results formed the original risk judgment matrix \(Z^{\prime}=\left(z^{\prime}_{ij}\right)_{n\times m}\), as in Formula (3): \[Z^{\prime}= \begin{bmatrix}z^{\prime}_{11}&z^{\prime}_{12}&\cdots&z^{\prime}_{ m}\\ z^{\prime}_{21}&z^{\prime}_{22}&\cdots&z^{\prime}_{2m}\\ \vdots&\vdots&&\vdots\\ z^{\prime}_{n1}&z^{\prime}_{n2}&\cdots&z^{\prime}_{mm}\end{bmatrix} \] (3)
2. Normalization processing of the original risk judgment matrix. In order to ensure that each indicator has the same importance in the evaluation process and to avoid the bias caused by the difference in the scale and the difference in the range of values, the raw data are normalized and converted into the [0, 1] interval as shown in Equation (4):  
\[z_{ij}=\frac{z_{ij}^{\prime}-\min\{z_{i}^{\prime}\}}{\max\{z_{i}^{\prime}\}-\min \{z_{i}^{\prime}\}} \tag{4}\]  
where \(z_{ij}\) denotes the normalized value of the jth sample of the ith indicator,\(z_{ij}^{\prime}\) denotes the original value of the jth sample of the ith indicator, and \(\min\)\(\left(z_{i}^{\prime}\right)\) and \(\max\)\(\left(z_{i}^{\prime}\right)\) denote the minimum and maximum values of the ith indicator, respectively. The normalized matrix is obtained after processing as Equation (5):  
\[Z= \begin{bmatrix}z_{11}&z_{12}&\cdots&z_{1m}\\ z_{21}&z_{22}&\cdots&z_{2m}\\ \vdots&\vdots&&\vdots\\ z_{n1}&z_{n2}&\cdots&z_{mm}\end{bmatrix}  \tag{5}\]  
3) Calculate the anti-entropy value of the indicators. Using the normalized data, calculate the anti-entropy value for each indicator, as Equation (6):  
\[H_{i}=-\sum\limits_{j=1}^{m}r_{ij}\ln(1-r_{ij}),1\leq j\leq m \tag{6}\]  
Among them, \(r_{ij}=\frac{z_{ij}}{\sum\limits_{j=1}^{n}z_{ij}}\) and m is the number of experts. 4) Calculate the objective weights of the indicators. The anti-entropy value is normalized to ensure that the sum of the weights of the indicators is 1. Equation (7) is as follows:  
\[w_{i}=\frac{H_{i}}{\sum\limits_{i=1}^{n}H_{i}},1\leq i\leq n \tag{7}\]  
where, \(w_{i}\) represents the normalized anti-entropy value of the ith indicator, and \(0\leq w_{i}\leq 1\). n represents the number of evaluation indicators. Through the above steps, the objective weights of each indicator can be obtained as follows in Equation (8):  
\[W^{(2)}=\{w_{i}\} \tag{8}\]  
(3) Combinatorial Empowerment Based on Game Theory  
Game theory is used to synthesize subjective and objective weights, considering them as two sides of the game, by solving the optimal solution (Nash equilibrium point) in order to obtain the combined weights [19]. The core of the methodology lies in the effective integration of subjective expert experience with data-driven objective analysis through a dynamic-balancing mechanism. Thus, it solves the inconsistency or conflict between the two in weight allocation, avoids the bias brought by a single assignment method, improves the stability and reliability of weights, and provides more comprehensive and accurate evaluation results. The specific steps are as follows:  
(1) Linear combination model construction. The subjective weight vector W\({}^{(1)}\) determined by the OPA method and the objective weight vector W\({}^{(2)}\) determined by the inverse entropy weight method are linearly combined, and the linear combination coefficients \(\alpha_{1}\) and \(\alpha_{2}\) are introduced as in Equation (9):  
\[W=\partial_{1}W^{(1)}+\partial_{2}W^{(2)} \tag{9}\]The aim is to minimize the deviation between subjective and objective weights by optimizing the combination coefficient \(\alpha\). 2. Optimizing objective function construction. In order to find the optimal weight vector \(\mathrm{W^{\prime}}\) in the set of possible weight vectors, the optimization objective function is constructed.