## 2 Materials and Methods
### Constructing a Game Theory-Combinatorial Empowerment Three-Dimensional Cloud Evaluation Model
#### 2.2.1 Basic Theory of the Cloud Model  
Cloud modeling is a theory and method that combines fuzzy mathematics and probability statistics to deal with complex systems with uncertainty and ambiguity, which can transform qualitative and quantitative problems into each other, and its core idea is to represent the ambiguity and randomness of concepts through cloud droplets, and to construct cloud diagrams to visualize the state of the system and use it to judge the level of risk. The cloud model has three numerical features for describing the morphological distribution of the cloud, which are Expectation \(\rm E_{x}\), Entropy \(\rm E_{n}\), and Hyper-entropy \(\rm H_{e}\)[20]. Expectation \(\rm E_{x}\) reflects the center value of the cloud model, which is the mean value of the cloud droplets, and it indicates the centralized tendency of things; entropy \(\rm E_{n}\) reflects the uncertainty and ambiguity of the cloud model, and the larger the entropy, the higher the ambiguity of the concepts; and hyper-entropy \(\rm H_{e}\) reflects the degree of dispersion of entropy, which is the standard deviation of entropy, and the larger the hyper-entropy is indicates the higher the uncertainty of the cloud droplets in the cloud model. The basic algorithm of the cloud model is implemented through cloud generators, which are categorized into Forward Cloud Transformation and Backward Cloud Transformation. The Backward Cloud Generator is responsible for converting cloud droplets \(\rm Drop(x_{i},\,\mu_{i})\) (\(i=1,\,2,\,3\ldots\)) into cloud numerical features (\(\rm E_{x}\), \(\rm E_{n}\), \(\rm H_{e}\)), which transforms quantitative data into qualitative concepts. The forward cloud transformation generator then converts cloud numerical features (\(\rm E_{x}\), \(\rm E_{n}\), \(\rm H_{e}\)) to cloud droplets \(\rm Drop(x_{i},\,\mu_{i})\) (\(i=1,\,2,\,3\ldots\)), realizing the conversion from qualitative concepts to quantitative data. In this paper, the inverse cloud transformation is carried out first to determine the cloud digital features of the quality risk indicators of the whole process of assembly building, and then the forward cloud transformation is carried out to draw the risk evaluation cloud diagram. The principle of the cloud generator is shown in Figure 3.  
Figure 2: Quality risk evaluation model for the whole process of assembled building. Each of the underlying variables of the three-dimensional cloud model, risk occurrence probability x, risk occurrence consequence y, and risk concealment z, are evaluated by constructing the model using three sets of numerical features of qualitative concepts, respectively. \[\left\{ \begin{aligned} (x,y,z)=T(E_{X,x},E_{X,y},E_{X,z},E_{n,x},E_{n,y},E_{n,z})\\ (D_{x},D_{y},D_{z})=T(E_{n,x},E_{n,y},E_{n,z},H_{e,x},H_{e,y},H_{e,z})\\ \mu=e^{-\frac{1}{2}[\frac{(x-E_{X,x})^{2}}{D_{x}^{2}}+\frac{(y-E_{ X,y})^{2}}{D_{y}^{2}}+\frac{(z-E_{X,z})^{2}}{D_{z}^{2}}]}\end{aligned} \right. \tag{13}\]  
where t represents a three-dimensional random obeying a normal distribution, \(E_{X,x}\), \(E_{X,y}\), \(E_{X,z}\) are the expectations of the three basis variables, \(E_{n,x}\), \(E_{n,y}\), \(E_{n,z}\) are the entropies of the three basis variables, and, \(H_{e,x}\), \(H_{e,y}\), \(H_{e,z}\) are the hyper-entropy of the three basis variables. \(\mu\) is the degree of certainty that a three-dimensional cloud droplet belongs to a certain characterization concept (\(\mu\in[0,1]\)). We call the cloud model consisting of cloud drops drop(x, y, z, \(\mu\)) conforming to the above equation a three-dimensional normal cloud model. The three-dimensional cloud model represents each risk indicator as a cloud containing a large number of cloud droplets that are distributed in three-dimensional space, reflecting the characteristics and uncertainties of the indicator. Standard clouds, on the other hand, are pre-defined to represent reference clouds for different risk levels.