## 2 Materials and Methods
### The Steps of the Multi-Attribute Weighted Intelligent Gray Target Decision-Making Algorithm  
Step 1: The entirety of events within a given research scope is referred to as the event set within that research scope, denoted as \(A=\{a_{1},a_{2},\ldots,a_{n}\}\), where \(a_{i}(i=1,2,\ldots,n)\) represents the \(i\)-th event. Correspondingly, the complete set of possible strategies forms the strategy set, denoted as \(B=\{b_{1},b_{2},\ldots,b_{m}\}\), where \(b_{j}(j=1,2,\ldots,n)\) represents the \(j\)-th decision. Based on the event and strategy sets, the decision-making solution set, \(S=\{s_{ij}=\big{(}a_{i},b_{j}\big{)}|a_{i}\in A,b_{j}\in B\}\), is constructed;  
Step 2: Determine the decision objectives \(k=1,2,\ldots,s\) based on actual needs;  
Step 3: Employ EEM to determine the indicator weights \(\eta_{1},\eta_{1},\ldots,\eta_{s}\);  
Step 4: For objectives \(k=1,2,\ldots,s\), compute the corresponding matrix of target effect samples.  
\[U^{k}=\Big{(}u^{(k)}_{ij}\Big{)}=\begin{bmatrix}u^{(k)}_{11}&u^{(k)}_{12}& \ldots&u^{(k)}_{1n}\\ u^{(k)}_{21}&u^{(k)}_{22}&\ldots&u^{(k)}_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ u^{(k)}_{n1}&u^{(k)}_{n2}&\ldots&u^{(k)}_{nm}\end{bmatrix}; \tag{1}\]  
Step 5: According to the different types of effect objectives (effect type objectives, cost type objectives, and moderate type objectives), set the critical value for the target effect, \(u^{(k)}_{ij}\).  
To ensure that the measures of various objective effects adhere to standardization, i.e., \(r^{(k)}_{ij}\in[-1,1]\), the following conditions are established:  
For the benefit type objective, set the decision-making gray target \(u^{(k)}_{ij}\geq 2u^{(k)}_{i_{0}j_{0}}-\underset{i}{max}\underset{j}{max}\Big{\{}u^{(k)}_ {ij}\Big{\}}\) for the objective \(k\);  
For the cost type objective, set the decision-making gray target \(u^{(k)}_{ij}\leq 2u^{(k)}_{i_{0}j_{0}}-\underset{i}{min}\underset{j}{\min}\Big{\{}u^{(k)}_ {ij}\Big{\}}\) for the objective \(k\);  
For the moderate type objective with the effect value falling below the lower limit critical value \(A-u^{(k)}_{i_{0}j_{0}}\), set the decision-making gray target \(u^{(k)}_{ij}\geq A-2u^{(k)}_{i_{0}j_{0}}\) for the objective \(k\);  
For the moderate type objective with the effect value exceeding the upper limit critical value \(A+u^{(k)}_{i_{0}j_{0}}\), set the decision-making gray target \(u^{(k)}_{ij}\leq A+2u^{(k)}_{i_{0}j_{0}}\) for the objective \(k\).  
Step 6: Compute the matrix of uniform effect measures for the objective \(k\).  
Due to the varying meanings, dimensions, and natures of the effect values for different objectives, direct comparison of these values is typically impractical. To facilitate the generation of comparable values, it is essential to convert the effect values of the objective samples, \(u_{ij}^{(k)}\), into uniform effect measure values. This conversion forms the basis for establishing the matrix of uniform effect measures for the objective samples.  
\[R^{(k)}=\left(r_{ij}^{(k)}\right)=\begin{bmatrix}r_{11}^{(k)}&r_{12}^{(k)}& \ldots&r_{1m}^{(k)}\\ r_{21}^{(k)}&r_{22}^{(k)}&\ldots&r_{2m}^{(k)}\\ \vdots&\vdots&\ddots&\vdots\\ r_{n1}^{(k)}&r_{n2}^{(k)}&\ldots&r_{nm}^{(k)}\end{bmatrix} \tag{2}\]  
Step 7: According to \(r_{ij}^{(k)}=\sum_{k=1}^{5}\eta_{k}*r_{ij}^{(k)}\), the matrix of synthetic effect measures is computed as follows:  
\[R=\left(r_{ij}\right)=\begin{bmatrix}r_{11}&r_{12}&\ldots&r_{1m}\\ r_{21}&r_{22}&\ldots&r_{2m}\\ \vdots&\vdots&\ddots&\vdots\\ r_{n1}&r_{n2}&\ldots&r_{nm}\end{bmatrix} \tag{3}\]  
Step 8: Determine whether the value of the synthetic effect measure hits the gray target. If \(r_{ij}^{(k)}\in[0,1]\), it indicates a hit on the gray target for the effect value of the objective \(k\); conversely, if \(r_{ij}^{(k)}\in[-1,0]\), it denotes a miss from the target for the effect value of the objective \(k\). Following this, the optimal strategy \(b_{j_{0}}\) or the optimal solution \(s_{i_{0}|_{0}}\) is identified in accordance with Definition 1.