## 4 Evaluation Model Based on AH-Entropy and Normal Cloud Model
### Determination of Combination Weights
#### 4.2.2 Deriving Objective Weights with Entropy  
In information theory, the definition of information entropy is as follows:  
\[H(x)=-\sum_{i}p(x_{i})\ln p(x_{i}) \tag{9}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Intensity of** & **Definition** & **Explanation** \\ \hline
1 & Equal Importance & Two indicators contribute equally to the objective \\ \hline
3 & Moderate importance & Experience and judgement slightly favor one indicator over another \\ \hline
5 & Strong importance & Experience and judgement strongly favor one indicator over another \\ \hline
7 & Very strong or demonstrated importance & An indicator is favored very strongly over another; its dominance is demonstrated in practice \\ \hline
9 & Extreme importance & The evidence favoring one indicator over another is of the highest possible order of affirmation \\ \hline
2, 4, 6, 8 & The median value of the above judgment & — \\ \hline  & If indicator \(i\) has one of the above & \\  & non-zero numbers assigned to it & \\ Reciprocals of above & when compared with indicator \(j\), then & — \\  & \(j\) has the reciprocal value when compared & \\  & with \(i\) & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Fundamental scale of absolute numbers.  
where, \(x\) represents a random variable, and \(p(x)\) represents the output probability function, \(p(x_{i})\in[0,1]\), \(\sum p(x_{i})=1\).  
According to Equation (9), a higher variability of the variable corresponds to a greater entropy, signifying that is carries a larger amount of information. Information entropy primarily indicates the extent of variation in indicators. A high entropy for a specific evaluation indicator suggests that the indicator provides more information in the comprehensive evaluation, resulting in a higher corresponding weight. Conversely, a lower weight is assigned. In cases where the evaluation values of an indicator are identical, it conveys no information and is assigned a weight of 0 in the comprehensive evaluation. The weight calculation process for the entropy method is detailed as follows [34; 38]:  
Step 1: Construct the original matrix. Assuming the number of evaluation indicators is \(n\) and the number of evaluation objects is \(m\), an original matrix \(B=(b_{\mathrm{ij}})_{m\times n}\) can be constructed as follows:  
\[B=\begin{bmatrix}b_{11}&b_{12}&\dots&b_{1n}\\ b_{21}&b_{22}&\dots&b_{2n}\\ \dots&\dots&\dots&\dots\\ b_{m1}&b_{m2}&\dots&b_{mn}\end{bmatrix} \tag{10}\]  
where \(b_{ij}\) represents the \(j\)th indicator detection result of the \(i\)th evaluation object.  
Step 2: The indices \(B_{ij}\) are normalized. The positive indicator is calculated as follows:  
\[{b^{\prime}}_{ij}=\frac{b_{ij}-\min\{b_{ij},\dots,b_{nj}\}}{\max\{b_{1j}, \dots,b_{nj}\}-\min\{b_{ij},\dots,b_{nj}\}} \tag{11}\]  
The negative indicator is calculated as follows:  
\[{b^{\prime}}_{ij}=\frac{\min\{b_{ij},\dots,b_{nj}\}-b_{ij}}{\max\{b_{1j}, \dots,b_{nj}\}-\min\{b_{ij},\dots,b_{nj}\}} \tag{12}\]  
Step 3: Determining the proportion of evaluation indicators in the scheme is achieved as follows:  
\[p_{ij}=\frac{{b^{\prime}}_{ij}}{\sum\limits_{i=1}^{m}{b^{\prime}}_{ij}},i=1,2,\dots,m;j=1,2,\dots,n \tag{13}\]  
Step 4: Calculate entropy. The computation of entropy for the evaluation indicators is as follows:  
\[H_{j}=-k\sum\limits_{i=1}^{m}{p_{ij}}\ln{p_{ij}},j=1,2,\dots,n,k=\frac{1}{\ln m} \tag{14}\]  
where \(H_{j}\) is the entropy of the \(j\)th indicator. If \(p_{ij}=0\), then \(p_{ij}\mathrm{ln}p_{ij}=0\), while ensuring \(H_{j}\in[0,1]\).  
Step 5: Calculate entropy weights. The calculation of the entropy weights for the evaluation indicators is as follows:  
\[\omega_{j}=\frac{1-H_{j}}{n-\sum\limits_{i=1}^{n}{H_{j}}},j=1,2,\dots,n \tag{15}\]  
where \(w_{j}\) is the entropy weight of the \(j\)th indicator, \(w_{j}\in[0,1]\), \(\sum\limits_{j=1}^{n}{\omega_{j}}=1\).