# Improved PBFT Algorithm Based on Comprehensive Evaluation Model
## 2 Preliminaries
### Entropy Weight Method  
There are two commonly used methods for calculating index weights: Subjective weighting and objective weighting, and the entropy weight method avoids the interference of subjective factors and has strong objectivity [25]. Entropy is a thermodynamic concept. [PERSON] first introduced it to information theory to measure the uncertainty of a system [22]. The entropy weight method determines the weight according to the information entropy of the evaluation index. The smaller the information entropy of the evaluation index is, the greater the fluctuation of the data of the evaluation index, and the greater the corresponding weight. In contrast, the greater the information entropy of the evaluation index is, the smaller the corresponding weight. Therefore, compared with the subjective weighting method, the entropy weight method has higher credibility and accuracy. The calculation steps are as follows:  
Step 1: Original data pre-processing. The evaluation expert \(K\) establishes an initial evaluation matrix according to the original data of the evaluation index, and the evaluation matrix \(X_{k}\) is as follows:  
\[X_{k}=\left(x_{ij}\right)_{m\times d}=\begin{bmatrix}x_{11}&x_{12}&\cdots&x_{1d }\\ x_{21}&x_{22}&\cdots&x_{2d}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m1}&x_{m2}&\cdots&x_{md}\end{bmatrix}. \tag{1}\]  
Noteworthily, \(X_{k}\) represents the initial evaluation matrix of evaluation expert \(k\), \(x_{ij}\) represents the original data of the _j_-th index of the _i_-th evaluation object, \(m\) represents the number of evaluation objects, and \(d\) represents the number of evaluation indices, where \(0\leq k\leq m,\;0\leq i\leq m,\;0\leq j\leq d\). For positive index and negative index, the standardized treatment formulas are obtained as follows:  
\[y_{ij}=\frac{x_{ij}-\min\limits_{1\leq i\leq m}\left\{x_{ij}\right\}}{\max \limits_{1\leq i\leq m}\left\{x_{ij}\right\}-\min\limits_{1\leq i\leq m}\left\{ x_{ij}\right\}}, \tag{2}\]  
\[y_{ij}=\frac{\max\limits_{1\leq i\leq m}\left\{x_{ij}\right\}-x_{ij}}{\max \limits_{1\leq i\leq m}\left\{x_{ij}\right\}-\min\limits_{1\leq i\leq m}\left\{ x_{ij}\right\}}. \tag{3}\]  
Figure 1: The consensus process of PBFT algorithm.  
where \(y_{ij}\) represents the standardized evaluation index data. The calculation process of \(y_{ij}\) normalization is as follows:  
\[p_{\bar{y}}=\frac{y_{ij}}{\sum\limits_{i=1}^{m}y_{ij}}. \tag{4}\]  
\(p_{ij}\) represents the normalized value of the evaluation object corresponding to index \(j\), and obtains a normalized matrix \(P_{k}\):  
\[P_{k}=\begin{bmatrix}p_{11}&p_{12}&\cdots&p_{1d}\\ p_{21}&p_{22}&\cdots&p_{2d}\\ \vdots&\vdots&\ddots&\vdots\\ p_{m1}&p_{m2}&\cdots&p_{md}\end{bmatrix}. \tag{5}\]  
where \(P_{k}\) denotes the standardized matrix of evaluation expert \(k\).  
Step 2: Calculate the information entropy of the evaluation index. The information entropy \(e_{j}\) of the evaluation index \(j\) is obtained:  
\[e_{j}=-\frac{1}{ln(m)}\sum\limits_{i=1}^{m}p_{ij}ln(p_{\bar{y}}). \tag{6}\]  
Step 3: Calculate the entropy weight of the evaluation index.  
\[\beta_{j}=\frac{1-e_{j}}{\sum\limits_{j=1}^{d}(1-e_{j})}. \tag{7}\]  
\(\beta_{j}\) represents the entropy weight of the evaluation index \(j\) in Equation (7), and then the weight vector \(\beta_{k}\) of the evaluation index \(d\) set of the evaluation objects \(m\) by evaluation expert \(k\) is:  
\[\beta_{k}=(\beta_{1},\,\beta_{2},\,\cdots,\beta_{d}). \tag{8}\]