# A Multi-Level Fuzzy Evaluation Method for the Reliability of Integrated Energy Systems
## 3 Methods  
### Combined Empowerment Based on AHP-EWM  
A correct evaluation system is needed to determine a reasonable weight, including the subjective weight and objective weight. Subjective weight factors can reflect intentions of decision makers, but they are susceptible to subjective arbitrariness. Objective weight methods can reflect the characteristics of data without considering subjective opinions, though the calculated weights may become divorced from reality. Therefore, hybrid indicators should be used to obtain the advantages of the two methods, which can not only reflect the preference of decision makers but also reflect the influence of objective factors. Several indicators such as network loss and response efficiency need to be considered when we evaluate the reliability of IES [35]. Firstly, the weight coefficient needs to be determined, since the importance of each indicator should be considered in the comprehensive evaluation. To make the weight coefficient more accurate, the analytic hierarchy process (AHP) proposed by [PERSON] is adopted to determine the weight [36; 37]. AHP is a multi-criteria decision-making method, and its execution process is shown in Figure 2.  
For the judgment process, the priorities of the two indicators are shown in Table 2. B\({}_{\rm i}\) and B\({}_{\rm j}\) represent the impact of the i\({}_{\rm th}\) and j\({}_{\rm th}\) indexes, respectively. The B\({}_{\rm i}\)/B\({}_{\rm j}\) becomes larger with the importance of the i\({}_{\rm th}\) index increasing relative to the j\({}_{\rm th}\) index [38]. Figure 2: Execution process of AHP. With the help of a two-by-two comparison between factors, it is transformed into mathematical information that can be processed by a computer. Hierarchical analysis requires experts to understand the nature of the development of things, but when the problem reaches a certain level of complexity, it is easy for error to occur [39]. In information theory, entropy is a measure of the uncertainty of a system. In general, the higher the value of entropy is, the greater the dispersion degree of the indicators [40]. If a random variable \(X\) may take the value of \(\left\{x1,\,x2,\,\dots\,xn\right\}\), its probability distribution is \(P(X=x_{i})=p_{i}(i=1,\,2,\dots,n)\), and the entropy value of random variable \(X\) is defined as shown in Equation (7). Entropy weight method (EWM) relies mainly on the calculation of indicator data, which sometimes results in missing errors and makes it difficult to obtain an idealized indicator ranking [41]. \[H(X)=-\sum_{x}P(x)\log p(x) \tag{7}\]  
In view of the fact that the relevant basic data of IES are unsatisfactory and incomplete, based on the above theory, this paper choose the combination of subjective and objective \"AHP-EWM\" method [42] to determine the indicator weights, in order to clarify the influence of different types of indicators on the assessment results. The following weighting algorithm is proposed. Step 1: Construct a judgment matrix \(M_{ij}\) for the number of states (i) and the number of indicators (j) of the system;  
\[M_{ij}=\prod_{j=1}^{n}B_{i}/B_{j} \tag{8}\]  
where \(M_{ij}\) donates the matrix line product, and the value of \(B_{i}/B_{j}\) is shown in Table 2.  
Step 2: The consistency ratio index (CR) was calculated to ensure the reliability of the AHP weights:  
\[CR=CI/RI \tag{9}\]  
RI donates the average value of the result consistency index according to the order of the matrix. CI is the consistency index and can be expressed as follows:  
\[CI=\frac{\lambda_{max}-m}{m-1} \tag{10}\]  
where \(\lambda_{max}\) donates the maximum or principal eigenvalue of the matrix. When CR was less than 0.10, the matrix had good consistency, otherwise it should be changed. Step 3: To avoid the influence of units of measure, the data were standardized as follows:  
\[Y_{ij}=\frac{x_{ij}-min\left\{x_{ij},x_{2j},\dots,x_{nj}\right\}}{\max\left\{ x_{ij},x_{2j},\dots,x_{nj}\right\}-min\left\{x_{ij},x_{2j},\dots,x_{nj}\right\}} \tag{11}\]  
Step 4: The information entropy was then calculated.