## 5 Construction of Fuzzy Comprehensive Evaluation
### Construction of Fuzzy Comprehensive Evaluation Model Based on Combination Weighting Method  
1. Decide the factor set and evaluation set  
Assume that \(N\) is the number of single factors belonging to the same level, \(u_{i}(i=1,2,\cdots,N)\) is the evaluation factor, \(U=\{u_{1},u_{2},\cdots,u_{N}\}\) is the factor set, \(n\) is the rating level, \(p_{j}(j=1,2,\cdots,n)\) is the evaluation rating standard, and \(V=\{v_{1},v_{2},\cdots,v_{n}\}\) is the determined evaluation set. Five levels for the evaluation of energy saving and emission reduction effects for electricity retailers are confirmed in this paper, namely, \(V=\{\)Excellent, Good, Moderate, Poor, Extremely poor\(\}\). \(H=\{5,4,3,2,1\}\) is the corresponding scale.  
2. Construct the fuzzy relation matrix  
The fuzzy relation matrix is also called the membership degree matrix, in which \(r_{ij}\) represents the relationship of the membership degree between the evaluation factor \(u_{i}\) and the corresponding evaluation level \(v_{j}\). In this paper, the membership degree is determined by the Delphi method. When considering the construction of the indicator system and the impact of specific indicators on the energy saving and emission reduction effects of electricity retailers, experts judge these indicators to determine the membership degree. The fuzzy vector \(R_{i}\) formed by the evaluation factor \(u_{i}\) is defined in Equation (12).  
\[R_{i}=\big{(}r_{i1},r_{i2},\cdots,r_{ij},\cdots r_{Nn}\big{)},\ i=1,2,\cdots,N, \ j=1,2,\cdots,n \tag{13}\]  
where \(0<r_{ij}<1\). The fuzzy relation matrix \(R\) composed of \(n\) elements is shown in Equation (13).  
\[R=\left[\begin{array}{cccc}r_{11}&r_{12}&\cdots&r_{1n}\\ r_{21}&r_{21}&\cdots&r_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ r_{N1}&r_{N1}&\cdots&r_{Nn}\end{array}\right] \tag{14}\]  
3. Determine the result set of fuzzy evaluation  
In order to make the best of all the information of the matrix, the \(M(\cdot,\oplus)\) operator is selected to determine the fuzzy judgment result set \(B\). The calculation of fuzzy judgment is given by Equation (14).  
\[B=w\circ R \tag{15}\]  
where \(w\) is the combination weights of the evaluation indicators given through AHP and the entropy weight method.  
4. Determine fuzzy comprehensive evaluation results  
In accordance with the principle of maximum membership degree, the energy saving and emission reduction effects of electricity retailers are classified into the corresponding evaluation levels based on the fuzzy evaluation results.  
In order to conduct a further assessment of the overall energy saving and emission reduction effects of electricity retailers, multi-level fuzzy evaluation should be carried out after obtaining the single-level fuzzy comprehensive evaluation results. In the light of the principle of maximum membership degree, the level of the maximum membership degree is the final evaluation level of the energy saving and emission reduction effects of the evaluated electricity retailers.  
The evaluation process of energy saving and emission reduction effects for electricity retailers based on fuzzy combination weighting method constructed in this paper is as shown in Figure 3.