## 2 Theoretical Background
### Fuzzy Theory
#### 2.1.5 Conducting the Fuzzy Evaluation  
If the fuzzy evaluation matrix of a certain scheme on the evaluation target is the R in Equation (7), the weighted comprehensive fuzzy evaluation and the product of fuzzy matrices to be considered is as follows:  
\[B=A\bullet R=\left[b_{1},b_{2},\ldots b_{j},\ldots b_{m}\right] \tag{8}\]  
where \(\bullet\) indicates the fuzzy synthetic operation. There are several synthetic approaches of fuzzy weight matrix \(A\) and factor judgment matrix R. In this study, four synthetic approaches were implemented for the analysis and comparison of the evaluation results. These four models are described as follows:  
Model 1: \(M\) (\(\wedge\), \(\vee\)) synthetic operation, in which  
\[b_{j}=\bigvee\limits_{i=1}^{m}(a_{i}\wedge r_{ij});j=1,2,\ldots,n, \tag{9}\]  
where \(\vee\) and \(\wedge\) indicate taking the maximum or minimum. When taking the minimum, the \(r_{ij}\) of all \(r_{ij}>w_{ij}\) is not taken into consideration. Therefore, when there are various factors, the values are bound to be large after the weighted coefficients are normalized. As a result, a lot of the single-factor evaluation information will be lost. When there are fewer factors, \(w_{ij}\) might be larger and therefore the \(w_{ij}\) of all \(w_{ij}>r_{ij}\) will not be taken into consideration. The influence of the main factors might be lost. Model 2: \(M\) (\(\bullet\), \(\vee\)) synthetic operation, in which  
\[b_{j}=\bigvee\limits_{i=1}^{m}(a_{i}r_{ij});j=1,2,\ldots,n. \tag{10}\]  
The feature of this model deals with no loss of any useful information. However, the operation of taking the maximum \(\vee\) could possibly lose a lot of useful information. It can still reflect the single-factor evaluation result and the degree of importance of each factor. Model 3: \(M\) (\(\wedge\), \(+^{\circ}\)) synthetic operation, in which  
\[b_{j}=\min\{1,\sum\limits_{i=1}^{m}(a_{i}\wedge r_{ij})\};j=1,2,\ldots,n, \tag{11}\]  
where \(+^{\circ}\) takes the summation with an upper limit of 1. The feature of this model deals with the loss of a lot of valuable information during the operation of taking the minimum. Therefore, it might not achieve the expected evaluation result. When the values of \(w_{i}\) and \(r_{ij}\) are larger, the resulting \(b_{j}\) could be equal to the upper limit of 1. When the values of \(w_{i}\) and \(r_{ij}\) are smaller, the resulting \(b_{j}\) could be equal to the summation of all \(w_{i}\) As a result, it could be more difficult to obtain the expected evaluation result. Model 4: \(M\left(\bullet,+\right)\) synthetic operation, in which  
\[b_{j}=\min\{1,\sum\limits_{i=1}^{m}a_{i}r_{ij}\};j=1,2,\ldots,n. \tag{12}\]  
This model is also called the weighted-average model and it deals with the situation when \(w_{i}\) is equipped with normalization, i.e., \(\sum\limits_{i=1}^{m}a_{i}=1\), \(\sum\limits_{i=1}^{m}a_{i}r_{ij}<<1\).This model can be restructured as \(M\left(\bullet,+\right)\), in which  
\[b_{j}=\sum\limits_{i=1}^{m}a_{i}r_{ij};j=1,2,\ldots,n;\sum\limits_{i=1}^{m}a_ {i}=1, \tag{13}\]  
where \(\sum\limits_{i=1}^{m}a_{i}=1\). This model not only considers the influence of all factors, but also keeps all the information of the single-factor evaluation. During its operation, it does not apply the upper limit to \(a_{i}\) and \(r_{ij}\) (\(i=1,\ 2,\ \cdots,\ m;j=1,\ 2,\ \cdots,\ n\)). However, \(a_{i}\) should be normalized. These are the significant characteristics and advantages of this model. When carrying out the fuzzy comprehensive evaluation and fuzzy optimization design on engineering design parameters, this model is implemented, since typically it can obtain better effects. This model not only considers the influence of all factors, but also keeps all information of the single-factor evaluation. During the operation, there is no upper limit on \(w_{i}\) and \(r_{ij}\). It is only required to carry out normalization on \(w_{i}\). This is the distinguishing feature and the main advantage of this model.