## 2 Multi-Objective Reliability Optimization Method
### Entropy Weighted Grey Relational Analysis  
Grey relational analysis (GRA) is an important branch of grey system theory and has been widely applied to decision-making problems. Its basic idea is to map the discrete data to the geometric shape of the space by using the linear interpolation between adjacent points of the sequence. It provides the grey relational grade to judge the close relationship between sequences based on the distance between reference sequence and comparison sequence [27; 28].  
In order to eliminate the non-commensurable caused by different dimensions, it is necessary to normalize the response values before grey relational analysis. The normalization methods corresponding to different response characteristics are different. If the response value has a smaller-the-better characteristic, that is, the smaller the response value is, the better the performance is, the normalization calculation method is as follows:  
\[x_{i}^{*}(k)=\frac{\max_{k}x_{i}(k)-x_{i}(k)}{\max_{k}x_{i}(k)-\min_{k}x_{i}(k)} \tag{7}\]  
Figure 1: Flowchart of MCS calculation.  
If the response value has a larger-the-better characteristic, that is, the larger the response value is, the better the performance is, the normalized calculation method can be expressed as follows:  
\[x_{i}^{*}(k)=\frac{x_{i}(k)-\min_{k}x_{i}(k)}{\max_{k}x_{i}(k)-\min_{k}x_{i}(k)} \tag{8}\]  
where \(x_{i}(k)\) and \(x_{i}^{*}(k)\) represent the original and normalized value of the \(i\)th response for the \(k\)th attribute, respectively; \(\max_{k}x_{i}(k)\) and \(\min_{k}x_{i}(k)\) are the maximum and minimum values of the \(k\)th attribute, respectively.  
The response results after normalization are converted to values between [0, 1]. The larger the value is, the better the performance is, and 1 represents the optimal value. Define \(x_{0}^{*}(k)=1\) as the reference sequence, and the normalized results as the comparison sequence. The deviation sequence of these two sequences is:  
\[\Delta_{0i}(k)=|x_{0}^{*}(k)-x_{i}^{*}(k)| \tag{9}\]  
The calculation formula of grey relational coefficient between comparison sequence and reference sequence is as follows:  
\[\left\{\begin{array}{l}\gamma\left(x_{0}^{*}(k),x_{i}^{*}(k)\right)=\frac{ \Delta_{\min}+\zeta\Delta_{\max}}{\Delta_{0i}(k)+\zeta\Delta_{\max}}\\ \Delta_{\min}=\min_{i}\min_{k}\Delta_{0i}(k)\\ \Delta_{\max}=\max_{i}\max_{k}\Delta_{0i}(k)\end{array}\right. \tag{10}\]  
where \(\Delta_{\min}\) and \(\Delta_{\max}\) are the minimum and maximum of the deviation sequence, respectively; \(\zeta\in[0,1]\) is the distinguishing coefficient, generally it equals to 0.5.  
The grey relational grade is calculated by weighted summation of the grey relational coefficient of each attribute. The calculation formula is:  
\[\Gamma_{i}=\sum\limits_{k=1}^{n}w_{k}\gamma(x_{0}^{*}(k),x_{i}^{*}(k)) \tag{11}\]  
where \(n\) is the number of attributes; \(w_{k}\) is the weight coefficient of the \(k\)th attribute, \(\sum\limits_{k=1}^{n}w_{k}=1\).  
Generally, the relative importance of each attribute may be different, and their weights can be calculated by information entropy which denotes the degree of uncertainty of a random variable [29; 30]. The projection value of the \(i\)th response for the \(j\)th attribute is formulated as:  
\[p_{ij}=\frac{x_{ij}}{\sum\limits_{i}^{n}x_{ij}} \tag{12}\]  
where \(i=1,\,\ldots,\,n,\,n\) is the number of responses; \(j=1,\,\ldots,\,m,\,m\) is the number of attributes; and \(x_{ij}\) represents the normalized value of the \(i\)th response for the \(j\)th attribute.  
The entropy value of the \(j\)th attribute is:  
\[e_{j}=-\frac{1}{\ln n}\sum\limits_{i=1}^{n}p_{ij}\ln p_{ij} \tag{13}\]  
Then, the weight coefficient of attribute can be calculated as:  
\[w_{j}=\frac{d_{j}}{\sum\limits_{j=1}^{m}d_{j}} \tag{14}\]  
where \(d_{j}=1-e_{j}\) represents the deviation degree of the \(j\)th attribute. Generally, a larger value of deviation degree indicates that it provides more information. Therefore, this attribute has a higher weight.