## 5 Solution Methodology Based on NSGA-II and TOPSIS
### Selection Based on TOPSIS  
The technique for order preference by similarity to ideal solution (TOPSIS) is a kind of comprehensive analysis method according to the distance between the evaluation object and the ideal solution. In this paper, TOPSIS is applied to select the compromise solution from the Pareto set. The process of TOPSIS algorithm is as follows:  
At first, the values of \(f_{1}(\mathbf{X})\) and \(f_{2}(\mathbf{X})\) are applied to construct a decision matrix, which can be defined as Equation (34):  
\[\mathbf{T}= \begin{bmatrix}t_{ij}\end{bmatrix} _{Q\times 2}= \begin{bmatrix}f_{1}(\mathbf{X} _{1})&f_{2}(\mathbf{X}_{1})\\ \vdots&\vdots\\ f_{1}\big{(}\mathbf{X}_{Q}\big{)}&f_{2}\big{(}\mathbf{X}_{Q}\big{)}\end{bmatrix} , \tag{34}\]  
where \(\mathbf{T}\) is the decision matrix, and \(t_{ij}\) represents the decision indicator obtained by the value of the objective function \(j\) of the solution \(i\); \(Q\) is the number of solutions in the Pareto set. By normalizing \(\mathbf{T}\), the standard matrix \(\mathbf{P}\) is obtained, which can be expressed as Equation (35):  
\[\mathbf{P}= \begin{bmatrix}p_{ij}\end{bmatrix} _{Q\times 2}= \begin{bmatrix} \frac{\underset{i}{\max}\Big{(}t_{ij}\Big{)}-t_{ij}}{\underset{i}{\max}\Big{(} t_{ij}\Big{)}-\underset{i}{\min}\Big{(}t_{ij}\Big{)}}\end{bmatrix} _{Q\times 2}. \tag{35}\]  
Then, the entropy values of the two objective functions can be calculated as Equation (36):  
\[e_{j}=-\frac{1}{\ln Q}\underset{i=1}{\sum} \begin{bmatrix}p_{ij}\\ \underset{i=1}{\sum}\begin{bmatrix}p_{ij}\\ \underset{i=1}{\sum}\end{bmatrix}_{i=1}^{Q}\frac{p_{ij}}{\underset{i=1}{\sum} \begin{bmatrix}p_{ij}\\ \underset{i=1}{\sum}\end{bmatrix}_{i=1}^{Q}p_{ij}}\end{bmatrix} j=1,\;2, \tag{36}\]  
where \(e_{j}\) represents the entropy value of the objective function \(f_{j}(\mathbf{X})\). According to the \(e_{j}\), the entropy weights of two objective functions can be calculated as Equation (37):  
\[\alpha_{j}=\frac{1-e_{j}}{\underset{i=1}{\sum} \begin{bmatrix}1-e_{i}\\ \end{bmatrix} }j=1,\;2, \tag{37}\]where \(\alpha_{j}\) represents the entropy weight of the objective function \(f_{j}(\mathbf{X})\). Considering the subjective weight of the planner, the weights of two objective functions can be calculated as Equation (38):  
\[w_{j}=\frac{\lambda_{j}\alpha_{j}}{\sum\limits_{i=1}^{2}\lambda_{i}\alpha_{i}}j= 1,\;2, \tag{38}\]  
where \(w_{j}\) represents the final weight of objective function \(f_{j}(\mathbf{X})\); \(\lambda_{j}\) is the subjective weight of the planner, the larger \(\lambda_{j}\) is, the more important \(f_{j}(\mathbf{X})\) is. According to \(w_{j}\) and \(\mathbf{P}\), the weighted decision matrix \(\mathbf{Z}\) is constructed, which can be defined as Equation (39):  
\[\mathbf{Z}= \begin{bmatrix}z_{ij}\end{bmatrix} _{Q\times 2}= \begin{bmatrix}w_{j}p_{ ij}\end{bmatrix} _{Q\times 2}. \tag{39}\]  
Next, the extreme values of the weighted decision matrix can be calculated as Equation (40):  
\[ \begin{array}{l}o_{j}^{+}=\max\limits_{1\leq i\leq Q}z_{ij}\\ o_{j}^{-}=\min\limits_{1\leq i\leq Q}z_{ij}\end{array} j=1,\;2, \tag{40}\]  
where \(o_{j}^{+}\) and \(o_{j}^{-}\) represent the maximum value and minimum value of the \(j-\text{th}\) column of the weighted decision matrix \(\mathbf{Z}\) respectively. After that, the positive ideal solution and the negative ideal solution can be calculated as Equation (41):  
\[ \begin{array}{l}\mathbf{Y}^{+}=\begin{pmatrix}o_{1}^{+},o_{2}^{+}\\ o_{1}^{-},o_{2}^{-}\end{pmatrix}\;,\end{array}  \tag{41}\]  
where \(\mathbf{Y}^{+}\) and \(\mathbf{Y}^{-}\) represent the positive ideal solution and the negative ideal solution respectively. For each solution in the set, the distances from the positive ideal solution and the negative ideal solution can be calculated as Equation (42):  
\[