## 3 Multi-Objective Optimization
### Multi-Objective Decision Making Based on TOPSIS Algorithm  
The optimal solution set with 2000 solutions is eventually obtained using NSGA-II, and the multi-objective decision making method can prefer the optimal solution from it. TOPSIS is one of the most commonly used multi-objective decision making methods for selecting the best compromise solution between the incommensurable and conflicting objective functions [22]. The optimal solution based on TOPSIS decision making is closest to the positive ideal solution and farthest from the negative ideal solution. The steps of TOPSIS multi-objective decision making are as follows:  
Step 1: Create a decision matrix \((a_{ij})m\times n\), where \(m\) is the decision point and \(n\) is the number of objective functions.  
Step 2: Normalize the decision matrix using the Euclidean method:  
\[x_{ij}\ =\ \frac{a_{ij}}{\sqrt{\sum_{j=1}^{m}a_{ij}^{2}}},\ i\ =\ 1,\ 2,\ \cdots,\ m;j\ =\ 1,\ 2,\cdots,\ n \tag{16}\]  
Step 3: Develop a weighted normalized decision matrix:  
\[X_{ij}\ =\ x_{ij}\times w_{j},\ i\ =\ 1,\ 2,\ \cdots,\ m;j\ =\ 1,\ 2,\cdots,\ n \tag{17}\]  
where\(\sum_{j=1}^{n}w_{j}\ =\ 1\).  
Step 4: Determine positive ideal solutions (\(X^{+}\)) and negative ideal solutions (\(X^{-}\)):  
\[X^{+}\ =\ \left[\left(\max_{i}X_{ij}\middle|j\in J_{+}\right),\left(\min_{i}X_{ ij}\middle|j\in J_{-}\right)\middle|\ i\ =\ 1,\ 2,\ \cdots,\ m\right]\ =\ \left\{X_{1}^{+},X_{2}^{+},\ldots,X_{n}^{+}\right\} \tag{18}\]\[X^{-}\ =\ \left[\left(\min_{i}X_{ij}\bigg{|}j\in J_{+}\right),\left(\max_{i}X_{ij} \bigg{|}j\in J_{-}\right)\bigg{|}\ i\ =\ 1,\ 2,\ \cdots,\ m\right]\ =\ \left\{X_{1}^{-},X_{2}^{-},\ldots,X_{n}^{-}\right\} \tag{19}\]  
where \(J_{+}\) is the indicator of positive standard, \(J_{-}\) is the indicator of negative standard, and \(X_{n}^{+}\) and \(X_{n}^{-}\) are the maximum and minimum values of each column, respectively.  
Step 5: Calculate the distance of each objective alternative from the positive ideal solution and the negative ideal solution:  
\[D_{i}^{+} =\ \sqrt{\sum\limits_{j\ =\ 1}^{n}\left(X_{ij}-X_{j}^{+}\right)^{2}},\ i \ =\ 1,\ 2,\ \ldots,\ m \tag{20}\] \[D_{i}^{-} =\ \sqrt{\sum\limits_{j\ =\ 1}^{n}\left(X_{ij}-X_{j}^{-}\right)^{2}},\ i \ =\ 1,\ 2,\ \ldots,\ m \tag{21}\]  
Step 6: Calculate the relative proximity of each objective alternative to the ideal state:  
\[C_{i}^{*}\ =\ \frac{D_{i}^{-}}{D_{i}^{-}+D_{i}^{+}},\ i\ =\ 1,\ 2,\ \ldots,\ m \tag{22}\]  
The solution of TOPSIS is the point with the largest value of \(C_{j}^{*}\) on the Pareto optimal solution set. If the solution of TOPSIS is not reasonable, the weights (\(w_{j}\)) of the objective function can be reassigned and the results are recalculated. In this study, the weights of all optimization objectives are set to be equal, i.e., the weights of heat transfer quantity, oil pressure drop, and oil outlet temperature are all \(1/3\). The TOPSIS algorithm is used to make decisions among 2000 optimal solutions, and the optimal result is preferentially selected.  
In summary, the overall optimization process for the plate-fin heat exchanger is shown in Figure 5. Firstly, the data of numerical simulation was obtained using the fluid-solid coupling model of the plate-fin heat exchanger, and then the regression models of heat transfer quantity, oil pressure drop, and oil outlet temperature with respect to the fin angle, oil flow rate, and water flow rate are obtained by SVR. Additionally, these regression models are used as the objective function for multi-objective optimization. NSGA-II is utilized to optimize on the objective function to obtain the Pareto optimal solution set. Finally, TOPSIS is applied to decide the optimal solution in the solution set.  
Figure 5: Flow chart of fluid-solid coupling simulation, SVR prediction model, and multi-objective optimization.