## 3 Optimization Method
### Optimum Solution Selection  
Once the NSGA-II program is completed, one optimum solution should be selected from Pareto frontier as the optimum design for PFHE. Selection criteria can depend on engineering experiences, relative importance of the objective functions, sensitivity of the optimum solutions to the parameters, etc. The technique for order of preference by similarity to ideal solution (TOPSIS) approach is applied in this paper for its wide utilization in choosing appropriate solutions from the Pareto frontier [32]. The principle of the TOPSIS approach is to evaluate every solution by calculating its comprehensive distance between the ideal optimal solution and the ideal worst solution. The solution with the highest score is considered as the optimum decision.  
Suppose there are m vectors as the optimization results of \(n\) objective functions, as shown in Equation (12).  
\[X_{i}=\left[X_{i1},\ X_{i2},\ldots,X_{in}\right]\,\left(i=1,2,\ldots,m\right) \tag{12}\]  
Firstly, positive processing is employed on solutions to achieve the largest value of each objective function, as the larger it is, the more favorable it is. As shown in Equation (13), the reciprocals of area and pressure drop were taken in the present paper.  
\[\overline{X_{ij}}=\frac{1}{X_{ij}},\left(i=1,2,\ldots,m\right),\left(j=1,2, \ldots,n\right) \tag{13}\]  
\[\overline{X_{i}}=\left[\overline{X_{i1}},\ \overline{X_{i2}},\ldots,\overline{X_{in}} \right]\,\left(i=1,2,\ldots,m\right) \tag{14}\]  
Secondly, \(\overline{X}_{i}\) are standardized by  
\[Z_{ij}=\frac{\overline{X_{ij}}}{\sqrt{\sum_{k=1}^{m}\left(\overline{X_{kj}} \right)^{2}}},\left(i=1,2,\ldots,m\right),\left(j=1,2,\ldots,n\right) \tag{15}\]  
\[Z_{i}=\left[Z_{i1},Z_{i2},\ldots Z_{in}\right],\left(i=1,2,\ldots,m\right) \tag{16}\]  
Then, the ideal optimal solution \(Z^{+}\) and the ideal worst solution \(Z^{-}\) are defined as the following:  
\[Z^{+}=\left[Z_{1}^{+},Z_{2}^{+},\ldots Z_{n}^{+}\right]=\left[\max\{Z_{11},Z_{ 21},\ldots,Z_{m1}\},\max\{Z_{12},Z_{22},\ldots,Z_{m2}\},\ldots,\max\{Z_{1n},Z_ {2n},\ldots,Z_{mn}\}\right] \tag{17}\]  
\[Z^{-}=\left[Z_{1}^{-},Z_{2}^{-},\ldots Z_{n}^{-}\right]=\left[\min\{Z_{11},Z_{ 21},\ldots,Z_{m1}\},\min\{Z_{12},Z_{22},\ldots,Z_{m2}\},\ldots,\min\{Z_{1n},Z_ {2n},\ldots,Z_{mn}\}\right] \tag{18}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Specified Range** \\ \hline \(N_{hot}\) & Number of fin layers on hot side & 2-200 \\ \(L_{w}\) (m) & Plate width & 0.1–5 \\ \(l_{s,hot}\) (m) & Offset fin length on hot side & 0.03–0.010 \\ \(l_{s,cold}\) (m) & Offset fin length on cold side & 0.03–0.010 \\ \(s_{hot}\) (m) & Spacing of fins on hot side & 0.001–0.005 \\ \(s_{cold}\) (m) & Spacing of fins on cold side & 0.001–0.005 \\ \(H_{hot}\) (m) & Height of fins on hot side & 0.002–0.020 \\ \(H_{cold}\) (m) & Height of fins on cold side & 0.002–0.020 \\ \(t_{hot}\) (m) & Thickness of fins on hot side & 0.0001–0.0005 \\ \(t_{cold}\) (m) & Thickness of fins on cold side & 0.0001–0.0005 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Decision variables and corresponding specified ranges for multi-objective optimization of plate-fin heat exchanger.  
Distance from solution \(Z_{i}\) to \(Z^{+}\) and \(Z^{-}\) can be calculated by the following equations, in which \(w_{j}\) is defined as the weight of target index \(j\).  
\[D_{i}^{+}=\sqrt{\sum_{j=1}^{n}w_{j}(Z_{j}^{+}-Z_{ij})^{2}},(i=1,2,\ldots,m) \tag{19}\]  
\[D_{i}^{-}=\sqrt{\sum_{j=1}^{n}w_{j}(Z_{j}^{-}-Z_{ij})^{2}},(i=1,2,\ldots,m) \tag{20}\]  
Therefore, the score of solution \(Z_{i}\) i.e., \(X_{i}\) is  
\[S_{i}=\frac{D_{i}^{-}}{D_{i}^{+}+D_{i}^{-}},(i=1,2,\ldots,m) \tag{21}\]  
The normalized score of solution \(Z_{i}\) i.e., \(X_{i}\) is calculated by  
\[S_{i,norm}=\frac{S_{i}}{\sum_{k=1}^{m}S_{k}},(i=1,2,\ldots,m) \tag{22}\]  
In the present paper, \(m\) is 150, \(n\) is 2, \(w_{1}\) is 0.8 and \(w_{2}\) is 0.2.