## 3 Research Methods
### TOPSIS Method to Evaluate Fitness Function  
This study used the TOPSIS method to convert the original multi-objective optimization problem into a single-objective optimization problem. The TOPSIS method has been widely used in multi-objective optimization problems in various fields to select the best solution in multi-objective optimization problems, and it has also been proved to be very effective [37]. In addition to reducing the calculation steps and increasing the efficiency of the solution, this method is characterized by being able to adjust the weights flexibly in line with the values of the decision-maker so as to produce a solution that meets the decision-maker's ideals. In this study, the weights of each objective value were calculated using the weights obtained in the literature [28], which uses the analytic hierarchy process (AHP) to calculate the relative weight of operating costs, carbon emissions, and primary energy usage rate. Therefore, in this study, the multi-objective optimization problem was transformed into a single-objective optimization problem by combining the TOPSIS method with the weights obtained in ref. [28], where \(w_{OP}=0.2725\), \(w_{CE}=0.353\), and \(w_{PE}=0.3745\) are the weighting parameters of operating costs, carbon emissions, and primary energy usage rate respectively. The concept of the TOPSIS method is to first define the positive ideal solution (positive ideal solution) and negative ideal solution (negative ideal solution), and its purpose is to find a set of solutions closest to the positive ideal solution and farthest from the negative ideal solution. For the purposes of this research, the so-called positive ideal solution was the best fitness values obtained in the form of a single-objective optimization, which are \(F_{1}^{\text{min}}\) (lowest operating cost), \(F_{2}^{\text{min}}\) (lowest carbon emission), and \(F_{3}^{max}\) (the highest primary energy usage rate). On the contrary, the negative ideal solution was its worst fitness values, which are \(F_{1}^{max}\), \(F_{2}^{max}\), and \(F_{3}^{\text{min}}\), respectively. The following takes three-objective optimization and three solutions, i.e., \(N_{sol}=3\), as an example to illustrate the steps to rewrite the fitness function using the TOPSIS method. 1. Establish a decision matrix, denoted as \(\mathbf{D}\): \[\mathbf{D}=\left| \begin{array}{ll}F_{11}=13000&F_{12}=2400&F_{13}=0.72\\ F_{21}=12500&F_{22}=2600&F_{23}=0.73\\ F_{31}=14800&F_{32}=2300&F_{33}=0.78\end{array} \right|\]where the first subscript \(i\) of \(F_{i\bar{y}}\) is the \(i\)th group of fitness solution and the second subscript \(j\) is the code of objective fitness, where \(j=1\) is the operating cost, \(j=2\) is the carbon emissions, and \(j=3\) is the primary energy usage rate. 2. Because \(F_{i3}\) is the energy-usage rate, the absolute value of which is small, the reciprocal was used here and then normalized. The overall normalization decision matrix \(\mathbf{R}\) is as follows: \[\mathbf{R}=\left| \begin{array}{ll}r_{11}=0.5572&r_{12}=0.5686&r_{13}=0.5949\\ r_{21}=0.5357&r_{22}=0.6160&r_{23}=0.5868\\ r_{31}=0.6437&r_{32}=0.5449&r_{33}=0.5492\\ \end{array} \right|\] where \(r_{ij}=\left\{ \begin{array}{ll}\frac{F_{\bar{y}}}{N_{\text{self}}^{\text{ self}}}&for\,j=1,2\\ \frac{1}{\sqrt{\sum\limits_{i=1}^{N_{\text{self}}}{1/F_{\bar{y}}}}}&for\,j=3 \\ \end{array} \right..\)
3. Establish a weighted normalized decision matrix \(\mathbf{V}\): \[\mathbf{V}=\left| \begin{array}{ll}v_{11}=\omega_{\text{OP}}\times r_{11}=0.1518 &v_{12}=\omega_{\text{CE}}\times r_{12}=0.2007&v_{13}=\omega_{\text{PE}} \times r_{13}=0.2228\\ v_{21}=\omega_{\text{OP}}\times r_{21}=0.1459&v_{22}=\omega_{\text{CE}}\times r _{22}=0.2174&v_{23}=\omega_{\text{PE}}\times r_{23}=0.2197\\ v_{31}=\omega_{\text{OP}}\times r_{31}=0.1754&v_{32}=\omega_{\text{CE}}\times r _{32}=0.1923&v_{33}=\omega_{\text{PE}}\times r_{33}=0.2056\\ \end{array} \right|\]
4.