# A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm
## 5 Simulation Experiment and Result Analysis
### Performance Metrics  
Performance metrics are often used to measure the effectiveness of an algorithm for MOO problems. As we know, the basic aim of multi-objective optimization is to obtain a Pareto set with better convergence, distribution and diversity of the Pareto set. According to whether the performance metrics can measure the convergence, the diversity or both of a solution set, they can be divided into three categories [42], and the following part describes some commonly used performance metrics of the three categories. Here we suppose the true Pareto optimal front is known as \(P^{*}\). (1) Generational Distance (\(GD\)): \(GD\) is used to evaluate the convergence of a solution set, the concept of \(GD\) was proposed by [PERSON] and [PERSON] [43]. It measures the distance between the obtained non-dominated front \(P\) and the true Pareto set \(PF^{*}\) as:  
\[GD=\frac{\sqrt{\sum_{i=1}^{|P|}minDis(X_{i},PF^{*})^{2}}}{|P|} \tag{16}\]  
where \(minDis(X_{i},~{}PF^{*})\) represents the minimum Euclidean distance between the obtained solution \(X_{i}\) and the solutions in \(PF^{*}\), and \(|P|\) is the number of the obtained optimal solutions. \(GD\) measures how far these non-dominated solutions \(X\) are from those in the true Pareto optimal front \(PF^{*}\). The smaller the \(GD\) value, the closer the Pareto solution set is to the real Pareto frontier and the better convergence the solution set has. If \(GD=0\), the solution set is on the real Pareto front, which is the most ideal situation. (2) Spacing metric (\(SP\)): The spatial evaluation method \(SP\)[44] proposed by [PERSON] is used to evaluate the distribution of individuals in the objective space. The function is defined as follows:  
\[SP=\sqrt{\frac{1}{|P|-1}\sum_{i=1}^{|P|}(\bar{d}-d_{i})^{2}} \tag{17}\]  
\[d_{i}=\min_{j}\left(|f_{1}^{i}(X)-f_{1}^{i}(X)|+|f_{2}^{i}(X)-f_{2}^{i}(X)| \right),~{}(i,j=1,2,~{}\ldots~{},|P|) \tag{18}\]  
where \(\bar{d}\) is the average of all \(d_{i}\) and \(|P|\) represents the size of Pareto solutions. A smaller value of \(SP\) demonstrates a better distribution of the obtained solution set. Here, \(SP=0\) when the algorithm obtained optimal solutions which are completely evenly distributed in the target space. (3) Spread indicator (\(SI\)): [PERSON] et al. [42] proposed the spread metric \(SI\), which is used for independently evaluating the breadth and the spread of the non-dominated solutions. \(SI\) can be defined mathematically as:  
\[SI=\frac{\sum_{i=1}^{m}d(e_{i},P)+\sum_{X\in P}|d(X,P)-\bar{d}|}{\sum_{i=1}^{m }d(e_{i},P)+(|P|-m)\times\bar{d}} \tag{19}\]  
\[d(X,P)=\min_{Y\in P,Y\
otin X}\|F(X)-F(Y)\| \tag{20}\]  
\[\bar{d}=\frac{1}{P}\times\sum_{X\in P}d(X,P) \tag{21}\]  
where \(P\) is optimal solution set, \(m\) represents the number of objectives, \(e_{i}~{}(i=1,2,~{}\ldots~{},m)\) are \(m\) extreme solutions, which belongs to the set of true Pareto front. As can be seen, a smaller \(SI\) means the optimal solution set has a better spread. (4) Pure diversity (\(PD\)): \(PD\) is used for measuring the diversity of species, it is defined as follows [45]:  
\[PD(X)=\max_{sp_{i}\in X}(PD(X-sp_{i})+d(sp_{i},X-sp_{i})) \tag{22}\]  
where\[d(sp,X)=\min_{sp_{i}\in X}(\mathit{diss}(sp,sp_{i})) \tag{23}\]  
Here, \(d(sp_{i},X-sp_{i})\) represents the dissimilarity \(d\) from one species \(sp_{i}\) to a community \(X\). The larger the \(PD\) value, the better the diversity of the solution set. (5) Hypervolume (\(HV\)): \(HV\) indicator [46] has become a popular evaluation index because of its good theoretical support. It evaluates the comprehensive performance of MOEAs by calculating the hypervolume value of the space enclosed by the non-dominated solution set and the reference point. The calculation formula is:  
\[HV=\lambda\left(\cup_{i=1}^{|P|}v_{i}\right) \tag{24}\]  
where \(\lambda\) represents the Lebesgue measure, \(v_{i}\) represents the hypervolume formed by the reference point and the non-dominated individual and \(P\) represents the optimal solution set.