## 5 Results  
In this section, we compare MOEA with PSO. PSO was used in article [30] to solve the PP problem. In order to compare the characteristics of algorithms, we used different quality metrics including the hypervolume indicator (HV) [43] and the set coverage metric (scm) [44]. In the objective space, the volume of non-dominated solutions can be measured by the HV. We first define the concepts of characters. Non-dominated solutions are denoted by \(A=\{a_{1},a_{2},...,a_{n}\}\). The reference point is represented by \(r=(r_{1},r_{2},...,r_{d})\). In the \(d\)-objective minimum optimization problem, the hypervolume region of \(A\) restricted by \(r\) can be calculated with Equation (9). \(L\) represents the Lebesgue measure [45]:  
\[ \begin{split} HV(A,r)=L_{\{i=1\}}^{\lfloor A\rfloor}h(a_{i})|a_{i }\in A)\\ h(a_{i})=[a_{i_{1}},r_{1}]\times[a_{i_{2}},r_{2}]\times...\times[ a_{i_{d}},r_{d}].\end{split}  \tag{9}\]  
Therefore, the dominant proportions of two groups of non-dominated solutions can be measured by the set coverage metric (scm). The two groups of solutions are denoted by \(A=\{a_{1},a_{2},...,a_{n}\}\) and \(B=\{b_{1},b_{2},\ldots,b_{m}\}\), respectively. Accordingly, \(scm(A,B)\) can be shown as  
\[scm(A,B)=\frac{\left|\{b\in B|\exists a\in A:a\preceq b\}\right|}{|B|}. \tag{10}\]  
If \(scm(A,B)=1\), any solution in \(A\) is better than the solution in \(B\). On the contrary, if \(scm(A,B)=0\), this implies that any solution in \(B\) is better than the solution in \(A\). Since relationship (\(\preceq\)) denotes a weak dominance relation and is not symmetric, \(scm(A,B)\) and \(scm(B,A)\) need to be calculated. In other words, the sum of them is not definitely equal to 1. The five scenarios shown in Figure 8 were used for testing. More maps are available from [[http://imr.ciirc.cvut.cz/planning/maps.xml](http://imr.ciirc.cvut.cz/planning/maps.xml)]([http://imr.ciirc.cvut.cz/planning/maps.xml](http://imr.ciirc.cvut.cz/planning/maps.xml)). In each workspace, obstacles were generated. The starting point and target point are presented in Table 1. For equitable contrast, the following PSO parameters were set: the swarm size was 80, the maximum generation number was 100, the size of the feasible archive was 20, and the size of the infeasible archive was 20, and the maximal sampling times was 20, respectively. For each scenario shown in Figure 8, 30 independent runs were performed. In each run, the optimal solutions obtained by two algorithms were analysed with quality metrics. Table 2 presentsthe results of the hypervolume median and interquartile range. The results show that the solutions obtained by MOEA were larger and denser in objective space. Furthermore, set coverage metric (scm) was also used to test the solutions. The results are presented in Table 3. It is obvious that most of solutions obtained by MOEA dominated the solutions by PSO. The average value of scm (MOEA,PSO) was 94.91%, while the average value of scm (PSO,MOEA) was 1.08%. The results indicate that solutions obtained by MOEA are closer to the approximate Pareto front. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Scenarios** & **Starting Point** & **Target Point** \\ \hline Figure 8A & \((1500,1200)\) & \((3900,3900)\) \\ Figure 8B & \((400,100)\) & \((1450,1450)\) \\ Figure 8C & \((253.6,403.5)\) & \((1895.3,1206.9)\) \\ Figure 8D & \((1007.2,1400)\) & \((1908.6,1008.1)\) \\ Figure 8E & \((120,120)\) & \((1800,1800)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Scenario initialization  
Figure 8: In this figure, five scenarios are shown and named \(A\)–\(E\) from left to right.