## 3 Results and Discussion
### Analysis of the Impact of the Rasterization Value  
While the number of evaluations has a linear influence on the computational time, it is expected that the rasterization value \(r_{v}\) has an exponential influence on the calculation. In this subsection, we examine the effects of the rasterization values on the computed coverage value \(C\). For all tests, the number of evaluations per independent variable is set to \(E_{v}=200\). Table 7 shows the descriptive statistics for using different rasterization values \(r_{v}\) of the optimization for sensor placement in the two-dimensional Environment 1. This table shows the average and standard deviation of the coverage \(C\) of 30 runs. Again, to statistically analyze the effects of the optimization algorithm used and the rasterization value on the average coverage, we performed a two-way RM ANOVA. Namely, a \(3\times 10\) ANOVA was utilized, with _Algorithms_ (three instances) and _Rasterization_ (10 instances, 0.1 m to 1.0 m, step 0.1 m) being the within-subjects factors. The test yielded the following results:  
* The mean coverage differed significantly between the optimization _Algorithms_: \[F(2,870)=402.097,p<0.001.\] \begin{table}
\begin{tabular}{r r r r r r r} \hline \hline \multirow{2}{*}{**Rasterization \(r_{v}\) [m]**} & \multicolumn{2}{c}{**ABC**} & \multicolumn{2}{c}{**FWA**} & \multicolumn{2}{c}{**PSO**} \\  & **Average** & **St. Dev.** & **Average** & **St. Dev.** & **Average** & **St. Dev.** \\ \hline
0.1 & 0.1444 & 0.0002 & 0.1437 & 0.0006 & 0.1445 & \textless{}0.0001 \\
0.2 & 0.1405 & 0.0002 & 0.1398 & 0.0006 & 0.1407 & \textless{}0.0001 \\
0.3 & 0.1370 & \textless{}0.0001 & 0.1359 & 0.0008 & 0.1370 & \textless{}0.0001 \\
0.4 & 0.1352 & 0.0004 & 0.1345 & 0.0006 & 0.1354 & 0.0001 \\
0.5 & 0.1303 & 0.0003 & 0.1291 & 0.0007 & 0.1303 & 0.0003 \\
0.6 & 0.1276 & 0.0006 & 0.1261 & 0.0009 & 0.1279 & 0.0007 \\
0.7 & 0.1294 & 0.0001 & 0.1268 & 0.0017 & 0.1292 & 0.0005 \\
0.8 & 0.1249 & 0.0002 & 0.1218 & 0.0021 & 0.1249 & \textless{}0.0001 \\
0.9 & 0.1200 & 0.0006 & 0.1167 & 0.0021 & 0.1198 & 0.0007 \\
1.0 & 0.1201 & 0.0005 & 0.1161 & 0.0026 & 0.1198 & 0.0017 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Coverage averages and standard deviations of the different algorithms for each rasterization for two-dimensional Environment 1. Figure 3: Computation time for each of the algorithms based on the number of evaluations \(E_{v}\) for Environment 1. A post hoc analysis with a Bonferroni adjustment showed that there is no statistical difference between the coverage achieved by ABC and PSO, while both achieve statistically higher coverage compared to FWA in these tests. The full results of the pairwise post hoc comparisons can be found in Table 8.
* Mean coverage differed significantly between the _Rasterization_: \[F(9,870)=7518.017,p<0.001.\] A post hoc analysis with a Bonferroni adjustment showed statistical significance between almost all _Rasterization_ values, with the only pair that yielded a statistically non-significant difference being \(r_{v}=0.9\) m and \(r_{v}=1.0\) m.
* The interaction between the factors _Algorithm \(*\) Rasterization_ is statistically significant: \[F(18,870)=15.191,p<0.001.\]  
Table 9 shows the descriptive statistics for using different rasterization values \(r_{v}\) of the optimization for the sensor placement in the three-dimensional representation of Environment 1. This table shows the average and standard deviation of the coverage \(C\) of all runs for each algorithm. Again, we performed a \(3\times 10\) two-way RM ANOVA with the same within-subjects factors. The test yielded the following results:  
* The mean coverage differed significantly between the optimization _Algorithms_: \[F(2,870)=683.308,p<0.001.\]
* Mean coverage differed significantly between the _Rasterization_: \[F(9,870)=28115.796,p<0.001.\]  
For both factors, a post hoc analysis with a Bonferroni adjustment yielded the same results as for the two-dimensional Environment.