# Topology Optimisation Using MPBILs and Multi-Grid Ground Element
## 5 Design Results  
The comparative performance of original MPBIL and the performance enhancements of MPBILs with MG and without MG approach for solving the design problems of MOP1-4 are given in Table 1, \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \multicolumn{2}{c}{**MOP1**} & \multicolumn{2}{c}{**MOP2**} & \multicolumn{2}{c}{**MOP3**} & \multicolumn{2}{c}{**MOP4**} \\ \cline{2-10}  & **WMG** & **WOMG** & **WMG** & **WOMG** & **WMG** & **WOMG** & **WMG** & **WOMG** \\ \hline MPBIL & 0.8553 & 0.8255 & 0.7255 & 0.6420 & 0.7951 & 0.7229 & 0.7195 & 0.6219 \\ OMPBIL & 0.8556 & 0.8426 & 0.7259 & 0.6430 & 0.7968 & 0.7438 & 0.6723 & 0.6403 \\ MPBILMLR & 0.8115 & 0.7739 & 0.7212 & 0.6285 & 0.7016 & 0.5976 & 0.6167 & 0.5651 \\ MPBILADLR & 0.8543 & 0.8371 & 0.7240 & 0.6385 & 0.7954 & 0.7407 & 0.6404 & 0.6292 \\ \hline \end{tabular}  
* WMG, with multi-grid approach; WOMG, without multi-grid approach; MPBIL, multi-objective population-based incremental learning; OMPBIL, opposite-based, multi-objective, population-based incremental learning; MPBIL MLR, MPBIL with multi-learning rate; MPBIL ADLR, multi-objective, population-based incremental learning with adaptive learning rate.  
\end{table} 
Table 1: Performance comparison based on hypervolume (HV) 1. Figure 3: Structural design domains: (**a**) design domain of MOP1 & MOP3; (**b**) design domain of MOP2; (**c**) design domain of MOP 4. Reproduce with permission from [8], Taylor & Francis, 2011.  
which compare based on HV indicator. It should be noted that all of the approximate Pareto fronts of the four design problems obtained from using the proposed MPBILs are normalized before calculating HV, as shown in Table 1. The highest mean of HV for each design problem is highlighted with grey color. The table shows OMPBIL promotes almost the best results for MOP1-4 except in case MOP4-WMG. Therefore, it is believed that the performance of OMPBIL is better result than the original MPBIL and their enhancements. In this study, the Friedman test and the Tukey-Kramer test are used for a statistical test to prove the significance of proposed algorithm. These tools are built-in functions in MATLAB/Octave. From our testing, the Friedman test gives OMPBIL has 1 st rank, whereas the second rank is MPBIL at \(p\)-value (0.0002) \(<\alpha\)(0.05) as shown in Table 2. It can be summarized that OMPBIL is the best performing algorithm for solving problem case MOP1-4. For multiple comparisons, we used the Tukey-Kramer test. The mean column ranks of OMPBIL are significantly different from MPBILMLR. The second best optimizer is MPBIL, whereas the third best is MPBILADLR. In addition, the worst optimizer for this design case is MPBILMLR. No questionable opposition concept is beneficial to improving the performance of MPBIL. The average HV for all optimizers of each problem with MG and WOMG approach is shown in Table 3, which is summed along each column from Table 1. This table shows that the design problems with MG approach give higher HV than the design problem without MG approach in all design problems. Friedman test of average result in Table 3 can prove that the design problem with MG technique significantly outperforms WOMG technique at \(p\)-value (0.0455) \(<\alpha\)(0.05). Furthermore, the best HVs of all cases give higher hypervolume than the previous work by [8] in all design cases, so OMPBIL with MG can improve the design results. Figures 4-7 shows some optimum topologies. The topologies in all figures are captioned with (a), which obtains from the best run of OMPBIL with multi-grid when solving each MOP with various r values. All figures are captioned with (b); they display the optimum topologies that are obtained from optimizing the design problem MOP1-4 with various \(r\) values by using MPBIL without multi-grid. These topologies are represented by the same technique from the previous work [8].