# A Novel Multi-Objective Harmony Search Algorithm with Pitch Adjustment by Genotype
## 5 Experimentation and Results
### Analysis of Results
#### 5.3.3 Problem P3  
As can be seen in Figure 6, problem P3 has a local PF and a global PF, which drives the algorithms to be trapped in the local optimum. Both fronts are discontinuous and are divided into four solution regions. In Figure 7, the local and global PSs for this problem can be seen, divided into four regions and with a linear behavior. The most populated PFs corresponded to MOHSg and MOHS2 with 3919 and 3863 points, respectively, while MOHS3 obtained a much lower front with 1013 points. The mentioned figures show the convergence of the three algorithms to global solutions. The lowest error ratio in Table 5 corresponds to MOHSg with 1.43%, followed by MOHS2 with 2.95 % and MOHS3 with an extremely unfavorable 77.9%. The fronts nearest to the real PF and PS also correspond to MOHSg followed by MOHS2 and MOHS3. In the case of distribution, MOHSg had the best average performance, although the lowest standard deviation corresponds to MOHS2. In this problem, MOHSg clearly presented the best performance.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **MOHS2** & **MOHS3** & **MOHSg** \\ \hline \multirow{4}{*}{**Error ratio**} & Average & \(\mathbf{7.650\times 10^{-2}}\) & \(1.283\times 10^{-1}\) & \(1.195\times 10^{-1}\) \\  & Best & \(\mathbf{4.000\times 10^{-2}}\) & \(8.000\times 10^{-2}\) & \(9.000\times 10^{-2}\) \\  & Worst & \(\mathbf{1.150\times 10^{-1}}\) & \(1.650\times 10^{-1}\) & \(1.650\times 10^{-1}\) \\  & Std. Dev. & \(\mathbf{1.623\times 10^{-2}}\) & \(2.551\times 10^{-2}\) & \(1.884\times 10^{-2}\) \\ \hline \multirow{4}{*}{_GDf_} & Average & \(2.635\times 10^{-3}\) & \(\mathbf{2.457\times 10^{-3}}\) & \(2.699\times 10^{-3}\) \\  & Best & \(2.287\times 10^{-3}\) & \(\mathbf{2.194\times 10^{-3}}\) & \(2.242\times 10^{-3}\) \\  & Worst & \(3.027\times 10^{-3}\) & \(\mathbf{2.850\times 10^{-3}}\) & \(3.303\times 10^{-3}\) \\  & Std. Dev. & \(2.208\times 10^{-4}\) & \(\mathbf{1.627\times 10^{-4}}\) & \(2.470\times 10^{-4}\) \\ \hline \multirow{4}{*}{_GDx_} & Average & \(2.948\times 10^{-2}\) & \(\mathbf{2.863\times 10^{-2}}\) & \(3.848\times 10^{-2}\) \\  & Best & \(\mathbf{2.594\times 10^{-2}}\) & \(2.640\times 10^{-2}\) & \(3.085\times 10^{-2}\) \\  & Worst & \(3.563\times 10^{-2}\) & \(\mathbf{3.199\times 10^{-2}}\) & \(4.948\times 10^{-2}\) \\  & Std. Dev. & \(2.441\times 10^{-3}\) & \(\mathbf{1.795\times 10^{-3}}\) & \(3.963\times 10^{-3}\) \\ \hline \multirow{4}{*}{**Spacing**} & Average & \(\mathbf{3.474\times 10^{-3}}\) & \(2.904\times 10^{-2}\) & \(1.639\times 10^{-2}\) \\  & Best & \(2.777\times 10^{-3}\) & \(\mathbf{2.735\times 10^{-3}}\) & \(2.757\times 10^{-3}\) \\ \cline{1-1}  & Worst & \(\mathbf{6.162\times 10^{-3}}\) & \(1.162\times 10^{-1}\) & \(9.262\times 10^{-2}\) \\ \cline{1-1}  & Std. Dev. & \(\mathbf{6.972\times 10^{-4}}\) & \(3.555\times 10^{-2}\) & \(2.535\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Statistic analysis for problem P2.  
Figure 5: Pareto sets for problem P2.  
Figure 6: Pareto fronts for problem P3.  
Figure 7: Pareto sets for problem P3.