# A Novel Multi-Objective Harmony Search Algorithm with Pitch Adjustment by Genotype
## 5 Experimentation and Results
### Analysis of Results
#### 5.3.2 Problem P2  
In Figure 4, the concave PF generated for problem P2 can be seen. The most populated front corresponds to the MOHS2 algorithm, followed by MOHS3 and MOHSg very close to each other. Note that in this problem, the concave front is made up of two PSs in the search space, as shown in Figure 5. The PFs and PSs obtained are more populated and better distributed than in P1, as can be seen in the results tabulated below.  
Figure 3: Pareto sets for problem P1.  
Figure 2: Pareto fronts for problem P1.  
Table 4 presents the statistical analysis of the algorithms. It can be observed that the best error ratio corresponds to the MOHS2 algorithm with 7.65% (mean value) followed by MOHSg with 11.95% and MOHS3 with 12.83%. The fronts and sets nearest to the real PF and PS correspond to those obtained by MOHS3, followed by the results of MOHS2 and MOHSg. The best distribution of the PFs corresponds to MOHS2, followed by the proposed algorithm MOHSg and MOHS3. In this problem, MOHS2 produced the most populated integrated front, and the fronts with the lowest error ratio and with the best uniform distribution of all the variants. Therefore, it can be concluded that MOHS2 had the best overall performance for this test function.  
Figure 4: Pareto fronts for problem P2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **MOHS2** & **MOHS3** & **MOHSg** \\ \hline \multirow{4}{*}{**Error ratio**} & Average & \(2.800\times 10^{-1}\) & \(4.105\times 10^{-1}\) & \(\mathbf{2.183\times 10^{-1}}\) \\  & Best & \(\mathbf{1.500\times 10^{-1}}\) & \(3.350\times 10^{-1}\) & \(\mathbf{1.500\times 10^{-1}}\) \\  & Worst & \(4.300\times 10^{-1}\) & \(4.950\times 10^{-1}\) & \(\mathbf{2.900\times 10^{-1}}\) \\  & Std. Dev. & \(7.222\times 10^{-2}\) & \(4.322\times 10^{-2}\) & \(\mathbf{3.945\times 10^{-2}}\) \\ \hline \multirow{4}{*}{_GDf_} & Average & \(1.145\times 10^{-2}\) & \(\mathbf{4.401\times 10^{-3}}\) & \(1.545\times 10^{-2}\) \\  & Best & \(6.990\times 10^{-3}\) & \(\mathbf{3.662\times 10^{-3}}\) & \(7.658\times 10^{-3}\) \\  & Worst & \(2.291\times 10^{-2}\) & \(\mathbf{5.560\times 10^{-3}}\) & \(2.469\times 10^{-2}\) \\  & Std. Dev. & \(3.748\times 10^{-3}\) & \(\mathbf{5.009\times 10^{-4}}\) & \(4.787\times 10^{-3}\) \\ \hline \multirow{4}{*}{_GDx_} & Average & \(18.21\times 10^{-1}\) & \(\mathbf{17.85\times 10^{-1}}\) & \(21.31\times 10^{-1}\) \\  & Best & \(11.47\times 10^{-1}\) & \(\mathbf{7.248\times 10^{-1}}\) & \(8.722\times 10^{-1}\) \\  & Worst & \(\mathbf{25.56\times 10^{-1}}\) & \(32.58\times 10^{-1}\) & \(36.73\times 10^{-1}\) \\  & Std. Dev. & \(3.746\times 10^{-1}\) & \(7.091\times 10^{-1}\) & \(7.283\times 10^{-1}\) \\ \hline \multirow{4}{*}{**Spacing**} & Average & \(1.389\times 10^{-2}\) & \(17.39\times 10^{-1}\) & \(\mathbf{1.151\times 10^{-2}}\) \\  & Best & \(5.869\times 10^{-3}\) & \(\mathbf{5.434\times 10^{-3}}\) & \(7.874\times 10^{-3}\) \\ \cline{1-1}  & Worst & \(5.500\times 10^{-2}\) & \(1.097\times 10^{-1}\) & \(\mathbf{2.276\times 10^{-2}}\) \\ \cline{1-1}  & Std. Dev. & \(1.269\times 10^{-2}\) & \(31.22\times 10^{-1}\) & \(\mathbf{3.980\times 10^{-3}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Statistic analysis for problem P1.