# A Novel Multi-Objective Harmony Search Algorithm with Pitch Adjustment by Genotype
## 6 Final Discussion  
In this work, a multi-objective HS algorithm (MOHSg) is proposed, whose fundamental contribution consists of the pitch adjustment based on the crowding distancing by  
Figure 13: Pareto sets for problem P6. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **MOHS2** & **MOHS3** & **MOHSg** \\ \hline \multirow{4}{*}{**Error ratio**} & Average & \(4.710\times 10^{-1}\) & \(\mathbf{3.495\times 10^{-1}}\) & \(4.998\times 10^{-1}\) \\  & Best & \(3.900\times 10^{-1}\) & \(\mathbf{2.950\times 10^{-1}}\) & \(4.400\times 10^{-1}\) \\  & Worst & \(5.300\times 10^{-1}\) & \(\mathbf{4.000\times 10^{-1}}\) & \(5.700\times 10^{-1}\) \\  & Std. Dev. & \(3.813\times 10^{-2}\) & \(\mathbf{2.946\times 10^{-2}}\) & \(3.299\times 10^{-2}\) \\ \hline \multirow{4}{*}{_GDf_} & Average & \(9.482\times 10^{-2}\) & \(\mathbf{9.229\times 10^{-2}}\) & \(9.464\times 10^{-2}\) \\  & Best & \(9.129\times 10^{-2}\) & \(\mathbf{8.838\times 10^{-2}}\) & \(9.073\times 10^{-2}\) \\  & Worst & \(1.003\times 10^{-1}\) & \(9.854\times 10^{-2}\) & \(\mathbf{9.848\times 10^{-2}}\) \\  & Std. Dev. & \(2.519\times 10^{-3}\) & \(2.913\times 10^{-3}\) & \(\mathbf{2.135\times 10^{-2}}\) \\ \hline \multirow{4}{*}{_GDx_} & Average & \(7.868\times 10^{-2}\) & \(8.357\times 10^{-2}\) & \(\mathbf{7.692\times 10^{-2}}\) \\  & Best & \(7.240\times 10^{-2}\) & \(7.698\times 10^{-2}\) & \(\mathbf{7.227\times 10^{-2}}\) \\  & Worst & \(8.891\times 10^{-2}\) & \(9.456\times 10^{-2}\) & \(\mathbf{8.118\times 10^{-2}}\) \\  & Std. Dev. & \(4.093\times 10^{-3}\) & \(4.468\times 10^{-3}\) & \(\mathbf{2.245\times 10^{-2}}\) \\ \hline \multirow{4}{*}{**Spacing**} & Average & \(8.465\times 10^{-2}\) & \(\mathbf{7.959\times 10^{-2}}\) & \(8.248\times 10^{-2}\) \\  & Best & \(7.110\times 10^{-2}\) & \(6.991\times 10^{-2}\) & \(\mathbf{6.080\times 10^{-2}}\) \\ \cline{1-1}  & Worst & \(1.024\times 10^{-1}\) & \(\mathbf{9.121\times 10^{-2}}\) & \(1.198\times 10^{-1}\) \\ \cline{1-1}  & Std. Dev. & \(9.119\times 10^{-3}\) & \(\mathbf{5.567\times 10^{-3}}\) & \(1.302\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Statistic analysis for problem P6. genotype, that is, the crowding distancing that works in the search space. This algorithm is capable of automatically adjusting the exploration and exploitation by adjusting the pitch, based on the distribution of solutions in the search space during the formation of the Pareto front. Therefore, MOHSg only needs the presetting of the harmonic memory and pitch adjustment parameters for its operation, without requiring the static bandwidth parameter of the MOHS2 variant nor the dynamic bandwidth and pitch adjustment parameters needed by the MOHS3 algorithm. For the test of the proposed algorithm, six multi-objective optimization problems were used with a diversity of characteristics regarding the shape of the Pareto-optimal fronts and Pareto-optimal sets, the coexistence of local and global solutions (multi-modality), and the number of decision variables and objective functions. MOHSg was able to produce similar or better results to those generated by the MOHS2 and MOHS3 algorithms, which constitute HS variants representative of the state-of-the-art in multi-objective optimization. Specifically, MOHSg produced the best results in three of the proposed problems, while in the rest, it registered competent results, excelling in some punctual performance indicators. From this comparative study, it can be concluded that the harmonic search algorithm based on pitch adjustment by genotype is an effective tool for solving multi-objective optimization problems. As part of future work, the application of the proposed algorithm to the solution of multi-objective problems with functional restrictions is proposed. Conceptualization and methodology, D.M.-P. and E.A.P.-F; validation and formal analysis, M.B.C.-Y. and G.S.-C.; investigation and software, D.M.-P.; writing--original draft preparation, D.M.-P; writing--review and editing, D.M.-P., E.A.P.-F., E.A.P.-F. and E.V.-A.