## 4 Results and Discussion  
Figure 7a,d represent sample pareto solutions obtained using different algorithms for crack length training and testing data sets. It is understood that the solution obtained by implementing DPGWO is better than the solution obtained by other algorithms, namely MFO and GWO. Figure 7b,c,e,f depict the quick convergence of R\({}^{2}\) and RMSE in both training and testing data sets of crack length in DPGWO as compared with MFO and GWO. It is observed from Figure A5a,d that DPGWO has higher R\({}^{2}\) and lower RMSE values as compared with MFO and GWO in both cases of crack depth training and testing data  
Figure 6: Statistical Analysis of Output using different algorithms for Crack Length. (**a**) Statistical analysis of output using MFO for Crack Length. (**b**) Statistical analysis of output using GWO for Crack Length. (**c**) Statistical analysis of output using DPGWO for Crack Length. sets. Early convergence is recorded in iteration numbers 16, 18, 30, and 28 in the cases of R\({}^{2}\)Tr, RMSETr, R\({}^{2}\)Tr, and RMSETt, respectively, in Figure A5b,c,e,f. Similar kinds of Pareto solution and convergence plots are obtained in the case of crack width, as shown in Figure A6a-f. It is also observed that a major difference in RMSE value is achieved in DPGWO as compared to MFO and GWO in Figure A6c,f. The \(p\)-value shown in Figure A4 confirms that the results obtained for 25 runs are normally distributed and acceptable. Performance indicators are used to compare the pareto solutions generated by the algorithms. In this work, two different performance indicators, namely inverted generational distance (IGD) and spacing (SP), are implemented to check the effectiveness of the algorithms. For problems with more than three objectives, IGD is used to measure the quality of pareto optimal solutions in terms of distribution and convergence, whereas SP is used to measure the distribution and spread of pareto optimal solutions. In both cases, the lower the value, the better the performance of the algorithm. From Table 12, it is confirmed that the DPGWO algorithm has lower values of IGD and SP as compared to the other two algorithms, hence it is outperformed. To further support the performance of the algorithms, [PERSON]'s test, a non-parametric test, has been implemented in this work. This test is used to determine whether statistically significant differences exist between the means of three or more groups. In this work, the Deng's values obtained by 25 runs using three algorithms are considered for ranking and the calculation of mean values. Using the \"Friedman (\(0\)\" function in Matlab, this test is conducted. Table 13 represents [PERSON]'s Test values and the mean ranking of each algorithm. All the probability values of crack dimensions are less than 0.05, which shows that the algorithm's performance differs from each other, i.e., a significant effect will be there on the selection of algorithms. Usually, a lower value will be allotted rank 1 in [PERSON]'s Test. But in this work, the higher the Deng's value, the better the solution; the higher the mean rank value, the higher the performance of the algorithm; hence, it is proved that the mean rank value of 2.24 in the case of crack length prediction using DPGWO outperformed as compared to the mean rank values of 2.2 and 1.56 of the MFO and GWO algorithms, respectively. Similarly, in the case of crack depth and width prediction, the mean rank of DPGWO will be 2.23 and 2.12, which are higher than the mean rank of MFO and GWO. \begin{table}
\begin{tabular}{l l l l l} \hline \multirow{2}{*}{**Crack Dimensions**} & \multicolumn{2}{c}{**Mean Rank**} & \multirow{2}{*}{**Probability**} \\ \cline{2-2} \cline{4-5}  & **MFO** & **GWO** & **DPGWO** \\ \hline Crack Length & 2.2 & 1.56 & 2.24 & 0.0263 \\ Crack Depth & 1.98 & 1.78 & 2.23 & 0.0049 \\ Crack Width & 1.84 & 2.04 & 2.12 & 0.0093 \\ \hline \end{tabular}
\end{table} 
Table 13: [PERSON]’s Test Values and Mean Ranking of Algorithms [25].