## 3 Results and Discussion
### Case Study 3  
In this case study, the simultaneous minimization of all three machinability indices is carried out using evolutionary algorithms. As in case study 2, the TOPSIS method is used to convert all three machinability indices to a single objective. Further, the quality indicators,  
Figure 5: Pareto optimality (**a–c**) Multi-objective for single run (**d–f**) Multi-Objective for 27 runs.  
namely Diversity (DIV), Inverted Generational Distance (IGD), and Hyper Volume (HV), are used to opt out the best evolutionary algorithm which provides minimum values of machinability indices along with the corresponding set of turning process parameters. The algorithm, which has a higher DIV and HV value and a lower IGD value, is to be considered as the best algorithm among others. The DIV and IGD values are calculated using Equations (4) and (5), respectively. The hypervolume is calculated based on the Pareto analysis. The calculated values of quality indicators for all the algorithms are presented in Table 6.  
\[DIV=\sqrt{~{}\sum_{j=1}^{k}{(f_{j}^{\max}-f_{j}^{\min}~{})}^{2}} \tag{4}\]  
\[IGD=\frac{\sqrt{\sum\limits_{i=1}^{n}{d_{i}^{2}}}}{n} \tag{5}\]  
\[d_{i}=\sqrt{\sum\limits_{j=1}^{n{o_{ij}}}{(o_{ij}-o_{bj})}^{2}} \tag{6}\]  
where,  
\(O_{ij}\)--ith run \(j\)th objective value;  
\(O_{bj}\)--Best \(j\)th objective value;  
\(d_{i}\)--Euclidean distance.  
Further the statistical analyses of the quality indicators are carried out using Minitab software to study the performance and consistency of the algorithms. The normal probability plots for the quality indicators IGD, DIV, and HV are given in Figure 6a-c, respectively, and the summary reports of the same are given in Figure 6d-f, respectively. From Figure 6d-f and as well as the findings (higher values of DIV and HV and Lower value of IGD) from Table 5, it is concluded that the MFO algorithm outperformed others.  
The reasons for the better performance of MFO compared with other algorithms are explained here. The GA became gradually a dominant optimization technique compared to deterministic approaches mainly due to the higher probability of local solutions avoidance. However, the main drawback of GA was the stochastic nature of this algorithm which resulted in finding different solutions in every run. Despite the relatively high convergence rate of PSO, it has the drawback of premature convergence to local optimal and ineffectiveness in exploring the whole search space. Similarly, the original version of the GWO algorithm has the drawbacks of low solving accuracy, bad local searching ability, and slow convergence rate. Similarly, the disadvantage of GHO was being easy to fall into the local optimum which prevented the search process from finding a better solution. On the other hand, MFO is able to locate the local and global optimal solutions accurately with less computational time [43].  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\begin{array}{c}\text{GHO}\\ \text{GA}\end{array}\) & \(7.98\) & \(233.85\) & \(0.273\) & \(71.00\) & \(0.051\) & \(3\) & \(193.36\) & \(2.44\) & \(85.25\) \\ \(\text{GA}\) & \(9.81\) & \(279.61\) & \(0.267\) & \(72.00\) & \(0.051\) & \(3\) & \(194.36\) & \(2.48\) & \(86.25\) \\ \(\text{PSO}\) & \(5.79\) & \(264.68\) & \(0.265\) & \(94.62\) & \(0.052\) & \(3\) & \(173.13\) & \(2.45\) & \(73.28\) \\ \(\text{MFO}\) & \(5.10\) & \(286.72\) & \(0.286\) & \(92.62\) & \(0.052\) & \(3\) & \(171.13\) & \(2.35\) & \(72.28\) \\ \(\text{GWO}\) & \(6.70\) & \(267.48\) & \(0.269\) & \(96.62\) & \(0.052\) & \(3\) & \(174.13\) & \(2.55\) & \(74.28\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Minimization of machinability indices based on IGD, DIV and HV for Case study 3.