## 6 Multi-Objective Optimization of ECM Process Parameters
### Applications of Meta-Heuristics for Optimization of the Process Parameters
The optimum process parameters obtained by different meta-heuristics and the RSM are tabulated in Table 13. The convergence plot of the Pareto optimal solutions is shown in Figure 16, and the Pareto optimal set of process parameters out of 12 runs is presented in Table 14.

Figure 16: Pareto optimal solutions for various algorithms. Figure 14: Implementation of PSO algorithm. Figure 15: Comparison of convergence plots for various algorithms for MRR and NP. Both the MFO and GWO algorithms produced the better trade-off performances with the same solutions of EC = 132.014 g/L, V = 13. 2406 V, and FR = 2.8455 L/min for MRR = 0.242 g/min and NP = 57.7202 PPM. ## 7 Conclusions

In the present study, ECM operations were carried out on Monel 400 alloys by varying the applied voltage (11-15 V), flow rate (1-3 L/min), and electrolyte concentration (130-190 g/L). The meta-heuristic algorithms (i.e., moth-flame optimization (MFO) algorithm, grey wolf optimizer (GWO), and particle swarm optimization (PSO)) were implemented to find out the optimum process parameters for producing the best performance measures for the material removal rate (MRR) and nickel presence (NP) in the sludge. The regression model equations were developed to determine the optimum parametric combination. The high coefficient of determination values (R2 = 90.34% for MRR and R2 = 92.29% for NP) for both responses confirmed their adequacy. The algorithms were tuned to obtain the best possible feasible solution. This study proved that the MFO and GWO algorithms could be successfully utilized to find the best process parameter setting for the ECM process for Monel 400 alloys. The TOPSIS algorithm was implemented to convert multiple objectives into a single objective used in meta-heuristic algorithms. It was observed from the surface, main, and interaction plots of this ECM experimentation that all the process variables significantly affected the performances. The PSO algorithm converged very quickly in 10 iterations, while GWO and MFO took 14 and 64 iterations, respectively, for convergence of the MRR. In the case of the NP model, the PSO tool took only six iterations to converge, whereas MFO and GWO took 48 and 88 iterations, respectively. However, both MFO and GWO obtained the better trade-off performances with the same solutions of EC = 132.014 g/L, V = 13. 2406 V, and FR = 2.8455 L/min for MRR = 0.242 g/min and NP = 57.7202 PPM. The Pareto optimal set of solutions provided by the GWO, MFO, and PSO algorithms for the ECM process was beneficial to the industries, as it contained a wide range of optimal values. A particular solution from the Pareto optimal set can be chosen according to the specific needs of the objectives. Hence, it was confirmed that the metaheuristic algorithms of MFO and GWO were suitable for finding the optimum process parameters for machining Monel 400 alloys with ECM. Future studies may be explored by combining the artificial neural network (ANN) and fuzzy logic (FL) concepts with MFO and GWO to determine the optimum ECM process parameters. Conceptualization and methodology, V.N., A.S., L.N. and S.K.M. ; writing--original draft preparation, V.N., A.S., L.N. and S.K.M. ; writing--review and editing, S.S., E.A.N., R.S. and H.M.A.M.H. All authors have read and agreed to the published version of the manuscript. The authors extend their appreciation to King Saud University for funding this work through Researchers Supporting Project number (RSP-2021/164), King Saud University, Riyadh, Saudi Arabia. Not applicable. \begin{table}
\begin{tabular}{l l l l l l l l l l l l l l l} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**R.No. **}} & \multicolumn{2}{c}{**GWO**} & \multicolumn{4}{c}{**MFO**} & \multicolumn{4}{c}{**PSO**} \\ \cline{2-13}  & **EC** & **V** & **FR** & **MRR** & **NP** & **EC** & **V** & **FR** & **MRR** & **NP** & **EC** & **V** & **FR** & **MRR** & **NP** \\ \hline
1 & 134.440 & 13. * [PERSON] and [PERSON] (2020) Kharwar, P.K.