## 6 Multi-Objective Optimization of ECM Process Parameters
### Applications of Meta-Heuristics for Optimization of the Process Parameters  
To apply the various meta-heuristic algorithms such as MFO, GWO, and PSO in this work, the TOPSIS method was adopted to convert the multiple objectives into a single objective. A performance comparison of these algorithms was carried out. The Algorithms 2-4 show the pseudo-codes of the MFO, GWO, and PSO respectively. ```
1: Initialize no. of grey wolf (\(X_{ij}\)--\(i\) =1,2,...\(nw\) and \(j\) =1,2,..nd)
2: While (\(it\) < \(nit\))
3: Determine the fitness function \(F_{ik}\)
4: Calculate Pareto optimal distance \(f_{i}\)
5: Sort \(\text{f}_{i}\) in descending order and set as \(S_{i}\)
6: Store the first wolf's data as \(X_{il}\). and \(F_{it}\). 7: Using the sorted data, assign \(X_{a}\). = \(X_{1}\), \(X_{b}\). = \(X_{2}\). and \(X_{d}\). = \(X_{3}\). 8: Compute \(a\) =


```
1:Initialize Particle Position \(P\)
2:For \(i=1\) to \(itr_{max}\)
3:For each particle \(p\) in \(P\)
4:Evaluate \(fp\) = \(f(p)\)
5:If \(fp\) is better than \(f(pB)\); \(pB\)= \(p\);
6:End
7:End
8:\(gB\)= best \(p\) in \(P\)
9:For each particle \(p\) in \(P\)
10:Compute \(v=v+c_{1}*rand()*(pB-p)+c_{2}*rand()*(gB-p)\)
11:Update \(p=p+v\)
12:End
13:End
14:Print the best solution
```

**Algorithm 4** Particle Swarm Optimization Algorithm

The terms in GWO and their equivalent meanings in the optimization problem are listed in Table 10.

Similarly, the equivalent terms for the MFO and PSO algorithms for an optimization problem are shown in Table 11. The algorithm parameters are shown in Table 12. \begin{table}
\begin{tabular}{l l} \hline \hline
**Optimization Problem** & **Grey Wolf Optimization (GWO) Algorithm** \\ \hline Number of solutions & Number of grey wolves (i = 1, 2,... nw) \\ Combination of parameters within & Position of grey wolf (Xij) \\ their bounds & Number of parameters, factors, and \\ independent variables & Number of dimensions involved in defining \\ Value of best parameters & the position of a wolf (j = 1, 2,... nd) \\ Response value, output, and & Position of prey (Xbest) \\ dependant variable & Fitness of grey wolf (Fik) \\ First best three solution’s parameters & Position of alpha, beta, and delta grey wolves (Xa., Xb., and Xd.) \\ First best three solution’s fitness values & Fitness of alpha, beta, and delta grey wolves (Fa., Fb., and Fd.) \\ Except for best first three solutions & Omega grey wolf \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Terms in GWO and optimization problems. The steps in implementing the GWO algorithm in this experimentation are shown in Figure 12. The flowchart for implementing the MFO algorithm in the present work is shown in Figure 13.

Figure 12: Implementation of GWO algorithm. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline
**GWO Algorithm** & \multicolumn{3}{c}{**MFO Algorithm**} & \multicolumn{3}{c}{**PSO Algorithm**} \\ \hline
**Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline Maximum number of grey wolves & 100 & Maximum number of months & 100 & Maximum number of particles & 100 \\ Constant a & 2 to 0 & Position of moth close to the flame (t) & From \(-1\) to \(-2\) & Learning factors (C\({}_{1}\) and C\({}_{2}\)) & 2 and 2 \\ Coefficient vectors A & \(-2\)a to \(2\)a & Update mechanism Adaptive number of flames & Logarithmic spiral round((mf-(itr*(mf-1)/max\_it))) & Inertia weight (\(\omega\)) & From 0.4 to 0.9 \\ No. of iterations (nitr) & 100 & No. of iterations (nitr) & 100 & Number of particles & 30 \\ \hline \end{tabular}
\end{table} 
Table 12: Algorithm parameters for GWO, MFO, and PSO algorithms. The PSO implementation is shown in Figure 14. The efficiency of these algorithms is compared in the convergence plot of Figure 15. In maximization of the MRR, the MFO convergence plot was attained after the 60 th iteration, while GWO and PSO performed better. In minimization of the NP, PSO converged well before MFO and GWO.