# Multi-Objective Optimisation of Welding Parameters for AZ91D/AA6082 Rotary Friction Welded Joints
## 4 Modeling and Optimisation
### Genetic Algorithms  
Recently, many works have been devoted to multi-criteria optimisation using genetic algorithms. The most popular are the multi-criteria genetic algorithms MOGA and NSGA. Both methods provide solutions in the form of a Pareto chart. These algorithms are an optimisation method based on the natural evolution of populations of living organisms. The algorithms conduct searches on populations built of chromosomes. These, in turn, are built of genes. Genetic algorithms search for the best solutions from a population and not from a single point. Genetic algorithms use random methods of chromosome selection in which crossover and mutation operators play an important role. Crossover is the exchange in a sequence of genes between two chromosomes. A mutation parameter involves changing the value of a gene in a single chromosome [48]. Genetic algorithms optimise tasks in a big search space for nonlinear functions of complex problems.  
A multi-objective genetic algorithm (MOGA) is employed to optimise engineering tasks where multiple criteria impact the quality of the solutions. MOGA identifies a set of decision variables that meet specific constraints to provide satisfactory values for all objective functions. The algorithm minimises or maximises several objective functions [46; 48]. MOGA produces a set of optimal solutions known as Pareto optimal solutions. The Pareto front represents all of these non-dominated solutions within the objectives space. This study used the Non-Sorting Genetic Algorithm II (NSGA-II) for the multi-objective optimisation of process parameters. The main steps of the NSGA-II have been demonstrated in previous papers [49; 50].  
Multi-objective optimisation was performed using the _Gamultiobj_ procedure in the Optimisation Tool of Matlab R2018a software. _Gamultiobj_ uses the NSGA II for multi-objective optimisation [47; 51]. The main parameters of the multi-objective genetic algorithm are presented in Table 8.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**i** & \(\mathbf{w_{1}}\) & \(\mathbf{w_{2}}\) & \(\mathbf{w_{3}}\) & \(\mathbf{b}\) \\ \hline
1 & 32.06 & 43.76 & 9.00 & \(-\)75.68 \\
2 & \(-\)11.04 & 1.27 & \(-\)22.02 & 28.69 \\
3 & \(-\)30.41 & 41.24 & \(-\)9.48 & 4.66 \\
4 & \(-\)23.66 & 19.73 & \(-\)27.44 & 25.19 \\
5 & \(-\)28.80 & 46.06 & 3.44 & \(-\)17.64 \\
6 & \(-\)22.36 & \(-\)33.61 & 22.25 & 33.11 \\
7 & \(-\)8.82 & 44.44 & \(-\)18.80 & \(-\)20.31 \\
8 & 6.98 & \(-\)37.82 & 16.70 & 14.74 \\
9 & \(-\)12.07 & 41.93 & 16.38 & \(-\)47.43 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Weights (\(w_{i}\)) and biases (bi) derived from input and hidden layer of neural network.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**GA Parameter** & **Symbol** & **Value** \\ \hline Number of generations & \(N_{g}\) & 500 \\ Population size & \(\text{Pop}_{\text{-size}}\) & 80 \\ Crossover function & \(\text{C}_{f}\) & heuristic \\ Crossover probability & \(\text{C}_{\text{prob}}\) & 0.9 \\ Mutation function & \(\text{M}_{\text{f}}\) & uniform \\ Mutation probability & \(\text{M}_{\text{prob}}\) & 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 8: _NSGA-2 main parameters._