## 2 Robust Design Optimization Methods and Challenges
### Multi-Objective Optimization Techniques
\begin{table}
\begin{tabular}{p{42.7 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Technique** & **Name** & **Description** \\ \hline NSGA-II & Non-dominated Sorting Genetic Algorithm II & It is a widely used optimization method. It uses the elitist strategy with the crowding distance operator to preserve diversity and the efficient non-dominated sorting operator to select the Pareto-dominant solutions [141]. \\ \hline NSGA-III & Reference point based Non-dominated Sorting Genetic Algorithm & This algorithm is designed for many-objective problems (more than two). It uses similar operators, like NSGA-II with reference points and the niche preservation operator, where the reference point can be associated with some solutions, and it keeps the solutions that are close to the reference point [142]. \\ \hline \(e\)-MOEA & \(e\)-dominance based Multiobjective Evolutionary Algorithm & It uses the epsilon-dominance concept with an epsilon archiving strategy for limiting and sorting the Pareto-dominant solutions, which can be faster than the non-dominated sorting, and it can be advantageous in many cases. However, the sorting time is not relevant in most cases of numerically expensive electrical machine design problems [143]. \\ \hline MOEA/D & Multiobjective Evolutionary Algorithm with Decomposition & This algorithm explicitly decomposes the problem into scalar optimization subproblems. It solves these subproblems simultaneously. At each generation, the population is composed of the best solution found so far for each subproblem. This algorithm can significantly reduce the computational complexity compared to NSGA-II [144]. \\ \hline GDES & Generalized Differential Evolution & GDE3 uses the weak-dominance concept to select the Pareto-dominant solutions and an improved crowding distance operator and non-dominated sorting for the results [145]. \\ \hline PAES & Pareto Archived Evolution Strategy & It uses a non-dominated bounded archive to maintain the Pareto-optimal solutions. There are three main versions: (1+1), (1+\(\lambda\)), and (9+\(\lambda\))-PAES [146], where (1+1)-ES means that during each iteration, one mutant is created from one parent, and their union is used in the selection. \\ \hline PESA2 & Pareto Envelope based Selection Algorithm & It uses a region based selection operator, instead of individual based ones, like NSGA-II. It requires only \(\mathcal{O}(k\cdot n)\) comparisons for every hyperbox [147]. \\ \hline SPEA2 & Strength based Evolutionary Algorithm & SPEA-2 uses the strength based diversity operator, the calculation and sorting time of which are more expensive than the case of NSGA-II; however, the diversity and convergence of the results can be better, which can be advantageous in the case of expensive optimization problems [148]. \\ \hline IBEA & Indicator Based Evolutionary Algorithm & It uses a flexible integration of preference information. Therefore, an arbitrary performance indicator can be used for the search. It does not need any diversity preservation techniques; moreover, the population size can be arbitrary [149]. \\ \hline MO-CMA-ES & Covariance Matrix Adaption & The adaptive grid archiving strategy, which was presented in PAES, is merged with the covariance matrix adaption evolutionary strategy. In contrast to most other evolutionary algorithms, the CMA-ES is quasi-parameter-free [150]. \\ \hline OMOPSO & Optimized Multiobjective Particle Swarm & Uses crowding distance to filter out leader solutions and the combination of two mutation operators to accelerate the convergence of the swarm and \(e\)-dominance based archiving strategy [135]. \\ \hline SMPSO & Speed-constrained Multicobjective Particle Swarm Optimization & It is very similar to the OMOPSO algorithm, and there are only three differences: the usage of the speed constriction factor and the polynomial mutation operator and velocity handling at the borders of the search space [134]. \\ \hline NSPSO & Non-dominated Sorting based multiobjective Particle Swarm Optimization & It uses the main mechanisms of NSGA-II (crowding distance, non-dominated sorting), and the global leader is selected randomly from the leaders’ archive. OMOPSO and SMPSO clearly outperform this variant [134]. \\ \hline AMOPSO & Another Multiobjective Particle Swarm Optimization & It selects leaders from a non-dominated external archive. Three different selection are techniques used: Roundly to preserve diversity, calledRandom to promote convergence, and called Prob, a weighted probability method [132]. \\ \hline MOFA & Multiobjective Firefly Algorithm & Uses random weights to select the best one from the Pareto-optimal solutions. Very quickly converges to the solution, however, it contains usually more than one function evaluation in every iteration step because the selected firefly makes one step with a new evaluation of a dominating one. This is a disadvantage in the case of expensive optimization problems [99]. \\ \hline \hline \end{tabular}
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