## 2 State-of-the-Art on Evolutionary Multi-Objective Optimization
### Many-Objective Optimization Methods  
Many-objective optimization algorithms are designed to manage the issues related to the exploitation and exploration concepts to preserve convergence and diversity in the search space. Table 2, presents several Many-Objective Evolutionary Algorithms (MaOEAs) and is classified into five classes, namely, decomposition-based methods, indicator-based approaches, diversity-based selection criterion, modified dominance relation-based approaches and preference-based approaches. Many Pareto-based approaches show their limits to rank and determine the set non-dominated solutions using the dominance operator since a high number of solutions have a large number of objectives leading to the poor convergence implied by the Active Diversity Promotion (ADP) phenomenon [19]. As a solution, a variety of enhancements have been adopted in the original MOEAs when solving MaOPs, including the decomposition-based and indicator-based approaches. Decomposition mechanisms combine multiple objectives into a single problem or a series of sub-problems. Some popular techniques of this type are Pareto sampling [20], improved Pareto sampling (MSOPS-II) [21] and Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) [13]. The decomposition-based approach becomes more effective with a set of sub-MOPs, such as those presented in the Reference Vector-Guided Evolutionary Algorithm (RVEA) [22], MOEA/D-M2M [23], NSGA-III [24], MOEA/DD [25] and the MOEA/D-ROD [26]. In addition, a set of performance metrics are considered to guide the optimization process over different indicator-based approaches, such as the fast hypervolume-based evolutionary algorithm (HypE) [27], S-metric-selection-based \begin{table}
\begin{tabular}{c c} \hline \hline Diversity-Based Approaches & DNSGA-II [1] \\ Memory-Based Approaches & SGEA [11] \\ Prediction-Based methods & PFS [14] \\ Parallel Approaches & MOEA/D [13] \\  & dCOEA [12] \\ Transfer Learning-Based Methods & MMTL-MOEA/D [15] \\  & RI-MOEA/D [15] \\  & SVR-MOEA/D [16] \\  & Tr-MOEA/D [17] \\  & KF-MOEA/D [18] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Classification of the MOEAs for DMOPs. evolutionary multi-objective algorithm (SMS-EMOA) [28], indicator-based evolutionary algorithm (IBEA) [29], and Evolutionary Many-Objective Optimization Algorithm based on an IGD Indicator with Region Decomposition [30] and MaOEA/IGD [31]. A set of dominance operators are proposed to deal with the issue of the ineffectiveness of sorting non-dominated solutions using the dominance operator proposed in the Pareto-based methods.The most know novel dominance operators as the following; L-optimality [32], \(\epsilon\)-dominance [33], fuzzy dominance [34], the Grid-based Evolutionary Algorithm (GrEA) [35], \(\theta\) Dominance-based Evolutionary Algorithm (\(\theta\)-DEA) [36] and the preference order ranking procedure [37]. Diversity management techniques are proposed to arrange a good balance between the convergence and the diversity when solving MaOPs. In [35], a three grid-based criterion was proposed to maintain diversity, including the grid crowding distance, the grid coordinate point distance and the grid ranking. A diversity promotion mechanism, DM, is introduced in [38] to activate or disactivate the diversity of the population based on the spread and the crowding distance of solutions. In NSGA-III algorithm [24], the reference point-based strategy is used to solve MaOPs. The shift-based density estimation (SDE) strategy [39] was utilized to replace the dominance operators of MOEAs. The knee point-driven evolutionary algorithm (KnEA) [40] was developed using both knee point-based selection and dominance-based selection. Three groups of preference-based approaches, including prior algorithms, interactive algorithms and posterior algorithms, were employed to deal with the issue of population size limitations regarding the large dimension of the objective space.