## 2 Preliminaries and Background
### C-Moea/Dd  
MOEA/DD is an evolutionary algorithm for many-objective optimization problems, drawing its strength from MOEA/D [44] and NSGA-III [45]. As it combines both the dominance-based and decomposition-based approaches, it implies an effective balance between the convergence and diversity of the evolutionary process. Decomposition is a popular method to break down a multiple objective problem into a set of scalar optimization subproblems. Here, the authors use the penalty-based boundary intersection approach,but they highlight that any approach could be applied. Subsequently, we briefly explain the general framework of MOEA/DD and expose its requisite modifications for solving constrained many-objective optimization problems. At first, a procedure generates \(N\) solutions to form the initial parent solutions and creates a weight vector set, \(W\), representing \(N\) unique subregions in the objective space. As the current problem does not exceed six objectives, only the one layer weight generation algorithm was used. The \(T\) closest weights for each solution are also extracted to form a neighborhood set of weight vectors, \(E\). The initial population, \(P\), is then divided into several non-domination levels using the fast non-dominated sorting method employed in NSGA-II. In the MOEA/DD main while-loop, a common process is applied for each weight vector in \(E\) until the termination criterion is reached. It consists of randomly choosing \(k\)-mating parents in the neighboring subregions of the weight vector considered. When no solution exists in the selected subregions, they are randomly selected in the current population. These \(k\)-solutions are then altered using genetic operators. For each offspring, an intricate update mechanism is applied on the population. First, the associated subregion of the offspring is identified. The considered offspring is then merged with the population in a temporary container, \(P^{\prime}\). Next, the non-domination level structure of \(P^{\prime}\) is updated. It is worthy to note that an ingenious method was employed to avoid full non-dominated sorting of \(P^{\prime}\). Since the population must preserve its size throughout the run of MOEA/DD, three cases may arise. When all solutions are non-dominated, the worst solution of the most crowded weight vector is deleted from the population. This function has been denominated LocateWorst. When there are multiple non-domination levels, the deletion of one solution depends on the number within the last non-domination level, \(F_{1}\). On the one hand, there is only one solution in \(F_{1}\), and the density of the associated subregion is investigated so as not to incorrectly alter the population diversity. LocateWorst is called in the case where the density contains only one element. When the most crowded subregion associated with each solution in \(F_{1}\) contains more than one element, the solution owning the largest scalarized value within it is deleted. Otherwise, LocateWorst is called so as not to delete isolated subregions. Since MOEA/DD is designed to solve unconstrained many-objective optimization problems, [PERSON] et al. (2018) also provided an extension for handling constrained many-objective optimization problems, which requires three modifications. First, a constraint violation value, \(CV(x)\), henceforth accompanies each solution x. It is determined as follows:  
\[CV(x)=\sum_{j=1}^{I}\langle g_{j}(x)\rangle+\sum_{k=1}^{K}|h_{k}(x)| \tag{3}\]  
where the function \(\langle\alpha\rangle\) returns the absolute value of \(\alpha\) if \(\alpha<0\) and returns \(0\) otherwise. Second, while the abovementioned update procedure is maintained for feasible solutions, the survey of the infeasible ones is dictated by their association with an isolated subregion. More precisely, a second chance of survival is granted to these infeasible solutions, and the solution with the largest \(CV\) or the one that is not associated with an isolated subregion is eliminated from the next population.