# An Improved MOEA/D with Optimal DE Schemes for Many-Objective Optimization Problems
## 1 Introduction  
In daily life and work, multi-objective optimization problems (MOPs) have a wide range of applications, such as water distribution systems [1], land use management problems [2], automotive engine calibration problems [3], and so on, which are all real-world applications of MOPs. All of these problems are needed to optimize two or more than two objectives simultaneously. Without a doubt, everyone wants to make all objectives achieve optimization. However, we cannot find a solution to achieve this goal, because all of the objectives are restricting each other. If one objective achieves the optimization, the other objectives must achieve more useless solutions, which the decision makers do not want to see. Due to the importance of MOPs, scholars have been trying to solve this effectively. In order to facilitate the study of MOPs, scholars summed up the general mathematical expression of the MOPs. Therefore, a multi-objective optimization problem (MOP) can be mathematically defined as follows:  
\[ \begin{array}{ll}\underset{\text{subject to}}{\text{minimize}}&F(x)=(f_{1}(x ),f_{2}(x),...,f_{m}(x))^{T}\\ \text{subject to}&x\in\Omega\end{array}  \tag{1}\]  
where \(F:\Omega\rightarrow\Re^{m}\) is made up of \(m\) real-valued objective functions, \(\Omega\) is the decision (variable) space and \(\Re^{m}\) is called the objective space. Given two objective vectors: \(u,v\in\Re^{m}\), vector \(u\) is said to dominate \(v\), which means if and only if \(u_{i}\leq v_{i}\) for every \(i\in\{1,\ldots,m\}\) and there must exist at least one \(j\in\{1,\ldots,m\}\) satisfied with \(u_{j}<v_{j}\). Additionally, if \(F(x)\) dominates \(F(y)\), a solution \(x\in\Omega\) is said to dominate another solution \(y\in\Omega\). A solution \(x^{*}\in\Omega\) is defined as a Pareto-optimal solution in the case that \(F(x^{*})\) is not dominated by any other solutions in the objective space. Then, \(F(x^{*})\) is called a Pareto-optimal objective vector. The set of all the Pareto-optimal solutions is known as the Pareto set (PS). In addition, the set of all of the Pareto-optimal objective vectors is the Pareto front (PF) [4; 5]. Additionally, any improvement of one objective in the Pareto-optimal solutions is bound to the decline of at least one other objective. In most real-world applications, because the objectives in (1) conflict with each other, there is no solution that could minimize all of the objectives simultaneously. Because of the complexity of MOPs, the traditional mathematical methods will meet different levels of difficulty while solving the MOPs. Nevertheless, the evolutionary multi-objective optimization (EMO) algorithm based on natural evolution can solve the MOPs exceedingly well to some extent. Over the past twenty years, EMO algorithms and their applications came into being and developed rapidly, which has attracted a large number of researchers to study them [6; 7; 8; 9; 10; 11; 12; 13]. Owing to the research of EMO, the evolutionary algorithm (EA) has some certain advantages in obtaining the multiple Pareto-optimal solutions; for example, it always can obtain more than one optimal solution in a single run. Therefore, it is very common to use multi-objective evolutionary algorithms (MOEAs) to solve MOPs. Many EMO algorithms did the fitness assignment through using Pareto dominance, which are called the domination-based algorithms. In these algorithms, they tried to find a set of solutions, which should be as approximate as possible to the true PF, and this is known as convergence. The allocation of the fitness scheme to solutions is based on the Pareto-dominance principle, and the convergence of the basic algorithm plays a pivotal role in this principle. On the other hand, these algorithms need an explicit diversity preservation scheme to maintain a diverse set of solutions. In this way, two of the most prevailing MOEAs are NSGA-II [14] and SPEA2 (the improved version of strength Pareto evolutionary algorithm) [15]. They have achieved remarkable results in solving MOPs.