## 1 Introduction  
During the last decade, a wide range of meta-heuristics have been designed to solve many complex problems based on Evolutionary Algorithms (EA), such as the Genetic Algorithm (GA) [1], and Swarm Intelligence (SI) approaches, such as the Particle Swarm Optimization (PSO) algorithm [2; 3; 4; 5]. Different Multi-Objective Evolutionary Algorithms (MOEAs) have been employed to solve static Single Objective Optimization Problems(SOPs) and static Multi-Objective Optimization Problems (MOPs), where the main challenge is to find the best solution for SOP and a set of optimal solutions when solving MOP to balance the convergence and diversity in the search space. However, this process becomes more challenging when solving Dynamic Multi-Objective Optimization Problems (DMOPs), characterized by several types of time-varying Pareto Optimal Sets (POSs) and Pareto Optimal Fronts (POFs) [6]. Equations (1) and (2) investigate the main differences between the static Multi-Objective Optimization Problem (MOP), static Many-Objective Optimization Problem (MaOP) and dynamic Multi-Objective Optimization Problem (DMOP). First, the common characteristics are as follows:  
* Each problem (MOP, MaOP, DMOP) has a fitness function \(F(X)\), which can be minimized or maximized with simultaneously performed \(M\) objectives. * Each problem has a set of solutions, \(X\), presented by a set of bounded decision variables, X, with a d-dimensional search space generated between the minimum (\(X_{min}\)) and the maximum (\(X_{max}\)) boundaries. * The search space of each problem can be limited by a set of inequality and equality constraint. \[F(X)= \begin{cases}F(X)=(f_{1}(X),\ldots,f_{M}(X))\\ \text{Subject to }:g(X_{i})\leq 0\text{ or }g(X_{i})\geq 0,h(X_{i})=0\\ \forall i=1,\ldots,d,x\in[X_{min},X_{max}]\\ 1<M\leq 3\end{cases} \] (1) where \(F(X)\) is the fitness function, \(M\) is the number of objectives, and \(g(X)\) and \(h(X)\) are the inequality and the equality constraints, respectively. The main difference between the static MOP, MaOP and DMOP is investigated by looking at the number of objectives, nature of decision space, objective space, and constraints. * For the MOP and the DMOP, the number of objectives (\(M\)) is fixed to 2 or 3 (\(1<M\leq 3\)) and (\(M\)) is upper than 3 for MaOP (\(M>3\)). * However, the DMOP has a time-varying decision variables, objectives and/or constraints, and is presented in Equation (2). * The DMOP has M dynamic objective functions \(F(X\), \(t)\) and a set of time-varying variables \(X(t)=X_{1}(t),\ldots,X_{n}(t)\), which are subject to different bounded constraints, limiting the search space. \[F(X,t)= \begin{cases}F(X,t)=(f_{1}(X,t),f_{2}(X,t),\ldots,f_{M}(X,t))\\ \text{Subject to }:g_{i}(X,t)\leq 0\text{ and }0h_{j}(X,t)=0\\ \forall i=1,\ldots,d_{g}(t)\text{ and },j=1,...,d_{h}(t)]\\ X\in[X_{min},X_{max}],t\in[t_{begin},t_{end}]\\ X\in\Omega_{X},t\in\Omega_{t}\end{cases} \] (2) where \(M\) is the number of conflicting objective functions, \(d_{g}(t)\) and \(d_{h}(t)\) are the number of inequality and quality constraints at time \(t\), respectively, and \(X\) is a set of bounded decision variables with a d-dimensional search space generated between minimum boundary (\(X_{min}\)) and maximum boundary (\(X_{max}\)). \(F(X,t)\) is the objective vector that optimizes solution \(X\) at time \(t\). \(\Omega_{X}\subseteq\mathbb{R}^{n}\) is the decision space, and \(\Omega_{t}\in\mathbb{R}\) is the time space, bounded between the starting time \(t_{begin}\) and the ending time \(t_{end}\). The objective vector is denoted by \(F(X,t):\Omega_{X}\times\Omega_{t}\rightarrow\mathbb{R}^{M_{t}}\), presenting the resulting values for each solution \(X\) at time \(t\). Generally speaking, Dynamic Multi-Objective Evolutionary Algorithms (DMOEAs) are designed to effectively detect and react to the changes that may affect the POS and POF, while conserving both convergence and diversity concepts [7,8].