# An Improved MOEA/D with Optimal DE Schemes for Many-Objective Optimization Problems
## 2 Preliminaries
### Decomposition Approaches Used in MOEA/D  
In the study of the original MOEA/D, a total of three kinds of decomposition approaches are introduced simply here. They are the weighted sum approach (WS), the Tchebycheff approach (TCH) and the penalty-based boundary intersection approach (PBI). A brief introduction of them is shown in the following:  
1. Weighted sum (WS) approach: The optimization problem of the WS approach is defined as: \[\begin{array}{ll}\text{minimize}&g^{\textit{ws}}(x|\lambda)=\sum_{i=1}^{m} \lambda_{i}f_{i}(x)\\ \text{subject to}&x\in\Omega\end{array}\] (2) where \(\lambda=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{N}\}\) is a set of weight vectors, for all \(i=1,\ldots,N\), \(\lambda_{i}=(\lambda_{i}^{1},\lambda_{i}^{2},\ldots,\lambda_{i}^{m})^{T}\), and subject to \(\lambda_{i}^{j}\geq 0\), \(\sum_{j=1}^{m}\lambda_{i}^{j}=1\) for all \(j=1,\ldots,m\). This approach could work well if the PF is convex (for minimization problems). Nevertheless, not every Pareto-optimal vector can be obtained through this approach if the PF is non-convex.
2. The Tchebycheff (TCH) approach: The optimization problem of the TCH approach is defined as: \[\begin{array}{ll}\text{minimize}&g^{\textit{tch}}(x|\lambda,z^{*})=\max_{1 \leq i\leq m}\{\lambda_{i}|f_{i}(x)-z_{i}^{*}|\}\\ \text{subject to}&x\in\Omega\end{array}\] (3) where \(\lambda\) is the same as above in (2), and \(z^{*}=(z_{1}^{*},...,z_{m}^{*})^{T}\) is the ideal reference point, i.e., for each \(i=1,\ldots,m\), \(z_{i}^{*}<\min\{f_{i}(x)|x\in\Omega\}\).  
As we know, there must exist a weight vector \(\lambda\) for every Pareto-optimal point \(x^{*}\), such that \(x^{*}\) is the optimal solution of (3). In addition, every optimal solution of (3) should be a Pareto-optimal solution of (1). Consequently, we can obtain different Pareto-optimal solutions by altering the weight vectors.
3. Penalty-based boundary intersection (PBI) approach: The optimization problem of the PBI approach is defined as: \[\begin{array}{ll}\text{minimize}&g^{pbi}(x|\lambda,z^{*})=d_{1}+\theta d_{ 2}\\ \text{where}&d_{1}=\frac{\|(F(x)-z^{*})^{T}\lambda\|}{\|\lambda\|}\\ &d_{2}=\|F(x)-(z^{*}-d_{1}\lambda)\\ \text{subjectto}&x\in\Omega\end{array}\] (4)  
In (4), \(\lambda\) is the weight vector as defined in (2), \(z^{*}\) is the ideal reference point as defined in (3) and \(\theta>0\) is a penalty parameter. From the previous study, using the same distributed weighted vectors, the PBI approach has some advantages over the TCH approach when solving an MOP whose number of objectives is more than two. However, the benefits must come with a price, which means the \(\theta\) as a penalty parameter requires proper adjusting.  
Some of these approaches have been successfully used in solving real-world problems. For example, in [8], [PERSON] et al. used the weighted sum approach to design a pattern reconfigurable array antenna and got some certain results. In [20], [PERSON] et al. used the Tchebycheff approach to solve the multi-objective traveling salesman problem (MTSP) and multiobjective 0-1 knapsack problem (MOKP) well.