# A Dynamic Parameter Tuning Strategy for Decomposition-Based Multi-Objective Evolutionary Algorithms
## 3 Suggested Strategies for Improvement
### MOEA/D-DPA Algorithm  
The MOEA/D-DPA algorithm is divided into five main steps: initialization, constructing the solution, selecting the children, updating the value of the penalty parameter \(\theta\), and updating the solution set in five parts. Its pseudo-algorithm is shown in Algorithm 1, and the algorithmic procedure is described in detail below.  
In the first step, an initialization operation is performed. Generate N vectors uniformly distributed in a multidimensional space. Ensure that the vectors are uniformly distributed and calculate their Euclidean distance. The Euclidean distance for each vector to select its nearest T vectors will be counted in the order in the middle \(\text{B}(i)\), \(\text{B}(i)=\{\text{i}_{1},\dots,\text{i}_{\text{T}}\}\), where this \(\text{w}^{\text{i}},\dots,\text{w}^{\text{iT}}\) that is \(\text{w}^{\text{i}}\) distance nearest T vectors.  
In the second step, an initial population \(\text{P}=\{x^{1},\dots,x^{\text{N}}\}\) is generated from the solution space in a randomized manner. \(z_{i}^{*}\) and \(z^{\text{nad}}\) will be computed at the same time,where \(z_{i}^{*}\) is the ideal value, and precise computation will be very time consuming, so it will be estimated by the minimum value of the current population, which is constantly updated as the population iterates; similarly \(z_{i}^{\text{nad}}\) of the minimum \(z^{\text{nad}}\) is estimated by the worst value achieved by the current population in the \(\text{f}_{i}\) direction, and continuously updated.  
The third step is to construct the solution. Constructing the solution in this algorithm is mainly divided into four parts, structured reference points, cross variation, normalization process, and clustering. The following algorithm outlines the steps to be taken.  
In the fourth step, the population is updated. The strategy used is to choose to replace \(x^{\text{j}}\) with a solution \(y\) if the resulting solution \(y\) is better than any of the solutions \(x\) in B(i), i.e., \(g^{\text{pbi}}(y|w^{\text{j}},x^{*})\leq g^{\text{pbi}}(x^{\text{j}}|w,x^{*})\), as judged by the PBI formula, this contributes to preserving the diversity of the population and gradually enhancing the quality of the solution.  
In the fifth step, the value of the penalty parameter \(\theta\) is adaptively adjusted according to the PBI tuning formula introduced in this chapter.  
Finally, keep repeating the above steps until the maximum number of iterations is reached.  
```
0: \(\bullet\)  maxIteration: a stopping criterion; \(\bullet\)  N: the quantity of subproblems evaluated in MOEA/D; \(\bullet\)  a uniform spread of N weight vector: \(\lambda^{1}\dots\lambda^{N}\); \(\bullet\)  T: the number of weight vectors in the neighborhood of each weight vector. Output: EP
1: Initialize: Generate a uniform spread of N weight vectors: \(w^{1},\dots,w^{N}\) and compute the T closest weight vectors to each weight vector by the distance. For each \(i=1,\dots,N\), set \(B(i)=\{i_{1},\dots,i_{T}\}\) where \(w^{1},\dots,w^{T}\) are the T closest weight vectors to \(w^{i}\).
2: Create the initial population \(\mathbb{P}\)
3: While the stopping criteria have not been met, do:
4: Randomly select two indexes \(r_{1}\)_and_\(r_{2}\)_from_\(B(i)\)
5: Solution construction
6: Select construction
7: Determine the value of parameter \(\theta\)
8: Update of EP
9: End while
```  
**Algorithm 1 MOEA/D\(-\)DPA**