# A Dynamic Parameter Tuning Strategy for Decomposition-Based Multi-Objective Evolutionary Algorithms
## 2 Related Work  
### Moea/d-Pbi  
The decomposition-based multi-objective optimization algorithm (MOEA/D) employs three decomposition methods. This section will focus on the penalty-based boundary cross-aggregation approach (PBI) as an example to provide a detailed introduction to the MOEA/D algorithm process.  
The input to the MOEA/D-PBI algorithm contains the following:  
1 Objective functions: Define multiple objective functions that need to be minimized or maximized; these functions are a measure of the performance of the optimization problem. 2 Constraints: Define possible constraints. Constraints are problem limitations that need to be satisfied, e.g., feasibility, resource constraints, etc. 3 Weight vector: Determines the weights of each objective function in the optimization problem. ss(4) Penalty factor: Gives the factor used to penalize the violation of constraints. Penalty factors are used to penalize solutions that do not meet the constraints and to motivate the algorithm to find a better solution if the constraints are met. 5 Algorithm parameters: Set the parameters required by the algorithm, including population size, crossover rate, mutation rate, etc., which will affect the speed of convergence and the final results of the algorithm.  
Output of the MOEA/D-PBI algorithm:  
1 Approximate Pareto optimal solution set: These solutions perform relatively well on multiple objective functions and can no longer be improved by improving one objective function without compromising the others. These solutions are distributed on the Pareto front and represent the best trade-off solution to the problem. 2 The objective function value for each solution: The MOEA/D-PBI algorithm provides the value of each objective function for every solution in the set of approximate Pareto optimal solutions. These values help us evaluate the performance of each solution on different objective functions. 3 Feasibility information: The MOEA/D-PBI algorithm also provides information on the feasibility of each solution. This indicates whether each solution satisfies the constraints. If the solution does not satisfy the constraints, the algorithm may apply a penalty factor to it to correct it.  
The MOEA/D-PBI algorithm steps include the following:  
1 Initialization: Set the number of populations N and generate the initial population P(0) = \(\{x_{1}(0),x_{2}(0),\ldots,x_{n}(0)\}\).  
2 Decoupling Decomposition: For each individual \(x_{(i)}(t)\in\) P(t), compute its decomposition weight vector \(\lambda(\text{i})\) = \((\lambda_{1}(i),\lambda_{2}(i),\ldots,\lambda_{m}(i)\), The variable'm' represents the number of targets, while 'i' denotes the index of the individual.  
3 Update the set of neighbors: For each individual \(x_{(i)}(t)\in\) P(t), find its set of neighbors. The selection of the set of neighbors can be based on the distance metric or topology.  
4 Evolutionary operations: For each individual \(x_{(i)}(t)\in\) P(t), perform the following steps: (a) Select two parent individuals (x(\(p_{1}\))x(\(p_{2}\)) \(\in\) N\({}_{(\text{i})}\). (b) Use crossover and mutation operations to generate a child individual x(off). (c) Compute the decomposition weight vector (off) for the child individuals. (d) Determine whether a child individual can replace a parent individual.  
If \(\lambda\)(off) is close to \(\lambda(\text{i})\), replace the parent individual. Otherwise, retain the parent individual.  
5 Renewal of stocks: For each individual \(x_{(i)}(t)\in\) P(t), it is updated according to the fitness value, which can be calculated using the Pareto boundary crossing (MOEA/D-PBI) function.  
\(\xi\) Termination judgment: Check whether the termination conditions are satisfied, such as reaching the preset number of iterations or the degree of convergence of the solution.  
\(\zeta\) Produce an approximate set of Pareto-optimal solutions: Upon satisfaction of the termination condition, the algorithm outputs the approximate set of Pareto-optimal solutions.