SUPPLEMENTARY DATA OF THE PAPER: DPb-MOPSO: A Dynamic Pareto bi-level Multi-Objective Particle Swarm Optimization Algorithm

This study proposes a Dynamic Pareto bi-level Multi-Objective Particle Swarm Optimization (DPb-MOPSO) algorithm including two parallel optimization levels. At the first level, all solutions are managed in a single search space. When a dynamic change is successfully detected in the objective values, the Pareto ranking operator is used to enable a multiple sub-swarm’ subdivisions and processing which drives the second level of enhanced exploitation. A dynamic handling strategy based on random detectors is used to track the changes of the objective function due to time-varying parameters. A response strategy consisting in re-evaluate all unimproved solutions and replacing them with newly generated ones is also implemented. The DPb-MOPSO system is tested on a set of DMOPs with different types of time-varying Pareto Optimal Set (POS) and Pareto Optimal Front (POF). Inverted generational distance (IGD), mean inverted generational distance (MIGD), and hypervolume difference (HVD) metrics are used to assess the DPb-MOPSO performances.

本研究提出了一种动态帕累托双层多目标粒子群优化算法（DPb-MOPSO），该算法包含两个并行优化层级。在第一层级中，所有解均被管理于单一搜索空间。一旦成功检测到目标函数的动态变化，便运用帕累托排名算子以实现多个子群分的细分与处理，从而驱动第二层级的增强探索。采用基于随机检测器的动态处理策略以追踪因时间变化参数而引起的目标函数的变化。同时，还实施了一种响应策略，该策略包括重新评估所有未改进的解，并将它们替换为新生成的解。DPb-MOPSO系统在一系列具有不同类型时间变化帕累托最优集（POS）和帕累托最优前沿（POF）的DMOPs上进行了测试。为了评估DPb-MOPSO的性能，使用了逆生成距离（IGD）、平均逆生成距离（MIGD）和超体积差异（HVD）等指标。