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.本研究提出一种包含双并行优化层级的动态帕累托双层多目标粒子群优化（Dynamic Pareto bi-level Multi-Objective Particle Swarm Optimization，DPb-MOPSO）算法。在第一层级中，所有解均于单一搜索空间内完成管理；当成功检测到目标值出现动态变化时，将启用帕累托排序算子以实现多子群的细分与处理，进而驱动第二层级的强化开发。本研究采用基于随机检测器的动态处理策略，追踪由时变参数引发的目标函数变化；同时还实施了一种响应策略：对所有未获得改进的解进行重新评估，并以新生成的解替换原有解。本研究在一组涵盖不同类型时变帕累托最优集（Pareto Optimal Set，POS）与时变帕累托最优前沿（Pareto Optimal Front，POF）的动态多目标优化问题（Dynamic Multi-Objective Problems，DMOPs）上对DPb-MOPSO算法进行了测试。采用反向世代距离（Inverted Generational Distance，IGD）、平均反向世代距离（Mean Inverted Generational Distance，MIGD）以及超体积差（Hypervolume Difference，HVD）三项指标，对DPb-MOPSO的性能开展评估。