SUPPLEMENTARY DATA OF 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. 

本研究提出一种动态帕累托双层多目标粒子群优化（Dynamic Pareto bi-level Multi-Objective Particle Swarm Optimization，简称DPb-MOPSO）算法，其包含两级并行优化层级。在第一层级中，所有解均在单一搜索空间内进行管理。当成功检测到目标值发生动态变化时，将采用帕累托排序算子实现多子群划分与处理，以此驱动第二层级的增强型开发搜索。本研究采用基于随机检测器的动态处理策略，以跟踪由时变参数引发的目标函数变化。此外，还实现了一种响应策略：对所有未得到改进的解进行重新评估，并以新生成的解替代这些未改进解。
本研究将DPb-MOPSO算法在一组动态多目标优化问题（Dynamic Multi-Objective Optimization Problems，简称DMOPs）上开展测试，这些问题涵盖不同类型的时变帕累托最优集（Pareto Optimal Set，简称POS）与时变帕累托最优前沿（Pareto Optimal Front，简称POF）。
本研究采用倒代距离（Inverted Generational Distance，简称IGD）、平均倒代距离（Mean Inverted Generational Distance，简称MIGD）与超体积差（Hypervolume Difference，简称HVD）三项指标，对DPb-MOPSO算法的性能进行评估。