SUPPLEMENTARY DATA OF THE PAPER: A Distributed Bi-behaviors Crow Search Algorithm for Dynamic Multi-Objective Optimization and Many-Objective Optimization Problems

This study proposes a Distributed Bi-behaviors Crow Search Algorithm (DB-CSA) with two different mechanisms, one corresponding to the search behavior and another to the exploitative behavior with a dynamic switch mechanism. The bi-behaviors CSA chasing profile is defined based on a large Gaussian-like Beta-1 function which ensures diversity enhancement, while the narrow Gaussian Beta-2 function is used to improve the solution tuning and convergence behavior. Two variants of the proposed DB-CSA approach are developed, the first variant is considered for solving a set of MaOPs with 2, 3, 5, 7, 8, 10,15 objectives and the second aims to solve several types of DMOPs with different time-varying Pareto optimal set and Pareto optimal front. The second variant of DB-CSA algorithm (DB-CSA-II) is proposed for solving DMOPs including a dynamic optimization process to detect and react effectively to the dynamic change. The Inverted General Distance, the Mean Inverted General Distance and the Hypervolume Difference are the main measurement metrics are used to compare the DB-CSA approach to the state-of-the-art MOEAs. All quantitative results are analyzed using the non-parametric Wilcoxon signed rank test with 0.05 significance level which proving the efficiency of the proposed method for solving 44 test beds (21 DMOPs and 23 MaOPS).