Artificial bee colony algorithm based on multiple indicators for many-objective optimization with irregular Pareto fronts

Artificial bee colony (ABC) algorithm has shown excellent performance over many single and multi-objective optimization problems (MOPs). However, ABC encounters some difficulties when solving many-objective optimization problems (MaOPs) with irregular Pareto fronts (PFs). The possible reasons include two aspects: (1) there are many non-dominated solutions in the population and the low selection pressure cannot move the population toward the PF; and (2) it is hard to maintain population diversity for PFs having irregular geometric structures. To address these issues, a new many-objective ABC variant based on multiple indicators (called MIMaOABC) is proposed in this paper. Firstly, a convergence indicator $I_{{\varepsilon }_{+}}$ and a diversity indicator ($Div$) based on parallel distance are utilized. A single indicator may have preferences and it easily causes the population to converge to a subregion of the PF. Then, a two-stage environmental selection method is designed based on the two indicators. In the first stage, the $I_{{\varepsilon }_{+}}$ based environmental selection is used to improve the convergence. In the second stage, the $Div$ based environmental selection is employed to maintain diversity and handle irregular PFs. To balance exploration and exploitation during the search, multiple search strategies are used in different search stages, respectively. In the onlooker bee stage, solutions with good convergence are chosen for further search based on a new selection mechanism. In order to verify the performance of MIMaOABC, a set of well-known benchmark problems with degenerate, discontinuous, inverted, and regular PFs are tested. Performance of MIMaOABC is compared with eight state-of-the-art algorithms. Computational results shows that the proposed MIMaOABC is competitive in solving MaOPs with both irregular and regular PFs.