An Improved Version of MMOEA/DC Based on Alternative Clustering Definitions

Multimodal multiobjective optimization problems (MMOPs) have recently received considerable attention since they emerge in many real-world applications (e.g., in multiobjective knapsack problems and flow shop scheduling). However, MMOPs constitute a very particular class of problem. Indeed, looking for an adequate Pareto front (PF) representation is insufficient. MMOPs contain multiple subsets within the Pareto optimal Set, each independently mapping to the same Pareto Front. So, traditional multiobjective evolutionary algorithms (MOEAs) are inappropriate for solving MMOPs. This has motivated the design of algorithms which are suitable for addressing MMOPs. This paper proposes modifying MMOEA/DC, which is an algorithm specifically designed for solving MMOPs, that adopts a dual clustering method in both decision and objective space. Particularly, we were interested in using clustering in decision space, which allows classifying solutions into multiple local clusters. Our proposed approach modifies the neighborhood definition in order to provide more robustness to the algorithm as well as to improve the clustering process in decision space and to reduce the influence of the control parameters. The efficiency of the proposed framework is validated by comparing its performance on test instances of two test suites (MMF and MMMPO) with respect to the original MMOEA/DC and other state-of-the-art algorithms designed for solving MMOPs.

I. Introduction

Multiobjective evolutionary algorithms (MOEAs) have been found to be very effective in solving a wide variety of complex multiobjective optimization problems [34]. However, for multimodal multiobjective problems (MMOPs), traditional MOEAs are insufficient. In MMOPs, an evolutionary algorithm must maintain diversity both in decision and in objective space while generating high-quality solutions. MOEAs usually are designed to focus on diversity in one space (typically, in objective space); therefore, they are inappropriate for MMOPs. Research on MMOPs is not only motivated by the need for diversity in decision space but also by the need to obtain sets of alternative solutions which map to the same Pareto front (PF). In fact, MMOPs hold different Pareto sets (PSs) mapping to the same PF or even maintain several local and global PSs simultaneously [30]. Moreover, MMOPs, compared to MOPs, can deliver the decision-maker a more extensive set of alternative solutions. In many cases, the obtained solutions provided by an MOEA may be unavailable or unusable due to environmental variances. The presence of alternatives (i.e., different PSs) can help the decision maker to eliminate uncertainties in practice [23].