Evolutionary Multimodal Multiobjective Optimization for Traveling Salesman Problems

Multimodal multiobjective optimization problems (MMOPs) are commonly seen in real-world applications. Many evolutionary algorithms have been proposed to solve continuous MMOPs. However, little effort has been made to solve combinatorial (or discrete) MMOPs. Searching for equivalent Pareto-optimal solutions in the discrete decision space is challenging. Moreover, the true Pareto-optimal solutions of a combinatorial MMOP are usually difficult to know, which has limited the development of its optimizer. In this article, we first propose a test problem generator for multimodal multiobjective traveling salesman problems (MMTSPs). It can readily generate MMTSPs with known Pareto-optimal solutions. Then, we propose a novel evolutionary algorithm to solve MMTSPs. In our proposed algorithm, we develop two new edge assembly crossover operators, which are specialized in searching for superior solutions to MMTSPs. Moreover, the proposed algorithm uses a new environmental selection operator to maintain a good balance between the objective space diversity and decision space diversity. We compare our algorithm with five state-of-the-art designs. Experimental results convincingly show that our algorithm is powerful in solving MMTSPs.