A reference set based many-objective co-evolutionary algorithm with an application to the knapsack problem

Despite the growing interest on many-objective evolutionary algorithms, studies on combinatorial problems are still rare. In this study, we choose many-objective knapsack problem (MaOKP) as the benchmark and target the challenges imposed by many-objectives in discrete search spaces by investigating several reference set handling approaches and combining several prominent evolutionary strategies in an innovative fashion. Our proposed algorithm uses elitist nondominated sorting and reference set based sorting, however reference points are mapped onto a fixed hyperplane obtained at the beginning of the algorithm. All evolutionary mechanisms are designed in a way to complement the reference set based sorting. Reference point guided path relinking is proposed as the recombination scheme for this purpose. Repair and local improvement procedures are also guided by reference points. Moreover, the reference set co-evolves simultaneously with the solution set, using both cooperative and competitive interactions to balance diversity and convergence, and adapts to the topology of the Pareto front in a self-adaptive parametric way. Numerical experiments display the success of the proposed algorithm compared to state-of-art approaches and yield the best results for MaOKP. The findings are inspiring and encouraging for the use of co-evolutionary reference set based techniques for combinatorial optimization.