A reinforcement learning-driven cooperative scatter search for the knapsack problem with forfeits

The knapsack problem with forfeits belongs to the NP-hard combinatorial optimization family and arises in various applications like resource allocation, finance, and logistics, where certain item combinations incur penalties. Efficiently solving such a problem is crucial for optimizing resources while minimizing penalties. This paper proposes a novel reinforcement learning-driven cooperative scatter search algorithm to solve it, combining the robust exploration capabilities of scatter search with the adaptive learning strengths of $Q$-learning. The algorithm starts by generating a diverse archive set to ensure broad exploration of the solution space. It then iteratively generates and combines subsets using path-relinking, followed by a two-stage improvement process: $Q$-learning for dynamic enhancement and a tabu-based local search for refinement. Experimental evaluations on benchmark instances highlight the proposed method’s competitiveness against state-of-the-art approaches. The method establishes new lower bounds on 22 instances and matches existing bounds on others.