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Abstract The knapsack problem (KP) with forfeits is a generalized KP that aims to select some items, among a set of candidate items, to maximize a profit function without exceeding the knapsack capacity. Moreover, a forfeit cost is incurred and deducted from the profit function when both incompatible items are placed in the knapsack. This problem is a relevant model for a number of applications and is however computationally challenging. We present a hybrid heuristic method for tackling this problem that combines the evolutionary search with adaptive feasible and infeasible search to find high‐quality solutions. A streamlining technique is designed to accelerate the evaluation of candidate solutions, which increases significantly the computational efficiency of the algorithm. We assess the algorithm on 120 test instances and demonstrate its dominance over the best performing approaches in the literature. Particularly, we show 94 improved lower bounds. We investigate the essential algorithmic components to understand their roles.
Zhou et al. (2024) studied this question.