A neural network-based iterative heuristic algorithm for the polynomial robust knapsack problem

Abstract

The polynomial robust knapsack problem (PRKP) is a variant of the classic knapsack problem by incorporating uncertain costs and benefits from item combinations, leading to a nonlinear objective function and exponential solution space. These complexities make the PRKP suitable for real-world scenarios where interactions between items unpredictably impact outcomes. However, existing algorithms struggle to efficiently solve large instances of the PRKP due to its computational complexity. Therefore, this paper presents an iterative heuristic algorithm leveraging a neural network (NN) to address the PRKP, reducing the solution space and enabling efficient resolution of subproblems. The framework integrates an NN trained in two steps: general training and fine-tuning. The trained model is then embedded in the iterative heuristic algorithm to tackle the PRKP. A synthetic dataset comprising 2500 instances, ranging from 100 to 1500 items, is created to train the NN. Comparative evaluations are conducted using 1600 benchmark instances from the literature and 140 larger instances containing between 2000 and 15,000 items. We compare our approach against two state-of-the-art algorithms for the PRKP: a genetic algorithm and a random forest-based heuristic. Computational results demonstrate that the proposed algorithm outperforms the genetic algorithm, providing superior solution quality with significantly reduced computing times. Meanwhile, against random forest-based heuristic, it delivers better solution quality with only a moderate increase in computing time. For larger instances, it maintains its advantage in solution quality while remaining computationally efficient. These results highlight the algorithm’s scalability, effectiveness, and potential to address the PRKP.