Optimizing multi-level shuttle-based puzzle storage systems with horizontal and vertical dynamics using integer programming and ALNS-IP

Efficient retrieval in automated storage systems is critical for warehouse performance. This study presents a novel mathematical model and a hybrid ALNS-IP heuristic for multi-level shuttle-based puzzle storage systems, explicitly incorporating both horizontal and vertical shuttle movements— addressing a gap in the literature where primarily horizontal movement has been considered. The model captures the operational characteristics of cube-shaped storage warehouses, and the heuristic is designed to optimize retrieval operations by minimizing the total number of moves. Numerical experiments show that the proposed approach achieves near-optimal solutions for small- and medium-sized instances and feasible, high-quality solutions for large-scale systems where exact optimization is computationally infeasible. The impact of vertical movement, as well as sensitivity to computation time, number of shuttles, and demand levels, was analyzed, demonstrating the robustness and adaptability of the heuristic. The results confirm that the proposed model and algorithm provide a practical, scalable, and efficient solution for complex automated storage and retrieval systems, bridging the gap between theoretical optimization and real-world application.