Retrieval scheduling in four-directional shuttle-based compact storage and retrieval systems with heterogeneous lifts

As an emerging tier-to-tier shuttle-based system, the four-directional shuttle-based compact storage and retrieval system with heterogeneous lifts utilizes four-directional shuttles for in-tier bin transport; inter-tier movements are executed by two heterogeneous lift types, with shuttle-lifts transporting shuttles and bin-lifts transporting bins. Since the bin-lift is typically less expensive than the shuttle-lift, employing two types of lifts can reduce both investment and operational costs. The execution of retrieval requests in this system requires coordinated operation of three types of equipment, making the scheduling optimization problem both complex and challenging. This paper investigates the retrieval scheduling problem, which focuses on determining the task sequences of shuttles and lifts to minimize the makespan. We formulate a mixed integer programming model and propose an exact logic-based Benders decomposition algorithm. Additionally, we develop an effective two-stage heuristic to provide near-optimal solutions for large-scale instances. The two-stage heuristic consistently outperforms alternative scheduling strategies, delivering shorter makespan, lower average retrieval time, and reduced capital investment. Numerical experiments reveal that increasing the number of shuttles and bin-lifts, rather than shuttle-lifts, leads to a more significant reduction in makespan. In addition, bin-lifts demonstrate greater cost-effectiveness compared to shuttle-lifts; partial substitution of shuttle-lifts with bin-lifts yields substantial economic benefits, with the highest benefits observed when bin-lifts account for 20% of the total lift investment. Furthermore, positioning lifts at the midpoint along the picking tracks produces the minimum makespan. These findings provide valuable insights into equipment configuration for optimizing the performance of such systems.