A branch-price-and-cut algorithm for the time-dependent multiple truck–drone routing problem

Time-varying traffic conditions are crucial features of urban logistics. Overlooking these conditions will pose a high coordination risk for drone-assisted routing problems. In this paper, a time-dependent multiple truck–drone routing problem (TD-MTDRP), which captures the time-varying traffic conditions as time-dependent travel times, is introduced. In the problem, trucks with time-dependent travel times travel through the urban roads, while drones fly over the urban areas, unaffected by the urban traffic, resulting in time-independent travel times. Our objective is to the total duration of all routes. We present the ready time function on a truck–drone path level, which is proven to satisfy the FIFO property. A route-based model is introduced based on this function to formulate the problem. The model decomposes the set-partitioning problem into a master problem (MP) and a time-dependent shortest truck–drone path problem with resource constraints (TDSTDPPRC). A branch-price-and-cut algorithm is also developed to solve the model. In this algorithm, a column-and-row generation is introduced to solve the MP, and a labeling algorithm incorporating new label extension and dominance rules is presented to address the TDSTDPPRC. For the numerical results, the approaches are evaluated on two sets of instances derived from the benchmark instances, demonstrating the effectiveness of the exact method in solving the TD-MTDRP.