Truck-and-drone routing problems have become an important research topic in the last decade because of their applications for last-mile deliveries. Despite the many publications in this area, the most efficient exact algorithms designed thus far struggle to solve the benchmark instances with 39 or more customers. This fact holds even for one of the simplest variants involving one truck and one drone whose routes must synchronize at customers’ locations: the basic traveling salesman problem with a drone (TSP-D). In this article, we devise a new algorithm for the TSP-D that solves every benchmark instance with up to 59 customers, and it scales up to 99 customers when the drone is much faster than the truck. The core of our method is a dynamic programming algorithm that is applied for column generation and variable fixing within tailored decremental state-space relaxation strategies. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by Fondo para la Investigación Científica y Tecnológica [Grant PICT-2018-2961] (Ministry of Science, Argentina). Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0390 .

The time-dependent traveling salesman problem with time windows (TDTSPTW) is a variant of the well-known traveling salesman problem with time windows, in which travel times are not assumed to be constant. The TDTSPTW accounts for the effects of congestion at the planning level, being particularly suited for distribution problems in large cities. In this paper we develop a labeling-based algorithm for the TDTSPTW that incorporates partial dominance and generalizes several state-of-the-art components from the time-independent related literature. We propose a framework general enough to be applied to the TDTSPTW and its variant without time windows, with the objective of minimizing the duration or the makespan. As part of the framework, we introduce a new state-space relaxation specifically designed for the time-dependent context. Extensive computational experiments show the effectiveness of the overall approach and the impact of the new relaxation, outperforming several recent algorithms proposed for these variants on more than 9,000 benchmark instances. In addition, we frame the minimum tour duration problem within the time-dependent literature and include it as a benchmark for our algorithm, obtaining improved computation times and 31 new optimal solutions. Summary of Contribution: In this paper, we study the time-dependent traveling salesman problem with time windows (TDTSPTW), a difficult single-vehicle routing problem that incorporates more realistic travel time functions than its classic time-independent counterpart. As a result, the TDTSPTW is harder to solve, as it requires more complex models and algorithms. Using state-of-the-art optimization techniques, we propose an efficient solution approach for the TDTSPTW and some related variants that outperforms the previous approaches in the literature. Our paper emphasizes the importance of algorithmic design and efficient implementations to tackle relevant practical combinatorial optimization problems—in particular, for time-dependent problems. Moreover, the resulting algorithm fosters a new research direction regarding exact algorithms for time-dependent problems using dynamic programming and relaxation techniques. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This research has been funded by Fondo para la Investigación Científica y Tecnológica (FONCyT) [Grants PICT-2016-2677 and PICT-2018-2961] from the Ministry of Science, Argentina, and by the Google Latin America Research Award (LARA) 2019. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1236 .

Abstract In this paper we implement a branch‐price and cut algorithm for a time dependent vehicle routing problem with time windows in which the goal is to minimize the total route duration. The travel time between two customers is given by a piecewise linear function on the departure time and, thus, it need not remain fixed along the planning horizon. We discuss different alternatives for the implementation of these linear functions within the labeling algorithm applied to solve the pricing problem. We also provide a tailored implementation for one of these alternatives, relying on efficient data structures for storing the labels, and show several strategies to accelerate the algorithm. Computational results show that the proposed techniques are effective and improve the column generation step, solving all instances with 25 customers, 49 of 56 with 50 customers, and many instances with 100 customers. Furthermore, heuristic adaptations are able to find good quality solutions in reasonable computation times.