Exact Methods and a Two-Stage Iterative Heuristic for the Carrier-Vehicle Traveling Salesman Problem

The carrier-vehicle traveling salesman problem (CVTSP) aims to optimize the routes of a larger, but slower, carrier and a smaller, but faster, vehicle to minimize the maximum completion time for visiting a set of targets. This paper introduces an enhanced formulation for the CVTSP using a set of new valid inequalities derived from structural properties. A logic-based Benders decomposition method is tailored to solve the problem by introducing various types of Benders cuts. In particular, a new analytical cut is developed based on valid bounds of the travel time between any two consecutive visited nodes. To handle practical instances, we design a simple, yet effective, two-stage iterative heuristic, which repeatedly solves a traveling salesman problem using an updated approximate travel time matrix. We conduct numerical experiments on 529 benchmark instances. Results show that the proposed formulation and exact method perform well, especially for the instances with small vehicle endurance and distant targets. The heuristic quickly achieves optimality for all instances with known optimal values. It finds new best solutions for most open instances, outperforming state-of-the-art heuristics in solution quality and efficiency. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: Financial support from the National Natural Science Foundation of China [Grant 72201044] and [Grant 72571037] are gratefully acknowledged. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2025.1140 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2025.1140 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .