An Exact Linearization-Based Refinement Algorithm to the Carrier-Vehicle Traveling Salesman Problem

Traditional single-vehicle systems often fail to provide efficient solutions in applications, such as rescue operations, postdisaster relief, environmental mapping, and last-mile delivery. Coordinated systems composed of two echelons and heterogeneous vehicles with complementary capabilities offer a promising alternative. This paper focuses on the carrier-vehicle traveling salesman problem, which involves a slow carrier vehicle (mothership) transporting a fast vehicle (e.g., a drone) that must take off from and return to the mothership to serve a set of customers. We propose a mixed-integer nonlinear programming formulation and an approximation that can be modeled as a mixed-integer linear programming model. An exact algorithm is proposed that corrects the approximation error of the nonlinear term. Combined with a set of valid inequalities, our branch-and-cut algorithm guarantees optimal solutions for the original nonlinear problem. We evaluate our algorithm on a wide range of benchmark instances from the literature, ranging from 10 to 200 customers. Our method outperforms all previous exact and heuristic approaches in the literature in terms of objective value or computational time. Moreover, we are the first to solve all benchmark instances with up to 25 customers optimally, and notably, we find a proven optimal solution for an instance with 35 customers, the largest one solved to optimality in the literature.

Funding: B. S. Vieira acknowledges institutional support from the Instituto Federal de Santa Catarina. The research of L. C. Coelho was partly funded by the Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada [Grant 2025-03964].