Benders decomposition for traveling salesman problem with drone: A space-time-state network perspective

As technology advances, the integration of diverse transportation modes is accelerating, with truck-drone collaboration emerging as a promising solution to urban last-mile delivery challenges. While prior research has extensively covered spatial synchronization in customer service networks, studies on temporal synchronization and time-dependence in real transportation networks remains limited, as does the consideration of candidate rendezvous points. To address this gap, we propose a novel modeling approach to the Traveling Salesman Problem with Drone, utilizing a Space-Time-State network framework, termed TSPD-STS. An arc-based integer linear programming (ILP) model is presented for single-truck single-drone delivery, incorporating time-dependent travel time influenced by varying speeds, candidate rendezvous, drone capacity, payload-dependent endurance and time windows. We solve the problem by a branch-and-Benders-cut approach based on generalized Benders decomposition, dividing it into an assignment problem (i.e., the master problem) and a coupled scheduling problem (i.e., the subproblem). The master problem is solved only once, while the subproblem dynamically provides cuts along the branching. To improve computational efficiency, we design five classes valid inequalities to strengthen the relaxed master problem, and propose a dynamically tightened optimal cut and a novel valid cut which, unlike traditional no-good cut, can handle clusters of variables in a single step. Extensive computational experiments on 170 instances with 4 to 20 space nodes, tested under four time-dependent patterns and compared against ten algorithms using branch-and-bound and operation-based dynamic programming as baselines, demonstrate the superior performance of the proposed method. Our approach reduces computation time by 38.15% to 79.43% and yields smaller optimality gaps, with an average of 0.46%. Sensitivity analyses provide practical insights for logistics planning and strategic decision-making.