The pickup and delivery problem with time windows and scheduling on the edges

We introduce the Pickup and Delivery Problem with Time Windows and Scheduling on the Edges (PDPTW-SE), a generalization of the PDPTW that integrates vehicle routing and machine scheduling. The problem involves defining routes for transportation requests with specific pickup and delivery locations using a heterogeneous vehicle fleet, while machines must be scheduled to traverse certain edges. The objective is to minimize the total completion time subject to capacity, time window, and precedence constraints. We propose a mixed-integer linear programming (MIP) formulation, including preprocessing and valid inequalities, and a multi-start heuristic with a linear programming (LP) improvement procedure. A benchmark set with two instance families is also introduced: (i) coordination of pickups and deliveries across islands requiring cargo ships, and (ii) transport across multiple floors, as in hospitals, requiring elevator scheduling. Computational experiments show that the solver on the MIP formulation solves instances with up to 12 requests and finds feasible solutions for 95.0% of the 320 instances with up to 12 requests. For these, the heuristic consistently provides feasible solutions with low deviations, often matching or outperforming the MIP results. For the remaining 160 instances with 40 and 60 requests, only the heuristic finds feasible solutions. We thus recommend the MIP for short-horizon instances (up to 12 requests) and the heuristic for larger or long-horizon instances. Results also highlight the LP improvement procedure’s relevance, reducing solution values by at least 5% on average in general. For larger, tightly constrained instances, an additional machine helps with feasibility and solution quality.