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

TL;DR

The paper addresses the PDPTW-SE, a generalization of the PDPTW that couples routing with scheduling on inter-region edges via dedicated machines. It proposes a compact arc-based MIP formulation with preprocessing and routing/scheduling tightening, complemented by a multi-start heuristic (MSLP) with an LP-based schedule improvement. Computational experiments on two instance families (multi-island and multi-floor) show that the baseline MIP solves up to 12 requests, while the heuristic effectively handles larger instances, often matching or exceeding MIP performance and achieving high feasibility. The LP improvement and the option of an extra machine meaningfully enhance feasibility and solution quality, underscoring practical guidance: use MIP for short horizons and MSLP for larger or long-horizon problems, with LP upgrades providing consistent gains.

Abstract

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.

Figures (21)

• Figure 1: PDPTW-SE solution example.
• Figure 2: Illustration map for the two instance families
• Figure 3: Box plots of the gap RPDs by instance type for the baseline MIP and the MIP with valid inequalities. Negative RPDs indicate that the version with valid inequalities outperforms the baseline MIP. Observations only for instances where a feasible solution was found, but the time limit was exceeded.
• Figure 4: Box plots of the solution value RPDs by instance type for the baseline MIP and the MIP with valid inequalities. Negative RPD indicate that the version with valid inequalities outperforms the baseline MIP. Observations only for instances where a feasible solution was found (optimal or not).
• Figure 5: Box plots of the solution value RPDs by instance type for the MIP with valid inequalities and the MSLP heuristic. Negative values indicate that the MSLP heuristic outperforms the MIP with valid inequalities. Observations only for instances with up to 12 requests where a feasible solution was found (optimal or not).