TLDR: This research introduces ‘sequence variables’ as a novel computational domain within Constraint Programming (CP) to address the limitations of existing models in solving Vehicle Routing Problems (VRPs). These new variables are designed to naturally handle optional visits and efficiently support insertion-based search strategies, including Large Neighborhood Search. The paper formalizes their domain, update operations, and consistency levels, and describes an implementation with global constraints tailored for VRPs. Demonstrated on the Dial-a-Ride Problem (DARP), sequence variables simplify modeling and achieve competitive computational performance, offering a more flexible and powerful approach to complex routing and sequencing challenges.
Vehicle Routing Problems (VRPs) are a cornerstone of modern logistics, impacting everything from package delivery to ride-sharing services. These complex challenges involve scheduling a fleet of vehicles to fulfill a set of transportation requests while minimizing costs and adhering to various constraints. Constraint Programming (CP) offers an intuitive way to model these problems, but traditional CP approaches have faced significant hurdles, particularly when dealing with optional stops or when trying to use efficient insertion-based search strategies.
Existing CP models, such as the ‘successor model’ (which defines the next stop for each vehicle) and ‘head-tail sequence variables’ (which manage growing sequences of visited nodes), have limitations. They often struggle to naturally represent optional visits – situations where a stop might or might not be part of a route. More critically, they don’t easily support ‘insertion-based heuristics.’ These heuristics are crucial for quickly finding high-quality solutions by intelligently inserting new stops into existing routes, rather than just appending them at the end. This is especially important for avoiding inefficient, long detours.
To overcome these challenges, a new computational domain called ‘sequence variables’ has been formalized. This innovative approach fundamentally changes how routing problems are modeled within CP. Unlike its predecessors, sequence variables are designed from the ground up to handle optional visits seamlessly and to fully support insertion-based search strategies. This includes advanced techniques like insertion-based Large Neighborhood Search (LNS), which iteratively refines solutions by relaxing parts of a route and then re-inserting stops optimally.
The paper provides a clear definition of how these sequence variables work, detailing their domain, how updates are performed, and introducing new ‘consistency levels’ for constraints that operate on this domain. These consistency levels help ensure that the solver efficiently prunes impossible solutions, speeding up the search process. An implementation of sequence variables is also described, including the underlying data structures needed to integrate them into existing CP solvers, making them practical for real-world use.
Furthermore, the research introduces global constraints specifically tailored for sequence variables and vehicle routing. These include constraints for managing travel distance, transition times between stops, specific precedence requirements (e.g., a pickup must occur before a drop-off), and cumulative capacity constraints (essential for problems involving goods or passengers). These specialized constraints allow for more accurate and efficient modeling of complex VRP scenarios.
The effectiveness of sequence variables is powerfully demonstrated through their application to the Dial-a-Ride Problem (DARP). DARP involves scheduling vehicles to pick up and drop off passengers, often with strict time windows, vehicle capacities, and ride-time limits for passengers. The paper shows that sequence variables not only simplify the modeling of DARP but also achieve competitive computational performance compared to other state-of-the-art solvers and commercial tools. Even with a relatively simple Large Neighborhood Search strategy, the sequence variable approach consistently delivers solutions close to the best-known results for DARP instances. The approach also proves adaptable to variations like the Pickup and Delivery Problem with Time Windows (PDPTW) and the basic Pickup and Delivery Problem (PDP), consistently yielding strong results.
Also Read:
- The Agentic Lakehouse: Enabling Safe AI-Driven Data Pipeline Management
- LiveOIBench: A New Benchmark for LLMs in Competitive Programming
In conclusion, sequence variables represent a significant advancement in Constraint Programming for vehicle routing and sequencing problems. By providing a robust theoretical foundation, a practical implementation, and specialized global constraints, they offer a more flexible and powerful framework for tackling complex logistics challenges. Their compatibility with optional visits, insertion-based heuristics, and Large Neighborhood Search makes them a valuable tool for researchers and practitioners aiming to optimize routing solutions. For more in-depth technical details, you can refer to the full research paper. Read the full paper here.
