[Paper Review] Finding optimal solutions for vehicle routing problem with pickup and delivery services with time windows: A dynamic programming approach based on state-space-time network representations

Monirehalsadat Mahmoudi, Xuesong Zhou|arXiv (Cornell University)|Jul 9, 2015

Transportation and Mobility Innovations55 references19 citations

TL;DR

This paper proposes a state-space-time network-based dynamic programming approach to solve the vehicle routing problem with pickup and delivery and time windows (VRPPDTW), enabling joint optimization of passenger-vehicle assignment and routing in congested networks. By leveraging a three-dimensional space-time representation and Lagrangian relaxation, the method achieves optimal solutions on medium- and large-scale networks like Chicago and Phoenix.

ABSTRACT

Optimization of on-demand transportation systems and ride-sharing services involves solving a class of complex vehicle routing problems with pickup and delivery with time windows (VRPPDTW). This paper first proposes a new time-discretized multi-commodity network flow model for the VRPPDTW based on the integration of vehicles carrying states within space-time transportation networks, so as to allow a joint optimization of passenger-to-vehicle assignment and turn-by-turn routing in congested transportation networks. Our three-dimensional state-space-time network construct is able to comprehensively enumerate possible transportation states at any given time along vehicle space-time paths, and further allows a forward dynamic programming solution algorithm to solve the single vehicle VRPPDTW problem. By utilizing a Lagrangian relaxation approach, the primal multi-vehicle routing problem is decomposed to a sequence of single vehicle routing sub-problems, with Lagrangian multipliers for individual passengers requests being updated by sub-gradient-based algorithms. We further discuss a number of search space reduction strategies and test our algorithms, implemented through a specialized program in C++, on medium-scale and large-scale transportation networks, namely the Chicago sketch and Phoenix regional networks.

Motivation & Objective

• Address the complexity of on-demand transportation and ride-sharing systems requiring efficient routing with time windows.
• Model the VRPPDTW as a multi-commodity flow problem in a time-discretized space-time network to capture dynamic vehicle states.
• Enable joint optimization of passenger-to-vehicle assignment and turn-by-turn routing in real-time traffic conditions.
• Develop a scalable dynamic programming algorithm for single-vehicle subproblems and extend it to multi-vehicle scenarios via decomposition.
• Improve computational efficiency through search space reduction strategies for large-scale networks.

Proposed method

• Construct a three-dimensional state-space-time network to represent vehicle positions, passenger loads, and time states simultaneously.
• Formulate the single-vehicle VRPPDTW as a shortest path problem in the state-space-time network using dynamic programming.
• Apply Lagrangian relaxation to decompose the multi-vehicle problem into independent single-vehicle subproblems.
• Use sub-gradient optimization to iteratively update Lagrangian multipliers associated with individual passenger requests.
• Integrate search space reduction techniques to limit state exploration and improve computational performance.
• Implement the algorithm in C++ and evaluate it on real-world transportation networks, including Chicago and Phoenix.

Experimental results

Research questions

• RQ1 How can vehicle routing with pickup and delivery and time windows be modeled to support joint optimization of passenger assignment and routing?
• RQ2 What is the effectiveness of a state-space-time network representation in capturing dynamic vehicle and passenger states over time?
• RQ3 Can dynamic programming on a time-discretized state-space-time network achieve optimal solutions for single-vehicle VRPPDTW instances?
• RQ4 How does Lagrangian relaxation with sub-gradient updates improve the solution quality and convergence for multi-vehicle VRPPDTW?
• RQ5 To what extent do search space reduction strategies enhance computational efficiency on large-scale networks?

Key findings

• The proposed state-space-time network model enables comprehensive enumeration of vehicle states across space and time, supporting exact optimization.
• The dynamic programming approach on the state-space-time network achieves optimal solutions for single-vehicle VRPPDTW instances.
• Lagrangian relaxation effectively decomposes the multi-vehicle problem into manageable single-vehicle subproblems with convergent dual bounds.
• Search space reduction strategies significantly decrease computational time without compromising solution quality.
• The algorithm successfully solved medium- and large-scale instances on the Chicago sketch and Phoenix regional networks, demonstrating scalability.
• The implementation in C++ confirms the feasibility of real-time application in congested urban transportation networks.