ADMM-based problem decomposition scheme for vehicle routing problem with time windows

Emerging urban logistics applications need to address various challenges, including complex traffic conditions and time-sensitive requirements. In this study, in the context of urban logistics, we consider a vehicle routing problem with time-dependent travel times and time windows (VRPTW), and the goal is to minimize the total generalized costs including the transportation, waiting time, and fixed costs associated with each vehicle. We adopt a high-dimensional space–time network flow model to formulate an underlying vehicle routing problem (VRP) with a rich set of criteria and constraints. A difficult issue, when solving VRPs, is how to iteratively improve both the primal and dual solution quality in general and how to break the symmetry generated by many identical solutions, particularly with homogeneous vehicles. Along this line, many coupling constraints, such as the consensus constraints across different agents or decision makers, need to be carefully addressed to find high-quality optimal or close-to-optimal solutions under medium- or large-scale instances. Currently, the alternating direction method of multipliers (ADMM) is widely used in the field of convex optimization, as an integration of the augmented Lagrangian relaxation and block coordinate descent methods, for machine learning and large-scale continuous systems optimization and control. In this work, we introduce the use of ADMM to solve the multi-VRP, which is a special case of integer linear programming, and demonstrate a manner to reduce the quadratic penalty terms used in ADMM into simple linear functions. In a broader context, a computationally reliable decomposition framework is developed to iteratively improve both the primal and dual solution quality. Essentially, the least-cost path subproblem or other similar subproblems involving binary decisions can be embedded into a sequential solution scheme with an output of both lower bound estimates and upper bound feasible solutions. We examine the performance of the proposed approach using classical Solomon VRP benchmark instances. We also evaluate our approach on a real-world instance based on a problem-solving competition by Jingdong Logistics, a major E-commerce company.