Branch-and-Price Approaches for Real-Time Vehicle Routing with Picking, Loading, and Soft Time Windows

Wölck M, Meisel S

Abstract

We propose and evaluate branch-and-price approaches for vehicle routing problems with picking, loading, and soft time windows. This general type of vehicle routing problem is of particular relevance in the same-day delivery context, in which fast routing algorithms are required because of the commitment to real-time delivery in the presence of high customer order frequencies. To boost the performance of the branch-and-price algorithms, we introduce the new method of tree-compatible labeling with nondominance trees. This method represents cost functions by a fixed number of breakpoints and uses a specialized tree-based data structure to store Pareto-optimal labels. We prove the theoretical soundness of the new method and evaluate its performance numerically with respect to pricing, column generation, and branch-and-price. Our numerical results show that the method yields substantial performance gains. In particular, we show that, with the new method, branch-and-price is able to reliably generate within a few minutes close to optimal solutions for problem instances with 50 customers. By additional experiments with classic vehicle routing problems with hard time windows, we show that the performance gains of our method result from its ability to handle cost functions in the pricing step. Our approach is the first branch-and-price approach for vehicle routing with picking, loading, and soft time windows. As such, it represents an exact routing algorithm that is able to reliably satisfy the runtime requirements of real-time delivery services.