A New Optimization Algorithm for the Vehicle Routing Problem with Time Windows

TLDR

The vehicle routing problem with time windows (VRPTW) generalizes the vehicle routing problem by allowing customer service to start within specified earliest and latest time windows. The study aims to develop a new optimization algorithm for solving the VRPTW. The algorithm solves the VRPTW by applying column generation to the LP relaxation of a set‑partitioning formulation, generating feasible columns via a shortest‑path dynamic‑programming subproblem, and using the resulting lower bound in a branch‑and‑bound scheme to solve the integer formulation. The algorithm successfully solved practical benchmark VRPTW instances, optimally handling 100‑customer problems—six times larger than any previously reported results.

Abstract

The vehicle routing problem with time windows (VRPTW) is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service. In this paper, we present the development of a new optimization algorithm for its solution. The LP relaxation of the set partitioning formulation of the VRPTW is solved by column generation. Feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming. The LP solution obtained generally provides an excellent lower bound that is used in a branch-and-bound algorithm to solve the integer set partitioning formulation. Our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark VRPTW test problems. The algorithm was capable of optimally solving 100-customer problems. This problem size is six times larger than any reported to date by other published research.