Branch-and-Price-and-Cut for the Periodic Vehicle Routing Problem with Flexible Schedule Structures

This paper addresses the periodic vehicle routing problem with time windows (PVRPTW). Therein, customers require one or several visits during a planning horizon of several periods. The possible visiting patterns (schedules) per customer are limited. In the classical PVRPTW, it is common to assume that each customer requires a specific visit frequency and offers all corresponding schedules with regular intervals between the visits. In this paper, we permit all kinds of schedule structures and the choice of the service frequency. We present an exact branch-and-price-and-cut algorithm for the classical PVRPTW and its variant with flexible schedules. The pricing problems are elementary shortest path problems with resource constraints. They can be based on one of two new types of networks and solved with a labeling algorithm, which uses several known acceleration techniques such as the ng-path relaxation and dynamic half-way points within bidirectional labeling. For instances whose schedule sets fulfill a certain symmetry condition, we present specialized improvements of the algorithm such as constraint aggregation and symmetry breaking. Computational tests on benchmark instances for the PVRPTW show the effectiveness of our algorithm. Furthermore, we analyze the impact of different schedule structures on run times and objective function values.