Summary: Travel times inside cities often vary quite a lot during a day and significantly impact the duration of commercial delivery routes. Several authors have suggested time-dependent variants of the most commonly encountered vehicle routing problems. In these papers, however, time-dependency is usually defined on customer-based graphs. Thus, one major impact of travel time variations is missed: in an urban environment, not only do travel times change, but also the paths used to travel from one customer to another. In fact, during a day, different paths may be used at different points in time. To address this issue, one possible approach is to work directly with the road network and consider travel time (or travel speed) variations on each road segment.
In this paper, we propose a solution approach for a time-dependent vehicle routing problem with time windows in which travel speeds are associated with road segments in the road network. This solution approach involves a tabu search heuristic that considers different shortest paths between any two customers at different times of the day. A major contribution of this work is the development of techniques to evaluate the feasibility and the approximate cost of a solution in constant time, which allows the solution approach to handle problem instances with up to 200 nodes and 580 arcs in very reasonable computing times. The performance of our algorithm is assessed by comparing it to an exact method on a set of benchmark instances. The results show that solutions of high quality are produced.