Efficient Move Evaluations for Time-Dependent Vehicle Routing Problems

In this paper we introduce several new methods for efficiently evaluating moves in neighborhood search heuristics for routing problems with time-dependent travel times. We consider both the case that route duration is constrained and the case that route duration appears in the objective. We observe that the composition of piecewise linear functions can be evaluated in various orders when computing the route duration. We use this to develop a new tree based data structure to improve the complexity of computations and memory usage. This also allows us to present methods that have the best known computational complexity, while they do not even require a lexicographic order of search. Our numerical experiments illustrate the trade-off between computation time and memory usage among the different methods. On 1000 customer instances, our methods are able to speed-up a construction heuristic by up to 8.89 times and an exchange neighborhood improvement heuristic by up to 3.94 times, without requiring excessive amounts of memory.