AbstractWe describe three simple heuristics for the vehicle routeing problem with customer time windows that can be violated by paying appropriate penalties. The customer demands are known, and a homogeneous fleet of vehicles stationed at a single depot is considered. The penalty payable to a customer is assumed to be a linear function of the amount of time window violation. Upper limits are imposed on both the penalty payable and the waiting time allowed at any customer. At each customer in a route, the PC-based heuristics simultaneously determine the actual time to begin service, and the next customer to serve. To achieve this, each heuristic uses different measures to compare the cost of waiting and penalty payable, with the benefit obtained by leaving immediately for the next customer. Computational results on a set of benchmark problems show that the procedure is efficient and enables significant reduction in the number of vehicles required and/or the total route distances while controlling both customer penalties and waiting times.