The present work deals with dynamic vehicle routing problems in which new customers appear during the design or execution of the routing. We propose a parallel Adaptive Large Neighborhood Search (pALNS) that produces high quality routes in a limited computational time. Then, we introduce the notion of driver inconvenience and define a bi-objective optimization problem that minimizes the cost of routing while maintaining its consistency throughout the day. We consider a problem setting in which vehicles have an initial routing plan at the beginning of the day, that is periodically updated by a decision maker. We introduce a measure of the driver inconvenience resulting from each update and propose a bi-objective approach based on pALNS that is able to produce a set of non-dominated solutions in reasonable computational time. These solutions offer different tradeoffs between cost efficiency and consistency, and can be used by the decision maker to update the routing of the vehicles introducing a controlled number of changes.