The vehicle routing problem (VRP) is a flourishing research area with clear applications to real-life distribution companies. However, most VRP-related academic articles assume the existence of a homogeneous fleet of vehicles and/or a symmetric cost matrix. These assumptions are not always reasonable in real-life scenarios. To contribute in closing this gap between theory and practice, we propose a hybrid methodology for solving the asymmetric and heterogeneous vehicle routing problem (AHVRP). In our approach, we consider: 1) different types of vehicle loading capacities (heterogeneous fleets); 2) asymmetric distance-based costs. The proposed approach combines a randomised version of a well-known savings heuristic with several local searches specifically adapted to deal with the asymmetric nature of costs. A computational experiment allows us to discuss the efficiency of our approach and also to analyse how routing costs vary when slight departures from the homogeneous fleet assumption are considered.