In this paper, we propose a relatively simple-to-implement procedure for solving the heterogeneous-fleet vehicle routing problem (HeVRP), in which different types of vehicle loading capacities are considered. Our approach is based on the so called successive approximations method (SAM), which is a multi-round process. At each round, a new subset of nodes and a new type of vehicle are selected following some specific criteria. Then, assuming an unlimited fleet of vehicles of this type, the associated homogeneous-fleet vehicle routing problem (HoVRP) is solved. After several rounds, a global solution for the HeVRP is obtained by merging routes from different HoVRP solutions. In the first part of the paper, we analyse how distance-based costs vary when slight deviations from the homogeneous fleet assumption are considered. In the second part of the article, the SAM approach is adapted so it can simultaneously deal with both fixed and variable costs in HeVRPs. An experimental comparison is then made with other HeVRP algorithms.