The Fermat-Weber problem seeks a point that minimizes the average distance from a given sample. The problem was studied by Lin and Yoshida (2018) using the standard tropical metric with the purpose of analyzing phylogenetic data. In this talk, we argue that using a related asymmetric distance we have better geometric and algorithmic properties. The new formulation is strongly related to tropical convexity and is equivalent to a transportation problem. This gives a geometric perspective to the transportation problem, which was exploited by Tokuyama and Nakano (1995) to obtain efficient algorithms. At the end, we will see an application to computational biology: a new method for computing consensus trees. The talk is based on joint work with Michael Joswig.