We consider a network model of a system of transportation links, with nodes representing locations of existing facilities, and study the problem of finding a new facility location on the network that minimizes the maximum of linear increasing functions of the “network distances” between the new facility and the existing facilities. The problem is formulated with respect to a metric space which is defined on the network, and a number of properties of the problem are developed. The properties lead to a new, simple algorithm for solving the problem when the network is a tree, and to a new, equivalent spanning tree problem for a general network.