The nestedness property of the convex ordered median location problem on a tree

This paper deals with the problem of locating an extensive facility of restricted length within a given tree network. Topologically, the selected extensive facility is a subtree. The nestedness property means that a solution of a problem with a shorter length constraint is part of a solution of the same problem with a longer length constraint. We prove the existence of a nestedness property for a common family of convex ordered median (COM) objective functions. We start with the proof of the nestedness property for a rooted tree problem, where the extended facility is a subtree of some tree network rooted at a specified node, and proceed to prove the nestedness property for the general location model on a tree.