A note on the nestedness property for ordered median problems in tree networks

The nestedness property has become an increasingly important means for devising efficient algorithms for network location problems. There have been attempts to explore the nestedness property of network location problems with some special cases of the convex ordered median objectives. However, there is little research on the nestedness property for those problems with the concave ordered median objectives. This paper constructs a tree network T and shows that the nestedness property cannot hold for the concave ordered median problem, which fills a gap in the research on the nestedness property. Finally, the authors pose an open problem on identifying the nestedness property for the continuous strategic ordered median problem.