In the last decade, several robustness approaches have been
developed to deal with uncertainty. In decision problems, and
particularly in location problems, the most used robustness
approach rely either on maximal cost or on maximal regret
criteria. However, it is well known that these criteria are too
conservative. In this paper, we present a new robustness approach,
called lexicographic α-robustness, which compensates
for the drawbacks of criteria based on the worst case. We apply
this approach to the 1-median location problem under uncertainty
on node weights and we give a specific algorithm to determine
robust solutions in the case of a tree. We also show that this
algorithm can be extended to the case of a general network
