  We consider a multi-stage facility reallocation problems on the real line,
where a facility is being moved between time stages based on the locations
reported by $n$ agents. The aim of the reallocation algorithm is to minimise
the social cost, i.e., the sum over the total distance between the facility and
all agents at all stages, plus the cost incurred for moving the facility. We
study this problem both in the offline setting and online setting. In the
offline case the algorithm has full knowledge of the agent locations in all
future stages, and in the online setting the algorithm does not know these
future locations and must decide the location of the facility on a
stage-per-stage basis. We derive the optimal algorithm in both cases. For the
online setting we show that its competitive ratio is $(n+2)/(n+1)$. As neither
of these algorithms turns out to yield a strategy-proof mechanism, we propose
another strategy-proof mechanism which has a competitive ratio of $(n+3)/(n+1)$
for odd $n$ and $(n+4)/n$ for even $n$, which we conjecture to be the best
possible. We also consider a generalisation with multiple facilities and
weighted agents, for which we show that the optimum can be computed in
polynomial time for a fixed number of facilities.
