  Dynamic facility location problems aim at placing one or more valuable
resources over a planning horizon to meet customer demand. Existing literature
commonly assumes that customer demand quantities are defined independently for
each time period. In many planning contexts, however, unmet demand carries over
to future time periods. Unmet demand at some time periods may therefore affect
decisions of subsequent time periods. This work studies a novel location
problem, where the decision maker relocates a single temporary facility over
time to capture cumulative customer demand. We propose two mixed-integer
programming models for this problem, and show that one of them has a tighter
continuous relaxation and allows the representation of more general customer
demand behaviour. We characterize the computational complexity for this
problem, and analyze which problem characteristics result in NP-hardness. We
then propose an exact branch-and-Benders-cut method, and show how optimality
cuts can be computed efficiently through an analytical procedure. Computational
experiments show that our method is approximately 30 times faster than solving
the tighter formulation directly. Our results also quantify the benefit of
accounting for cumulative customer demand within the optimization framework,
since the corresponding planning solutions perform much better than those
obtained by ignoring cumulative demand or employing myopic heuristics.
