  This paper deals with a bilevel approach of the location-allocation problem
with dimensional facilities. We present a general model that allows us to
consider very general shapes of domains for the dimensional facilities and we
prove the existence of optimal solutions under mild, natural assumptions. To
achieve these results we borrow tools from optimal transport mass theory that
allow us to give explicit solution structure of the considered lower level
problem. We also provide a discretization approach that can approximate, up to
any degree of accuracy, the optimal solution of the original problem. This
discrete approximation can be optimally solved via a mixed-integer linear
program. To address very large instance sizes we also provide a GRASP heuristic
that performs rather well according to our experimental results. The paper also
reports some experiments run on test data.
