  We analyze the asymptotic properties of a Euclidean optimization problem on
the plane. Specifically, we consider a network with three bins and $n$ objects
spatially uniformly distributed, each object being allocated to a bin at a cost
depending on its position. Two allocations are considered: the allocation
minimizing the bin loads and the allocation allocating each object to its less
costly bin. We analyze the asymptotic properties of these allocations as the
number of objects grows to infinity. Using the symmetries of the problem, we
derive a law of large numbers, a central limit theorem and a large deviation
principle for both loads with explicit expressions. In particular, we prove
that the two allocations satisfy the same law of large numbers, but they do not
have the same asymptotic fluctuations and rate functions.
