  We take the classic facility location problem and consider a variation, in
which each agent's individual cost function is equal to their distance from the
facility multiplied by a scaling factor which is determined by the facility
placement. In addition to the general class of continuous scaling functions, we
also provide results for piecewise linear scaling functions which can
effectively approximate or model the scaling of many real world scenarios. We
focus on the objectives of total and maximum cost, describing the computation
of the optimal solution. We then move to the approximate mechanism design
setting, observing that the agents' preferences may no longer be single-peaked.
Consequently, we characterize the conditions on scaling functions which ensure
that agents have single-peaked preferences. Under these conditions, we find
results on the total and maximum cost approximation ratios achievable by
strategyproof and anonymous mechanisms.
