Location Problems with Cutoff

In this paper, we study a generalized version of the Weber problem of finding a point that minimizes the sum of its distances to a finite number of given points. In our setting, these distances may be cut off at a given value C > 0, and we allow for the option of an empty solution at a fixed cost C′. We analyze under which circumstances these problems can be reduced to the simpler Weber problem, and also when we definitely have to solve the more complex problem with cutoff. We furthermore present adaptions of the algorithm of Drezner, Mehrez and Wesolowsky (1991 [The facility location problem with limited distances. Transportation Science, 25(3), 183-187, INFORMS]) to our setting, which in certain situations are able to substantially reduce computation times as demonstrated in a simulation study. The sensitivity with respect to the cutoff value is also studied, which allows us to provide an algorithm that efficiently solves the problem simultaneously for all C > 0.