Location problems with cutoff

TL;DR

This paper extends the Weber problem by incorporating distance cutoffs and an option for empty solutions, analyzing when these modifications simplify the problem and adapting algorithms for efficient computation.

Contribution

It introduces a generalized Weber problem with cutoff distances and empty solution options, providing reduction criteria and an adapted algorithm for faster solutions.

Findings

Reduction to classical Weber problem under certain conditions

Algorithm adaptations significantly reduce computation times

Efficient solution method for all cutoff values C > 0

Abstract

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 et al., 1991, $Transportation$ $Science$ 25(3), 183--187] 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 $C>0$.