Geometrical properties of the symmetrical single facility location problem (Q2708295)

In this paper a general single facility location problem in a real normed space \(X\) with a finite set of existing facilities \(A=\{a_1, \ldots, a_n\}\) is presented. For \(x \in X\), let \(d(x)\) be the \(n\)-dimensional vector of distances from \(x\) to each \(a_i\) measured by the norm in \(X\). The objective function to be minimized is given by \(F(x) := g(d(x))\), where \(g\) is assumed to be monotone and symmetric. The latter property means that the value of \(g(d(x))\) must not depend on the order in which the elements of \(d(x)\) are given. The set of optimal solutions to this problem is characterized. Also bounds are proved and localization results are given. Moreover, the relationship to the ordered Weber (or median) location problem is established. The ordered Weber problem contains most of the classical location problems as special cases and the authors prove that the symmetrical single facility location problem can be seen as a special type of the ordered Weber problem (where the components of the \(n\)-dimensional vector responsible for choosing a specific objective function are given in non-decreasing order).