A flexible approach to location problems (Q1974022)

The authors consider a very general location problem in the plane. Each of the \(M\) given existing facilities \(a_1, \dots,a_M\) measures its distance to other points in the plane by a gauge, and has associated with it two scalars \(w_i\) and \(\lambda_i\), \(i=1, \dots,M\). The objective function to be minimized is \(F(x)= \sum^M_{i=1} \lambda_i\gamma (x-A)_{(i)}\), where \(\gamma(x-A)_{(i)}\) is the \(i\)-th smallest of the values \(w_{\sigma_i} \gamma(x-a_{ \sigma_i})\) with respect to a permutation \(\sigma\). The resulting problem is called ordered Weber problem. It is a framework including the well-known median, center and centdian problems as special cases, but also allows enhanced modeling of other location problems. The authors develop efficient solution algorithms and investigate the structure of the problem. Further generalizations including multiple facilities and restrictions of the regions in which new facilities can be sited are considered as well.