Locating a facility outside the transportation network. Localization results (Q2708101)

The paper considers the problem of locating a semiobnoxious facility in the plane which provides service to a set of points being vertices of a network embedded in the plane. Since negative effects of the new facility like pollution, noise, etc. are not respecting the transportation network it might be useful to locate the new facility even outside the network. Therefore the objective is to locate a new facility \(F\) in a feasible subset of the plane and simultaneously a point \(T\) on the network linking the new facility and the existing transportation network. The objective function consists of three main building blocks. First, the sum of distances of costumers to \(T\) receiving service from \(F\) has to be taken into account. Secondly, we look at the distance between \(F\) and \(T\), which might be measured by any norm. This distance is multiplied by a nonnegative weight \(K\) representing the building and transportation costs for the link between \(F\) and \(T\). In the last block the (planar) distance between \(F\) and each existing facility which is negatively effected by \(F\) is used as argument of a non-increasing convex function, representing the negative effect. The effects are summed up for all existing facilities. Of course the objective function should be minimized.NEWLINENEWLINENEWLINEAfter having introduced this model the authors state localization results for this general model, where they mainly focus on the issue of finiteness and existence. More specific results are obtained for the case where the negative effect is measured by a linear function of the distance between \(F\) and \(T\). For the low repulsive case (\(K\) is not smaller than the cumulative negative effect of the new facility) localization results can be obtained using the concept of visibility. The optimal location of the new facility (in the finite case) is either on the boundary of the feasible region of the plane or on a node of the transportation network. Similar results can be obtained when looking at the high repulsive case.NEWLINENEWLINENEWLINEThe paper is well structured and nicely written.