Heuristic algorithms for general k‐level facility location problems

In a general k‐level uncapacitated facility location problem (k‐GLUFLP), we are given a set of demand points, denoted by D, where clients are located. Facilities have to be located at a given set of potential sites, which is denoted by F in order to serve the clients. Each client needs to be served by a chain of k different facilities. The problem is to determine some sites of F to be set up and to find an assignment of each client to a chain of k facilities so that the sum of the setup costs and the shipping costs is minimized. In this paper, for a fixed k, an approximation algorithm within a factor of 3 of the optimum cost is presented for k‐GLUFLP under the assumption that the shipping costs satisfy the properties of metric space. In addition, when no fixed cost is charged for setting up the facilities and k=2, we show that the problem is strong NP‐complete and the constant approximation factor is further sharpen to be 3/2 by a simple algorithm. Furthermore, it is shown that this ratio analysis is tight.