Hop Constrained Connected Facility Location

The Connected Facility Location (ConFL) problem combines facility location and Steiner trees: given a set of customers, a set of potential facility locations and some inter-connection nodes, ConFL searches for the minimum-cost way of assigning each customer to exactly one open facility, and connecting the open facilities via a Steiner tree. The costs needed for building the Steiner tree, facility opening costs and the assignment costs need to be minimized. In some telecommunications applications, the number of nodes between the root and open facilities is limited. This leads to the Hop Constrained Facility Location Problem. This is a new problem not studied in the literature yet. We develop 17 mixed integer programming models for this problem. Some of these models are extensions from corresponding models for the ConFL, some extend ideas for related problems like the Minimum Spanning or Steiner Tree problem with hop constraints. In branch-and-bound frameworks the quality of linear programming lower bounds of these formulations is of particular interest. We compare the relative quality of these relaxations and provide a hierarchy of the corresponding models. We also show how the Hop Constrained ConFL can be modelled as ConFL or the Steiner Tree problem on three different layered graphs. Finally we prove, that a disaggregation of variables indicating facilities or customers in the layered graphs does not improve the quality of the lower bounds.