A Lagrangian Decomposition Based Heuristic for Capacitated Connected Facility Location

We consider a generalized version of the rooted Connected Facility Location Problem (ConFL) with capacities and prizes on clients as well as capacity constraints on potential facilities. Furthermore, we are interested in selecting and connecting the most profitable client subset (i.e. a prize collecting variant) instead of mandatorily connecting all clients. Connected Facility Location Problems occur for instance when increasing the bandwidth of existing networks to meet growing bandwidth requirements of customers [12, 13]. In such scenarios new routes are installed between some source and so called facilities acting as mediation points between the so far existing and the newly installed network. Each facility is able to meet the demands of several assigned customers up to some maximum available capacity (capacity constraints). Furthermore, next to costs for installing routes to facilities, facility installation costs (opening costs) as well as costs for assigning a customer to a facility (assignment costs) might occur. On the other hand for each supplied customer a return of invest (customer prize) is obtained. Formally, we are given an undirected graph G = (V,E) with a dedicated root node 0 ∈ V and edge costs ce ≥ 0, ∀e = (u, v) ∈ E, corresponding to the costs of installing a new route between u and v. Furthermore, we are given a set of potential facility locations F ⊆ V with associated opening costs fi ≥ 0 and maximum assignable demands Di ∈ N0, ∀i ∈ F , as well as clients C with individual demands dk ∈ N0 and prizes pk ≥ 0, ∀k ∈ C, (i.e. the expected return on invest). Finally we are given costs aik ≥ 0, ∀i ∈ F, ∀k ∈ C for assigning the complete demand of client k to facility i. If a client k may not be assigned to a potential facility i we assume aik = ∞. A solution to this Capacitated Connected Facility Location Problem (CConFL) S = (R, T, F ′, C ′, α) consists of a Steiner tree (R, T ) with R ⊆ V , T ⊆ E connecting the set of opened facilities F ′ ⊆ F