Discrete Optimization Combining very large scale and ILP based neighborhoods for a two-level location problem Bernardetta Addis b,⇑ , Giuliana Carello a , Alberto Ceselli c a Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy b Dipartimento di Informatica, Università degli Studi di Torino, C.So Svizzera 185, 10149 Torino, Italy c Dipartimento di Informatica, Polo didattico e di ricerca di Crema, Università degli Studi di Milano, via Bramante 65, 26013 Crema, Italy article info Article history: Received 1 August 2012 Accepted 8 June 2013 Available online 18 June 2013 Keywords: Location Local search Variable neighborhood search Very large scale neighborhood search Matheuristics abstract In this paper we tackle a generalization of the Single Source Capacitated Facility Location Problem in which two sets of facilities, called intermediate level and upper level facilities, have to be located; the dimensioning of the intermediate set, the assignment of clients to intermediate level facilities, and of intermediate level facilities to upper level facilities, must be optimized, as well. Such problem arises, for instance, in telecommunication network design: in fact, in hierarchical networks the traffic arising at client nodes often have to be routed through different kinds of facility nodes, which provide different services. We propose a heuristic approach, based on very large scale neighborhood search to tackle the problem, in which both ad hoc algorithms and general purpose solvers are applied to explore the search space. We report on experimental results using datasets from the capacitated location literature. Such results show that the approach is promising and that Integer Linear Programming based neighborhoods are significantly effective. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction In this paper we consider a generalization of the classical Single Source Capacitated Facility Location Problem in which a set of cli- ents and two different sets of facilities are considered, intermedi- ate level and upper level facilities. The location of each facility must be chosen among a set of candidate sites, requiring different setup costs. Each client must be assigned to exactly one intermedi- ate level facility, and each intermediate level facility must be as- signed to exactly one upper level facility. The network is assumed to have a star–star topology, and therefore the assign- ment costs are proportional to the distance between clients, inter- mediate level and upper level facilities. A demand amount to be served for each client, and a capacity for each facility, are given: the amount of demand assigned to each facility cannot exceed its capacity. Moreover, each intermediate level facility must be dimensioned, i.e. it must be equipped with one device, chosen among a set of possible types. Different devices provide different capacities at different costs. We denote this problem as Two Level Capacitated Facility Location Problem (TLCFLP). The TLCFLP belongs to the family of multi-level facility location problems, which may arise in several fields, such as telecommuni- cation or transportation. They may take into account different features and have been widely addressed in the literature. A recent review on multi-level hierarchical facility location problems, cov- ering papers since the mid-80s, can be found in Sahin and Süral (2007). In such survey, hierarchical facility location problems are classified according to features such as flow pattern and service availability; applications, models and approaches are described. In a classical generalization of the facility location problem, the so-called Multi-level Facility Location Problem, a set of clients is gi- ven together with k sets of facilities, where each set represents a different facility level. Each client must be assigned to a path of k facilities, and its demand must be routed through a facility of each level following a hierarchical order. Star–star topology which is ad- dressed in this paper is not considered. For the uncapacitated ver- sion of the problem, heuristic and exact approaches (Tcha & Lee, 1984) as well as approximation properties (Aardal, Chudak, & Shm- oys, 1999) have been investigated. Another similar generalization of the facility location problem is the Two-level Simple Plant Loca- tion Problem addressed in Chardaire, Lutton, and Sutter (1999): each client must be assigned to one and only one facility of the intermediate level which, in turns, must be assigned to one and only one facility of the high level. In this case star–star topology is considered, but facilities of both levels are uncapacitated. The problem we considered shares some features also with the Two-echelon Single Source Capacitated Facility Location Problem described in Tragantalerngsak, Holt, and Ronnqvist (2000), where each client must be assigned to exactly one intermediate level 0377-2217/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2013.06.010 ⇑ Corresponding author. Tel.: +39 011 670 6813; fax: +39 011 751603. E-mail address: bernardetta.addis@gmail.com (B. Addis). European Journal of Operational Research 231 (2013) 535–546 Contents lists available at SciVerse ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor