Discrete Optimization Enhanced formulations and branch-and-cut for the two level network design problem with transition facilities Stefan Gollowitzer a , Luis Gouveia b,1 , Ivana Ljubic ´ a,⇑,2 a Department of Statistics and Operations Research, University of Vienna, Brünnerstraße 72, A-1210 Vienna, Austria b DEIO/CIO, Faculdade de Ciências da Universidade de Lisboa, Bloco C2, Campo Grande, 1749-016 Lisboa, Portugal article info Article history: Received 22 November 2011 Accepted 23 September 2012 Available online 2 October 2012 Keywords: OR in telecommunications Integer programming Linear programming relaxations Hierarchical network design Tree–tree networks Network design and facility location abstract We consider a new combinatorial optimization problem that combines network design and facility loca- tion aspects. Given a graph with two types of customers and two technologies that can be installed on the edges, the objective is to find a minimum cost subtree connecting all customers while the primary cus- tomers are served by a primary subtree that is embedded into the secondary subtree. In addition, besides fixed link installation costs, facility opening costs, associated to each node where primary and secondary subtree connect, have to be paid. The problem is called the Two Level Network Design Problem with Transition Facilities (TLNDF). We first model the problem on an extended graph where an additional set of arcs corresponds to the installation of node facilities and propose a cut set based model for the TLNDF that is defined on this extended graph. We present several theoretical results relating families of cut set inequalities on the extended graph with subfamilies of cut set inequalities on the original graph. We then show how a stan- dard multi-commodity flow model defined on the original graph can be strengthened using disaggrega- tion ‘‘by technology’’. We prove that the disaggregated compact formulation on the original graph provides the same lower bound as the cut set formulation on the extended graph. We develop a branch-and-cut algorithm for solving the TLNDF. The performance of this algorithm is improved by separating subfamilies of cut set inequalities on the original graph. Our computational study confirms the efficiency and applicability of the new approach. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction The Multi-Level Network Design Problem (MLND) has been orig- inally defined by Balakrishnan et al. [1]: We are given an undi- rected graph with a set of nodes partitioned into L levels and a set of edges such that along each edge one of L different technolo- gies can be installed, with higher grade technologies inducing higher fixed costs. The goal is to find a (spanning) subtree and decide which technology to install along each edge, so that all cus- tomers at level ‘ can communicate with each other along a path using technology of grade ‘ or higher. We extend the definition of the MLND by introducing the fixed costs for transition nodes, i.e., the nodes where a change of technology takes place, and by considering so-called potential Steiner nodes which are nodes that need not be included in the solution. In this problem, which we denote by Multi-Level Network Design Problem with Transition Facil- ities, the overall goal is to find a MLND subtree that minimizes the sum of fixed edge and facility installation costs. In this paper we study the case with L = 2, which we will denote by the Two Level Network Design Problem with Transition Facilities (TLNDF). TLNDF arises in the topological design of hierarchical communi- cation, transportation, and electric power distribution networks. One of the most important applications of TLNDF is in the context of telecommunication networks, where networks with two cable technologies, fiber optic and copper, are built. Telecommunication companies distinguish between primary and secondary customers. The switching centers, important infrastructure nodes and small businesses are considered as primary customers (i.e., those to be served by fiber optic connections). Single households are not con- sidered as being consumers of a high potential and hence they only need to be supplied using copper cables. The secondary technology is much cheaper, but the guaranteed quality of the connections and bandwidth is significantly below the quality provided by the primary technology. The goal is to build a network (with tree topology) such that there is a fiber optic connection between each primary customer and a designated root node (e.g., a central office), and each secondary customer is connected to the root along 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.09.040 ⇑ Corresponding author. Tel.: +43 1 4277 38661; fax: +43 1 4277 38699. E-mail addresses: stefan.gollowitzer@univie.ac.at (S. Gollowitzer), legouveia@fc. ul.pt (L. Gouveia), ivana.ljubic@univie.ac.at (I. Ljubic ´). 1 Supported by National Funding from FCT – Fundação para a Ciência e Tecnologia, under the project: PEst-OE/MAT/UI0152. 2 Supported by the APART Fellowship of the Austrian Academy of Sciences (OEAW). European Journal of Operational Research 225 (2013) 211–222 Contents lists available at SciVerse ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor