We study a version of the capacity expansion problem (CEP) that arises in telecommunication network design. Given a capacitated network and a traffic demand matrix, the objective in the CEP is to add capacity to the edges, in batches of various modularities, and route traffic, so that the overall cost is minimized. We study the polyhedral structure of a mixed-integer formulation of the problem and develop a cutting-plane algorithm using facet defining inequalities. The algorithm produces an extended formulation providing both a very good lower bound and a starting point for branch and bound. The overall algorithm appears effective when applied to problem instances using real-life data.

We present a branch-and-cut algorithm to solve capacitated network design problems. Given a capacitated network and point-to-point traffic demands, the objective is to install more capacity on the edges of the network and route traffic simultaneously, so that the overall cost is minimized. We study a mixed-integer programming formulation of the problem and identify some new facet defining inequalities. These inequalities, together with other known combinatorial and mixed-integer rounding inequalities, are used as cutting planes. To choose the branching variable, we use a new rule called "knapsack branching". We also report on our computational experience using real-life data.

We present several classes of facet-defining inequalities to strengthen polyhedra arising as subsystems of network design problems with survivability constraints. These problems typically involve assigning capacities to a network with multicommodity demands, such that after a vertex- or edge-deletion at least some prescribed fraction of each demand can be routed.

This paper provides an analysis of capacitated network design cut--set polyhedra. We give a complete linear description of the cut--set polyhedron of the single commodity -- single facility capacitated network design problem. Then we extend the analysis to single commodity -- multifacility and multicommodity -- multifacility capacitated network design problems. The valid inequalities described here have coefficients for both inflow and outflow arcs of a cut--set and are applicable to network design problems with an arbitrary number of facility types and arbitrary capacities. We report a computational study to test the effectiveness of the new inequalities. 1 Introduction Given a network and a set of demands on the vertices of the network, the capacitated network design problem is to install integer multiples of capacities on the arcs of the network and route the flow so that the total capacity installation and flow routing costs are minimized. For instance, installing or leasing fiber...

We study the mixed--integer knapsack polyhedron, that is, the convex hull of the mixed--integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet--defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable. These inequalities strengthen and/or generalize known inequalities for several special cases. We report computational results on using the inequalities as cutting planes for mixed--integer programming.

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We describe an upper-bound algorithm for multicommodity network design problems that relies on new results for approximately solving certain linear programs, and on the greedy heuristic for set-covering problems. 1 Introduction. Network design problems are mixed-integer programs that have the following broad structure. Given a graph, and a set of "demands" -- positive amounts to be routed between pairs of vertices -- capacity must be added to the edges and/or vertices of the graph, in discrete amounts, and at minimum cost, so that a feasible routing is possible. Problem of this form are increasingly important in telecommunications applications, because of the great expense inherent in maintaining and upgrading metropolitan networks. A wide variety of special cases have been studied. For example, one may be constrained to using a fixed family of paths to carry out the routing, or to using a single path for each demand, or to using integral flows. The precise manner in which capacit...

We consider the design and capacity expansion of ATM networks as an optimization problem in which flows representing end-to-end variable bit-rate services of different classes are to be multiplexed and routed over ATM trunks and switches so as to minimize the costs of additional switches and transport pipes while meeting service quality and survivability constraints. After discussing the underlying fractional Brownian motion models for flows, a non-linear multicommodity optimization problem is formulated and heuristics for its approximate solutions are described. Finally, computational results are produced that demonstrate realistic size problems can be solved with the proposed method to shed light on key economic characteristics of ATM traffic, such as safe levels of statistical multiplexing, as well as robust and efficient design alternatives.

We present a scenario approach to capacity planning in semiconductor manufacturing under demand uncertainty. We formulate an integer programming model in which we minimize the expected value of the unmet demand, subject to capacity and budget constraints, to arrive at a tool set that does well across all of the scenarios. This is a dicult two-stage stochastic integer program that is tackled with a heuristic approach. Analysis of the results in some real-life situations are presented. 1

We study 0-1 reformulations of the multicommodity capacitated network design problem, which is usually modeled with general integer variables to represent, design decisions on the number of facilities to install on each arc of the network. The reformulations are based on the multiple choice model, a generic approach to represent piecewise linear costs using 0-1 variables. This model is improved by the addition of extended linking inequalities, derived from variable disaggregation techniques. We show that these extended linking inequalities for the 0-1 model are equivalent to the residual capacity inequalities, a class of valid inequalities derived for the model with general integer variables. In this paper, we compare two cutting-plane algorithms to compute the same lower bound on the optimal value of the problem: one based on the generation of residual capacity inequalities within the model with general integer variables, and another based on the addition of extended linking inequalities to the 0-1 reformulation. To further improve the computational results of the latter approach, we develop a column-and-row generation approach; the resulting algorithm is shown to be competitive with the approach relying on residual capacity inequalities.