OR-239-91 - MODELING AND SOLVING THE CAPACITATED NETWORK...

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Unformatted text preview: MODELING AND SOLVING THE CAPACITATED NETWORK LOADING PROBLEM Thomas. L. Magnanti, Prakash Mirchandani, and Rita Vachani OR 239-91 January 1991 MODELING AND SOLVING THE CAPACITATED NETWORK LOADING PROBLEM Thomas L. Magnanti Sloan School of Management, MIT, Cambridge, MA 02139 Prakash Mirchandani Katz Graduate School of Business, University of Pittsburgh, Pittsburgh, PA 15260 Rita Vachani GTE Laboratories, Waltham, MA 02254 January 1991 1_1____1 1 II II ___ ______I _ II II__11 LILLII_~1 ~ -------- I-- _r I _ Abstract This paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry. In this problem, given point-to- point demand between various pairs of nodes of a network must be met by installing (loading) capacitated facilities on the arcs. The facilities are chosen from a small set of alternatives and loading a particular facility incurs an arc specific and facility dependent cost. The problem is to determine the configuration of facilities to be loaded on the arcs of the network that will satisfy the given demand at minimum cost. Since we need to install (load) facilities to carry the required traffic, we refer to the problem as the network loading problem. In this paper, we develop modeling and solution approaches for the problem. We consider two approaches for solving the underlying mixed integer programming model: (i) a Lagrangian relaxation strategy, and (ii) a cutting plane approach that uses three classes of valid inequalities that we identify for the problem. In particular, we show that a linear programming formulation that includes the valid inequalities always approximates the value of the mixed integer program at least as well as the Lagrangian relaxation bound (as measured by the gaps in the objective functions). We also examine the computational effectiveness of these inequalities on a set of prototypical telecommunications data. The computational results show that the addition of these inequalities considerably improves the gap between the integer programming formulation of the problem and its linear programming relaxation: for 6 - 15 node problems from an average of 25% to an average of 8%. These results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry. I_ __LII _____ I ____ _1__1 _ _I _ L_ 1·_1_·__11____1_1___ 111-··--·-·-^·-pll._. .. . II..-I-r- Y- ·l-·l---*-ll--^YI_-nI___IIIIC In this paper, we study a problem that is becoming increasingly important in the telecommunications industry: given an organization's forecast for data and voice traffic between its various locations, what configuration of transmission facilities between the locations (nodes) will provide the necessary link capacities to carry this traffic at minimum cost? A similar problem arises in the context of transportation planning; in this setting, the traffic corresponds to freight and the transmission facilities to different types...
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OR-239-91 - MODELING AND SOLVING THE CAPACITATED NETWORK...

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