Bektas, T., Crainic, T.G. and Gendron, B.
Lagrangean decomposition for the fixed charge multicommodity network design problem , Southampton, UK University of Southampton
(Discussion Papers in Centre for Operational Research, Management Science and Information Systems, CORMSIS-08-17).
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Traditional Lagrangean relaxations for the multicommodity capacitated network design problem (MCNDP) involve dualizing either arc capacity or flow conservation constraints. The former (shortest-path relaxation) results in loosing the capacity structure whereas the latter (knapsack relaxation) does not maintain any information related to the network structure. Furthermore, both relaxations yield bounds that are at best equal to the value of the LP relaxation. This paper describes a new relaxation for the MCNDP, based on Lagrangean decomposition, which allows one to decompose the problem by nodes, and the subproblems partially preserve both the network and the capacity structure. This is, to the best of the authors' knowledge, the first relaxation for the MCNDP that theoretically yields better bounds than the LP relaxation.
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