Abstract
In this paper we obtain Lower Bounds (LBs) to concave cost network flow problems. The LBs are derived from state space relaxations of a dynamic programming formulation, which involve the use of non-injective mapping functions guaranteing a reduction on the cardinality of the state space. The general state space relaxation procedure is extended to address problems involving transitions that go across several stages, as is the case of network flow problems. Applications for these LBs include: estimation of the quality of heuristic solutions; local search methods that use information of the LB solution structure to find initial solutions to restart the search (Fontes et al., 2003, Networks, 41, 221–228); and branch-and-bound (BB) methods having as a bounding procedure a modified version of the LB algorithm developed here, (see Fontes et al., 2005a). These LBs are iteratively improved by penalizing, in a Lagrangian fashion, customers not exactly satisfied or by performing state space modifications. Both the penalties and the state space are updated by using the subgradient method. Additional constraints are developed to improve further the LBs by reducing the searchable space. The computational results provided show that very good bounds can be obtained for concave cost network flow problems, particularly for fixed-charge problems.
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References
Beasley, J.E. (OR-L), ‘Or-Library’, http://www.brunel.ac.uk/depts/ma/research/jeb/ info.html.
R.E. Burkard H. Dollani P.H. Thach (2001) ArticleTitleLinear approximations in a dynamic programming approach for the uncapacitated single-source minimum concave cost network flow problem in acyclic networks Journal of Global Optimization. 19 121–139 Occurrence Handle10.1023/A:1008379621400
N. Christofides E. Hadjiconstantinou (1995) ArticleTitleAn exact algorithm for orthogonal 2-D cutting problems using guillotine cuts European Journal of Operational Research. 83 21–38 Occurrence Handle10.1016/0377-2217(93)E0277-5
N. Christofides A. Mingozzi P. Toth (1981) ArticleTitleState space relaxation procedures for the computation of bounds to routing problems Networks. 11 145–164
N. Christofides J. Paixão (1993) ArticleTitleAlgorithms for large scale set covering problems Annals of Operations Research. 43 261–277 Occurrence Handle10.1007/BF02025297
D.B.M.M. Fontes (2000) Optimal Network Design Using Nonlinear Cost Flows The Management School, Imperial College of Science Technology and Medicine London, U.K
D.B.M.M. Fontes E. Hadjiconstantinou N. Christofides (2003) ArticleTitleUpper bounds for single source uncapacitated minimum concave-cost network flow problems Networks. 41 221–228 Occurrence Handle10.1002/net.10076
Fontes D.B.M.M., Hadjiconstantinou E., Christofides N. (2005a). A branch-and-bound for the uncapacitated single source minimum concave cost network flow problem. This Journal.
Fontes D.B.M.M., Hadjiconstantinou E., Christofides N. (2005b). A dynamic programming approach for solving single-source uncapacitated concave minimum cost network flow problems. European Journal of Operational Research. in Press
G. Gallo C. Sandi C. Sodini (1980) ArticleTitleAn algorithm for the min concave cost flow problem European Journal of Operational Research. 4 249–255 Occurrence Handle10.1016/0377-2217(80)90109-5
G.M. Guisewite (1994) Network problems R. Horst P.M. Pardalos (Eds) Handbook of Global Optimization. Kluwer Academic Dordrecht 609–648
G.M. Guisewite P.M. Pardalos (1991a) ArticleTitleAlgorithms for the single-source uncapacitated minimum concave-cost network flow problem’ Journal of Global Optimization. 3 245–265
G.M. Guisewite P.M. Pardalos (1991b) ArticleTitleGlobal search algorithms for minimum concave-cost network flow problems Journal of Global Optimization. 1 309–330
E. Hadjiconstantinou N. Christofides A. Mingozzi (1995) ArticleTitleA new exact algorithm for the vehicle routing problem based on q-paths and k-shortest paths relaxations Annals of Operations Research. 61 21–43 Occurrence Handle10.1007/BF02098280
M. Held R.M. Karp H.P. Crowder (1974) ArticleTitleValidation of subgradient optimization Mathematical Programming. 6 62–88 Occurrence Handle10.1007/BF01580223
D.S. Hochbaum A. Segev (1989) ArticleTitleAnalysis of a flow problem with fixed charges Networks. 19 291–312
R. Horst N.V. Thoai (1998) ArticleTitleAn integer concave minimization approach for the minimum concave cost capacitated flow problem on networks OR Spectrum. 20 45–53 Occurrence Handle10.1007/s002910050051
D. Kim P.M. Pardalos (1999) ArticleTitleA solution approach to the fixed charge network flow problem using a dynamic slope scaling procedure Operations Research Letters. 24 195–203 Occurrence Handle10.1016/S0167-6377(99)00004-8
H.-J. Kim J. Hooker (2002) ArticleTitleSolving fixed-charge network flow problems with a hybrid optimization and constraint programming approach Annals of Operations Research. 115 95–124 Occurrence Handle10.1023/A:1021145103592
B.W. Lamar (1993) A method for solving network flow problems with general nonlinear arc costs D.-Z. Du P.M. Pardalos (Eds) Network Optimization Problems. World Scientific Singapore
F. Ortega L.A. Wolsey (2003) ArticleTitleA branch-and-cut algorithm for the single-commodity, uncapacitated, fixed-charge network flow problem Networks. 41 143–158 Occurrence Handle10.1002/net.10068
W.I. Zangwill (1968) ArticleTitleMinimum concave cost flows in certain networks Management Science. 14 429–450 Occurrence Handle10.1287/mnsc.14.7.429
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Fontes, D.B.M.M., Hadjiconstantinou, E. & Christofides, N. Lower Bounds from State Space Relaxations for Concave Cost Network Flow Problems. J Glob Optim 34, 97–125 (2006). https://doi.org/10.1007/s10898-005-1657-y
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DOI: https://doi.org/10.1007/s10898-005-1657-y