Abstract
In this paper a Branch-and-Bound (BB) algorithm is developed to obtain an optimal solution to the single source uncapacitated minimum cost Network Flow Problem (NFP) with general concave costs. Concave NFPs are NP-Hard, even for the simplest version therefore, there is a scarcity of exact methods to address them in their full generality. The BB algorithm presented here can be used to solve optimally single source uncapacitated minimum cost NFPs with any kind of concave arc costs. The bounding is based on the computation of lower bounds derived from state space relaxations of a dynamic programming formulation. The relaxations, which are the subject of the paper (Fontes et al., 2005b) and also briefly discussed here, involve the use of non-injective mapping functions, which guarantee a reduction on the cardinality of the state space. Branching is performed by either fixing an arc as part of the final solution or by removing it from the final solution. Computational results are reported and compared to available alternative methods for addressing the same type of problems. It could be concluded that our BB algorithm has better performance and the results have also shown evidence that it has a sub-exponential time growth.
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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 A. Mingozzi P. Toth (1981) ArticleTitleState space relaxation procedures for the computation of bounds to routing problems Networks. 11 145–164
C. Cordier H. Marchand R. Laundy L.A. Wolsey (1999) ArticleTitlebc-opt: A branch and cut code for mixed integer programs Mathematical Programming. 86 335–353 Occurrence Handle10.1007/s101070050092
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. and Christofides, N. (2005a), A dynamic programming approach for solving single-source uncapacitated concave minimum cost network flow problems, European Journal of Operational Research, in Press.
Fontes D.B.M.M., Hadjiconstantinou, E. and Christofides, N. (2005b), Lower bounds from state space relaxations for concave network flow problems. This Journal.
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
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
D. Kim P.M. Pardalos (2000a) ArticleTitleA dynamic domain contraction algorithm for nonconvex piecewise linear network flow problems Journal of Global Optimization. 17 225–234 Occurrence Handle10.1023/A:1026502220076
D. Kim P.M. Pardalos (2000b) ArticleTitleDynamic slope scaling and trust interval techniques for solving concave piecewise linear network flow problems Networks. 35 216–222 Occurrence Handle10.1002/(SICI)1097-0037(200005)35:3<216::AID-NET5>3.0.CO;2-E
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 non-linear 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. A Branch-and-Bound Algorithm for Concave Network Flow Problems. J Glob Optim 34, 127–155 (2006). https://doi.org/10.1007/s10898-005-1658-x
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DOI: https://doi.org/10.1007/s10898-005-1658-x