Skip to main content
SpringerLink
Log in

Solving Fixed-Charge Network Flow Problems with a Hybrid Optimization and Constraint Programming Approach

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

We apply to fixed charge network flow (FCNF) problems a general hybrid solution method that combines constraint programming and linear programming. FCNF problems test the hybrid approach on problems that are already rather well suited for a classical 0–1 model. They are solved by means of a global constraint that generates specialized constraint propagation algorithms and a projected relaxation that can be rapidly solved as a minimum cost network flow problem. The hybrid approach ran about twice as fast as a commercial mixed integer programming code on fixed charge transportation problems with its advantage increasing with problem size. For general fixed charge transshipment problems, however, it has no effect because the implemented propagation methods are weak.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. U. Akinc and B.M. Khumawala, An efficient branch and bound algorithm for the capacitated warehouse location problem, Management Science 23 (1977) 585-594.

    Google Scholar 

  2. A. Balakrishnan, Lp extreme points and cuts for the fixed-charge network design problem, Mathematical Programming 39 (1987) 263-284.

    Google Scholar 

  3. M.L. Balinkski, Fixed cost tranportation problems, Naval Research Logistics Quarterly 8(1) (1961) 41-54.

    Google Scholar 

  4. F. Glover, R.S. Barr and D. Klingman, A new optimization method for large scale fixed charge transportation problems, Operations Research 29(3) (1981) 448-463.

    Google Scholar 

  5. W.J. Baumol and P.Wolfe, A warehouse-location problem, Operations Research 6(2) (1958) 252-263.

    Google Scholar 

  6. A.V. Cabot and S.S. Erenguc, Some branch-and-bound procedures for fixed-cost transportation problems, Naval Research Logistics Quarterly 31 (1984) 145-154.

    Google Scholar 

  7. L. Cooper and C. Drebes, An approximate solution method for the fixed-charge problem, Naval Research Logistics Quarterly 14(1) (1968) 101-113.

    Google Scholar 

  8. G. Cornuejols and J.M. Thizy, Some facets of the simple plant location polytope, Mathematical Programming 23 (1982) 50-74.

    Google Scholar 

  9. G.L. Nemhauser, G. Cornuejols and L.A.Wolsey, Worst-case and probabilistic analysis of algorithms for a location problem, Operations Research 28 (1980) 847-858.

    Google Scholar 

  10. M.L. Fisher, G. Cornuejols and G.L. Nemhauser, Location of bank accounts to optimize floats: An analytic study of exact and approximate algorithms, Management Science 23 (1977) 789-810.

    Google Scholar 

  11. M.L. Fisher, G. Cornuejols and G.L. Nemhauser, On the uncapacitated plant location problem, Annals of Discrete Mathematics 1 (1977) 163-177.

    Google Scholar 

  12. G. Daeninck and Y. Smeers, Using shortest paths in some transshipment problems with concave costs, Mathematical Programming 12 (1977) 18-25.

    Google Scholar 

  13. D.R. Denzler, An approximate algorithm for the fixed charge problems, Naval Research Logistics Quaterly 16(3) (1969) 411-416.

    Google Scholar 

  14. J. Gottlieb, Fctp instances, http://www.in.tu-clausthal.de/~gottlieb/benchmarks/fctp/ (1998).

  15. P. Gray, Exact solution of the fixed-charge transportation problem, Operations Research 19(6) (1971) 1529-1538.

    Google Scholar 

  16. M. Guignard, Fractional vertices, cuts and facets of the simple plant location problem, Mathematical Programming 12 (1980) 150-162.

    Google Scholar 

  17. D.S. Hochbaum and A. Segev, Analysis of a flow problem with fixed charges, Networks 19 (1989) 291-312.

    Google Scholar 

  18. J.N. Hooker, Logic-based methods for optimization, in: Principles and Practice of Constraint Programming, Lecture Notes in Computer Science, Vol. 874 (1994) pp. 336-349.

  19. J.N. Hooker, Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction (Wiley, New York, 2000).

    Google Scholar 

  20. J.N. Hooker, Logic, optimization and constraint programming, INFORMS Journal on Computing, to appear.

  21. J.N. Hooker, H.-J. Kim and G. Ottosson, A declarative modeling framework that integrates solution methods, Annals of Operations Research 104 (2001) 141-161.

    Google Scholar 

  22. J.N. Hooker and M.A. Osorio, Mixed logical/linear programming, Discrete Applied Mathematics 96-97 (1999) 395-442.

    Google Scholar 

  23. J.N. Hooker, G. Ottosson, E. Thorsteinsson and H.-J. Kim, On integrating constraint propagation and linear programming for combinatorial optimization, in: Proceedings of the 16th National Conference on Artificial Intelligence (AAAI-99) (AAAI/MIT Press, Cambridge, MA, 1999) pp. 136-141.

    Google Scholar 

  24. J.N. Hooker, G. Ottosson, E. Thorsteinsson and H.-J. Kim, A scheme for unifying optimization and constraint satisfaction methods, The Knowledge Engineering Review 15(1) (2000) 11-30.

    Google Scholar 

  25. V. Jain and I.E. Grossmann, Algorithms for hybrid milp/clp models for a class of optimization problems, INFORMS Journal on Computing 13(4) (2001) 258-276.

    Google Scholar 

  26. J.J. Jarvis, V.E. Unger, R.L. Rardin and R.W. Moore, Optimal design of regional wastewater systems: A fixed charge network flow model, Operations Research 26(4) (1978) 538-550.

    Google Scholar 

  27. J. Kennington and E. Unger, The group-theoretic structure in the fixed-charge transportation problem, Operations Research 21(5) (1973) 1142-1152.

    Google Scholar 

  28. J. Kennington and E. Unger, A new branch-and-bound algorithm for the fixed-charge transportation problem, Management Science 22(10) (1976) 1116-1126.

    Google Scholar 

  29. H.W. Kuhn and W.J. Baumol, An approximate algorithm for the fixed-charge transportation problem, Naval Research Logistics Quarterly 9(1) (1962) 1-16.

    Google Scholar 

  30. T.L. Magnanti, A. Balakrishnan and R.T.Wong, A dual ascent procedure for large-scale uncapacitated network design, Operations Research 37(5) (1989) 716-740.

    Google Scholar 

  31. T.L. Magnanti and R.T. Wong, Network design and transportation planning: Models and algorithms, Transportation Science 18 (1984) 1-55.

    Google Scholar 

  32. P. McKeown, A vertex ranking procedure for solving the linear fixed-charge problem, Operations Research 23(6) (1975) 1183-1191.

    Google Scholar 

  33. M. Minoux, Network synthesis and optimum network design problems: Models, solution methods and applications, Network 19 (1989) 313-360.

    Google Scholar 

  34. K.G. Murty, Solving the fixed charge problem by ranking the extreme points, Operations Research 16(2) (1968) 268-279.

    Google Scholar 

  35. A. Napier, D. Klingman and J. Stutz, Netgen: A program for generating large scale capacitated assignment, transportation, and minimum cost flow network problems, Management Science 20(5) (1974) 814-821.

    Google Scholar 

  36. U.L. Ogbu, S.C. Narula and H.M. Samuelsson, An algorithm for the p-median problem, Operations Research 25 (1977) 709-712.

    Google Scholar 

  37. G. Ottosson, E. Thorsteinsson and J.N. Hooker, Mixed global constraints and inference in hybrid clp-ip solvers, in: CP99 Post-Conference Workshop on Large Scale Combinatorial Optimization and Constraints (1999) pp. 57-78, http://www.dash.co.uk/wscp99.

  38. R.L. Rardin and U. Choi, Tighter relaxations of fixed charge network flow problems, Technical Report J-79-18, School of Industrial and System Engineering, Georgia Institute of Technology, Atlanta (May 1979).

    Google Scholar 

  39. T.J. Roy, M.W. Padberg and L.A. Wolsey, Valid linear inequalities for fixed charge problems, Technical Report, CORE Discussion Paper No. 8232, Center for Operations Research and Econometrics, Universite Catholique de Louvain (1982).

  40. B. Sanso and P. Soriano (eds.), Telecommunication Network Planning (Kluwer, Dordrecht, 1999).

    Google Scholar 

  41. E.S. Thorsteinsson, Branch-and-check: a hybrid framework integrating mixed integer programming and constraint logic programming, in: Seventh International Conference on Principles and Practice of Constraint Programming (CP2001) (2001).

  42. W.E. Walker, A heuristic adjacent extreme point algorithm for the fixed charge problem, Management Science 22(5) (1976) 587-596.

    Google Scholar 

  43. L.A. Wolsey, Integer Programming (Wiley, New York, 1998).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Business Administration, Yonsei University, Seoul, 120-749, Korea

    Hak-Jin Kim

  2. Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    John N. Hooker

Authors
  1. Hak-Jin Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. John N. Hooker
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kim, HJ., Hooker, J.N. Solving Fixed-Charge Network Flow Problems with a Hybrid Optimization and Constraint Programming Approach. Annals of Operations Research 115, 95–124 (2002). https://doi.org/10.1023/A:1021145103592

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021145103592

Keywords

Search

Navigation