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The fixed charge transportation problem: a strong formulation based on Lagrangian decomposition and column generation

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Abstract

A new and strong convexified formulation of the fixed charge transportation problem is provided. This formulation is obtained by integrating the concepts of Lagrangian decomposition and column generation. The decomposition is made by splitting the shipping variables into supply and demand side copies, while the columns correspond to extreme flow patterns for single sources or sinks. It is shown that the combination of Lagrangian decomposition and column generation provides a formulation whose strength dominates those of three other convexified formulations of the problem. Numerical results illustrate that our integrated approach has the ability to provide strong lower bounds. The Lagrangian decomposition yields a dual problem with an unbounded set of optimal solutions. We propose a regularized column generation scheme which prioritizes an optimal dual solution with a small \(l_1\)-norm. We further demonstrate numerically that information gained from the strong formulation can also be used for constructing a small-sized core problem which yields high-quality upper bounds.

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References

  1. Aguado, J.S.: Fixed charge transportation problems: a new heuristic approach based on Lagrangean relaxation and the solving of core problems. Ann. Oper. Res. 172, 45–69 (2009)

    Article  MathSciNet  Google Scholar 

  2. Avella, P., Boccia, M., Sforza, A., Vasilev, I.: An effective heuristic for large-scale capacitated facility location problems. J. Heuristics 15(6), 597–615 (2008)

    Article  Google Scholar 

  3. Balas, E., Zemel, E.: An algorithm for large zero–one knapsack problems. Oper. Res. 28(5), 1130–1154 (1980)

    Article  MathSciNet  Google Scholar 

  4. Balinski, M.L.: Fixed-cost transportation problems. Nav. Res. Logist. Q. 8(1), 41–54 (1961)

    Article  Google Scholar 

  5. Barr, R.S., Glover, F., Klingman, D.: A new optimization method for large scale fixed charge transportation problems. Oper. Res. 29(3), 448–463 (1981)

    Article  MathSciNet  Google Scholar 

  6. Bell, G., Lamar, B., Wallace, C.: Capacity improvement, penalties, and the fixed charge transportation problem. Nav. Res. Logist. 46(4), 341–355 (1999)

    Article  MathSciNet  Google Scholar 

  7. Brenninger-Göthe, M.: Two vehicle routing problems—mathematical programming approaches, Linköping studies in science and technology. Dissertations No. 200, Department of Mathematics, Linköping University, Linköping, Sweden (1989)

  8. Buson, E., Roberti, R., Toth, P.: A reduced-cost iterated local search heuristic for the fixed-charge transportation problem. Oper. Res. 62(5), 1095–1106 (2014)

    Article  MathSciNet  Google Scholar 

  9. Caprara, A., Fischetti, M., Toth, P.: A heuristic method for the set covering problem. Oper. Res. 47(5), 730–743 (1999)

    Article  MathSciNet  Google Scholar 

  10. Ceria, S., Nobili, P., Sassano, A.: A Lagrangian-based heuristic for large-scale set covering problems. Math. Program. 81(2), 215–228 (1998)

    Article  MathSciNet  Google Scholar 

  11. Cooper, L., Drebes, C.: An approximate solution method for the fixed charge problem. Nav. Res. Logist. Q. 14(1), 101–113 (1967)

    Article  Google Scholar 

  12. du Merle, O., Villeneuve, D., Desrosiers, J., Hansen, P.: Stabilized column generation. Discret. Math. 194(1–3), 229–237 (1999)

    Article  MathSciNet  Google Scholar 

  13. Eremin, I.I.: Theory of Linear Optimization. VSP, Utrecht (2002)

    Google Scholar 

  14. Fisk, J., McKeown, P.G.: The pure fixed charge transportation problem. Nav. Res. Logist. Q. 26(4), 631–641 (1979)

    Article  Google Scholar 

  15. Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming. Duxbury Press, Pacific Grove (2002)

    MATH  Google Scholar 

  16. Glover, F., Amini, M., Kochenberger, G.: Parametric ghost image processes for fixed-charge problems: a study of transportation networks. J. Heuristics 11(4), 307–336 (2005)

    Article  Google Scholar 

  17. Gray, P.: Exact solution of the fixed-charge transportation problem. Oper. Res. 19(6), 1529–1538 (1971)

    Article  Google Scholar 

  18. Guignard, M.: Lagrangean relaxation. TOP 11(2), 151–228 (2003)

    Article  MathSciNet  Google Scholar 

  19. Guignard, M., Kim, S.: Lagrangean decomposition: a model yielding stronger Lagrangean bounds. Math. Program. 39(2), 215–228 (1987)

    Article  MathSciNet  Google Scholar 

  20. Göthe-Lundgren, M., Larsson, T.: A set covering reformulation of the pure fixed charge transportation problem. Discret. Appl. Math. 48(3), 245–249 (1994)

    Article  MathSciNet  Google Scholar 

  21. Hirsch, W.M., Dantzig, G.B.: The fixed charge problem. RAND Corporation Rept RM-1383 (1954) (Published 1968 in Naval Research Logistics Quarterly 15(3):413–424)

    Article  MathSciNet  Google Scholar 

  22. Hoffman, K.L.: A method for globally minimizing concave functions over convex sets. Math. Program. 20(1), 22–32 (1981)

    Article  MathSciNet  Google Scholar 

  23. Hultberg, T.H., Cardoso, D.M.: The teacher assignment problem: a special case of the fixed charge transportation problem. Eur. J. Oper. Res. 101(3), 463–473 (1997)

    Article  Google Scholar 

  24. IBM ILOG CPLEX Optimizer. http://www-01.ibm.com/software/integration/optimization/cplex-optimizer/ (2009). Accessed 20 Oct 2017

  25. Jörnsten, K., Näsberg, M.: A new Lagrangian relaxation approach to the generalized assignment problem. Eur. J. Oper. Res. 27(3), 313–323 (1986)

    Article  MathSciNet  Google Scholar 

  26. Kennington, J., Unger, E.: A new branch-and-bound algorithm for the fixed-charge transportation problem. Manag. Sci. 22(10), 1116–1126 (1976)

    Article  MathSciNet  Google Scholar 

  27. Kim, D., Pardalos, P.M.: A solution approach to the fixed charge network flow problem using a dynamic slope scaling procedure. Oper. Res. Lett. 24(4), 195–203 (1999)

    Article  MathSciNet  Google Scholar 

  28. Kim, D., Pan, X., Pardalos, M.P.: An enhanced dynamic slope scaling procedure with tabu scheme for fixed charge network flow problems. Comput. Econ. 27(2), 273–293 (2006)

    Article  Google Scholar 

  29. Klingman, D., Napier, A., Stutz, J.: NETGEN: a program for generating large scale (un)capacitated assignment, transportation, and minimum cost flow network problems. Manag. Sci. 20(5), 814–821 (1974)

    Article  Google Scholar 

  30. Klose, A.: Algorithms for solving the single-sink fixed-charge transportation problem. Comput. Oper. Res. 35(6), 2079–2092 (2008)

    Article  MathSciNet  Google Scholar 

  31. Larsson, T., Migdalas, A., Rönnqvist, M.: A Lagrangian heuristic for the capacitated concave minimum cost network flow problem. Eur. J. Oper. Res. 78(1), 116–129 (1994)

    Article  Google Scholar 

  32. Larsson, T., Patriksson, M., Rydergren, C.: A column generation procedure for the side constrained traffic equilibrium problem. Transp. Res. Part B Methodol. 38(1), 17–38 (2004)

    Article  Google Scholar 

  33. Lawphongpanich, S.: Dynamic slope scaling procedure and Lagrangian relaxation with subproblem approximation. J. Glob. Optim. 35(1), 121–130 (2006)

    Article  MathSciNet  Google Scholar 

  34. Letocart, L., Nagih, A., Touati-Moungla, N.: Dantzig–Wolfe and Lagrangian decompositions in integer linear programming. Int. J. Math. Oper. Res. 4(3), 247–262 (2012)

    Article  MathSciNet  Google Scholar 

  35. Li, X., Lin, G., Shen, C., Hengel, A.V.D., Dick, A.: Learning hash functions using column generation. In: Proceedings of the 30th International Conference on Machine Learning, Atlanta, Georgia, USA, vol. 28 (2013)

  36. Lübbecke, M.E., Desrosiers, J.: Selected topics in column generation. Oper. Res. 53(6), 1007–1023 (2005)

    Article  MathSciNet  Google Scholar 

  37. Maniezzo, V., Mendes, I., Paruccini, M.: Decision support for siting problems. Decis. Support Syst. 23(3), 273–284 (1998)

    Article  Google Scholar 

  38. Manimaran, P., Selladurai, V., Ranganathan, R., Sasikumar, G.: Genetic algorithm for optimisation of distribution system in a single stage supply chain network with fixed charges. Int. J. Ind. Syst. Eng. 7(3), 292–316 (2011)

    Google Scholar 

  39. Marsten, R.E., Hogan, W.W., Blankenship, J.W.: The boxstep method for large-scale optimization. Oper. Res. 23(3), 389–405 (1975)

    Article  MathSciNet  Google Scholar 

  40. McKeown, P.G.: A vertex ranking procedure for the linear fixed charge problem. Oper. Res. 23(6), 1183–1191 (1975)

    Article  MathSciNet  Google Scholar 

  41. Mingozzi, A., Roberti, R.: An exact algorithm for the fixed charge transportation problem based on matching source and sink patterns. Transp. Sci., articles in advance, published online April 27, 2017 (2017)

  42. Murty, K.G.: Solving the fixed charge problem by ranking the extreme points. Oper. Res. 16(2), 268–279 (1968)

    Article  Google Scholar 

  43. Pimentel, C.M.O., Alvelos, F.P., de Carvalho, J.M.V.: Comparing Dantzig–Wolfe decompositions and branch-and-price algorithms for the multi-item capacitated lot sizing problem. Optim. Methods Softw. 25(2), 299–319 (2010)

    Article  MathSciNet  Google Scholar 

  44. Roberti, R., Bartolini, E., Mingozzi, A.: The fixed charge transportation problem: an exact algorithm based on a new integer programming formulation. Manag. Sci. 61(6), 1275–1291 (2015)

    Article  Google Scholar 

  45. Stroup, J.W.: Allocation of launch vehicles to space missions: a fixed-cost transportation problem. Oper. Res. 15(6), 1157–1163 (1967)

    Article  Google Scholar 

  46. Sun, M., Aronson, J.E., McKeown, P.G., Drinka, D.: A tabu search heuristic procedure for the fixed charge transportation problem. Eur. J. Oper. Res. 106(2–3), 441–456 (1998)

    Article  Google Scholar 

  47. Wilhelm, W.E.: A technical review of column generation in integer programming. Optim. Eng. 2(2), 159–200 (1995)

    Article  MathSciNet  Google Scholar 

  48. Wright, D.D., von Lanzenauer, C.H.: Solving the fixed charge problem with Lagrangian relaxation and cost allocation heuristics. Eur. J. Oper. Res. 42(3), 305–312 (1989)

    Article  MathSciNet  Google Scholar 

  49. Yang, L., Feng, Y.: A bicriteria solid transportation problem with fixed charge under stochastic environment. Appl. Math. Model. 31(12), 2668–2683 (2007)

    Article  Google Scholar 

  50. Yang, L., Liu, L.: Fuzzy fixed charge solid transportation problem and algorithm. Appl. Soft Comput. 7(3), 879–889 (2007)

    Article  Google Scholar 

  51. Zhao, Y.: On the integration of heuristics with column-oriented models for discrete optimization, Linköping studies in science and technology. Dissertations No. 1764, Department of Mathematics, Linköping University, Linköping, Sweden (2016)

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Acknowledgements

The work by Yixin Zhao is partly supported by 1035 Equipment Pre-Research Field Foundation (Project ID 61403120401) of China and partly supported by the Fundamental Research Funds for the Central Universities (Project No. 30918011333). The work by Elina Rönnberg is supported by the Center for Industrial Information Technology (CENIIT) at Linköping University (Project ID 16.05). The authors acknowledge Prof. Jesùs Sàez Aguado, Prof. Roberto Roberti and Prof. Minghe Sun for sending us the instances and other related information on their published work on FCTP. We thank the reviewers and editor for valuable comments that improved the presentation of the paper.

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Authors and Affiliations

  1. School of Automation, Nanjing University of Science and Technology, Nanjing, China

    Yixin Zhao

  2. Department of Mathematics, Linköping University, Linköping, Sweden

    Torbjörn Larsson & Elina Rönnberg

  3. Department of Industrial and System Engineering, University of Florida, Gainesville, USA

    Panos M. Pardalos

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  1. Yixin Zhao
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  2. Torbjörn Larsson
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  4. Panos M. Pardalos
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Correspondence to Yixin Zhao.

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Zhao, Y., Larsson, T., Rönnberg, E. et al. The fixed charge transportation problem: a strong formulation based on Lagrangian decomposition and column generation. J Glob Optim 72, 517–538 (2018). https://doi.org/10.1007/s10898-018-0661-y

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