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.
Similar content being viewed by others
References
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)
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)
Balas, E., Zemel, E.: An algorithm for large zero–one knapsack problems. Oper. Res. 28(5), 1130–1154 (1980)
Balinski, M.L.: Fixed-cost transportation problems. Nav. Res. Logist. Q. 8(1), 41–54 (1961)
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)
Bell, G., Lamar, B., Wallace, C.: Capacity improvement, penalties, and the fixed charge transportation problem. Nav. Res. Logist. 46(4), 341–355 (1999)
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)
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)
Caprara, A., Fischetti, M., Toth, P.: A heuristic method for the set covering problem. Oper. Res. 47(5), 730–743 (1999)
Ceria, S., Nobili, P., Sassano, A.: A Lagrangian-based heuristic for large-scale set covering problems. Math. Program. 81(2), 215–228 (1998)
Cooper, L., Drebes, C.: An approximate solution method for the fixed charge problem. Nav. Res. Logist. Q. 14(1), 101–113 (1967)
du Merle, O., Villeneuve, D., Desrosiers, J., Hansen, P.: Stabilized column generation. Discret. Math. 194(1–3), 229–237 (1999)
Eremin, I.I.: Theory of Linear Optimization. VSP, Utrecht (2002)
Fisk, J., McKeown, P.G.: The pure fixed charge transportation problem. Nav. Res. Logist. Q. 26(4), 631–641 (1979)
Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming. Duxbury Press, Pacific Grove (2002)
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)
Gray, P.: Exact solution of the fixed-charge transportation problem. Oper. Res. 19(6), 1529–1538 (1971)
Guignard, M.: Lagrangean relaxation. TOP 11(2), 151–228 (2003)
Guignard, M., Kim, S.: Lagrangean decomposition: a model yielding stronger Lagrangean bounds. Math. Program. 39(2), 215–228 (1987)
Göthe-Lundgren, M., Larsson, T.: A set covering reformulation of the pure fixed charge transportation problem. Discret. Appl. Math. 48(3), 245–249 (1994)
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)
Hoffman, K.L.: A method for globally minimizing concave functions over convex sets. Math. Program. 20(1), 22–32 (1981)
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)
IBM ILOG CPLEX Optimizer. http://www-01.ibm.com/software/integration/optimization/cplex-optimizer/ (2009). Accessed 20 Oct 2017
Jörnsten, K., Näsberg, M.: A new Lagrangian relaxation approach to the generalized assignment problem. Eur. J. Oper. Res. 27(3), 313–323 (1986)
Kennington, J., Unger, E.: A new branch-and-bound algorithm for the fixed-charge transportation problem. Manag. Sci. 22(10), 1116–1126 (1976)
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)
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)
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)
Klose, A.: Algorithms for solving the single-sink fixed-charge transportation problem. Comput. Oper. Res. 35(6), 2079–2092 (2008)
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)
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)
Lawphongpanich, S.: Dynamic slope scaling procedure and Lagrangian relaxation with subproblem approximation. J. Glob. Optim. 35(1), 121–130 (2006)
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)
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)
Lübbecke, M.E., Desrosiers, J.: Selected topics in column generation. Oper. Res. 53(6), 1007–1023 (2005)
Maniezzo, V., Mendes, I., Paruccini, M.: Decision support for siting problems. Decis. Support Syst. 23(3), 273–284 (1998)
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)
Marsten, R.E., Hogan, W.W., Blankenship, J.W.: The boxstep method for large-scale optimization. Oper. Res. 23(3), 389–405 (1975)
McKeown, P.G.: A vertex ranking procedure for the linear fixed charge problem. Oper. Res. 23(6), 1183–1191 (1975)
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)
Murty, K.G.: Solving the fixed charge problem by ranking the extreme points. Oper. Res. 16(2), 268–279 (1968)
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)
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)
Stroup, J.W.: Allocation of launch vehicles to space missions: a fixed-cost transportation problem. Oper. Res. 15(6), 1157–1163 (1967)
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)
Wilhelm, W.E.: A technical review of column generation in integer programming. Optim. Eng. 2(2), 159–200 (1995)
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)
Yang, L., Feng, Y.: A bicriteria solid transportation problem with fixed charge under stochastic environment. Appl. Math. Model. 31(12), 2668–2683 (2007)
Yang, L., Liu, L.: Fuzzy fixed charge solid transportation problem and algorithm. Appl. Soft Comput. 7(3), 879–889 (2007)
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)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-018-0661-y
Keywords
- Fixed charge transportation problem
- Lagrangian decomposition
- Column generation
- Regularization
- Core problem