Abstract
The single-sink fixed-charge transportation problem is known to have many applications in the area of manufacturing and transportation as well as being an important subproblem of the fixed-charge transportation problem. However, even the best algorithms from the literature do not fully leverage the structure of this problem, to the point of being surpassed by modern general-purpose mixed-integer programming solvers for large instances. We introduce a novel reformulation of the problem and study its theoretical properties. This reformulation leads to a range of new upper and lower bounds, dominance relations, linear relaxations, and filtering procedures. The resulting algorithm includes a heuristic phase and an exact phase, the main step of which is to solve a very small number of knapsack subproblems. Computational experiments are presented for existing and new types of instances. These tests indicate that the new algorithm systematically reduces the resolution time of the state-of-the-art exact methods by several orders of magnitude.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Herer, Y.T., Rosenblatt, M.J., Hefter, I.: Fast algorithms for single-sink fixed charge transportation problems with applications to manufacturing and transportation. Transp. Sci. 30, 276–290 (1996)
Klose, A.: Algorithms for solving the single-sink fixed-charge transportation problem. Comput. Oper. Res. 35(6), 2079–2092 (2008)
Görtz, S., Klose, A.: The single-sink fixed-charge transportation problem: Applications and solution methods. In: Günther, H.-O., Mattfeld, D.C., Suhl, L. (eds.) Management logistischer Netzwerke, pp. 383–406. Physica-Verlag HD, Heidelberg (2007)
Mingozzi, A., Roberti, R.: An exact algorithm for the fixed charge transportation problem based on matching source and sink patterns. Transp. Sci. 52(2), 229–238 (2018)
Christensen, T.R.L., Andersen, K.A., Klose, A.: Solving the single-sink, fixed-charge, multiple-choice transportation problem by dynamic programming. Transp. Sci. 47(3), 428–438 (2013)
Kazemzadeh, M.R.A., Bektaş, T., Crainic, T.G., Frangioni, A., Gendron, B., Gorgone, E.: Node-based Lagrangian relaxations for multicommodity capacitated fixed-charge network design. Discrete Appl. Math. 308(C), 255–275 (2022)
Haberl, J.: Exact algorithm for solving a special fixed-charge linear programming problem. J. Optim. Theory Appl. 69, 489–529 (1991)
Alidaee, B., Kochenberger, G.A.: A note on a simple dynamic programming approach to the single-sink, fixed-charge transportation problem. Transp. Sci. 39(1), 140–143 (2005)
Martello, S., Toth, P.: An upper bound for the zero-one knapsack problem and a branch and bound algorithm. Euro. J. Oper. Res. 1(3), 169–175 (1977)
Martello, S., Toth, P.: Upper bounds and algorithms for hard 0–1 knapsack problems. Oper. Res. 45(5), 768–778 (1997)
Pisinger, D.: A minimal algorithm for the 0–1 knapsack problem. Oper. Res. 45(5), 758–767 (1997)
Pisinger, D.: A minimal algorithm for the bounded knapsack problem. INFORMS J. Comput. 12(1), 75–82 (2000)
Rosenblatt, M.J., Herer, Y.T., Hefter, I.: Note: an acquisition policy for a single item multi-supplier system. Manag. Sci. 44(11–part–2), S96–S100 (1998)
Balas, E., Zemel, E.: An algorithm for large zero-one knapsack problems. Oper. Res. 28(5), 1130–1154 (1980)
Görtz, S., Klose, A.: Analysis of some greedy algorithms for the single-sink fixed-charge transportation problem. J. Heuristics 15(4), 331–349 (2009)
Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. Wiley, New York (1990)
Martello, S., Pisinger, D., Toth, P.: Dynamic programming and strong bounds for the 0–1 knapsack problem. Manage. Sci. 45(3), 414–424 (1999)
Pisinger, D.: Where are the hard knapsack problems? Comput. Oper. Res. 32(9), 2271–2284 (2005)
Acknowledgements
This work is dedicated to our colleague Bernard Gendron who was taken far too soon, and will be deeply missed. Financial support for this work was provided by the Canadian Natural Sciences and Engineering Research Council (NSERC) under Grants 2017-06054 and 2021-04037. This support is gratefully acknowledged. We thank Andreas Klose for making available the source code of his algorithms.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Bernard Gendron: Deceased.
This research was supported by the Natural Sciences and Engineering Research Council of Canada through the Discovery Grants 2017-06054 and 2021-04037.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Legault, R., Côté, JF. & Gendron, B. A novel reformulation for the single-sink fixed-charge transportation problem. Math. Program. 202, 169–198 (2023). https://doi.org/10.1007/s10107-023-01930-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-023-01930-y