The Maximum Flow Problem with Minimum Lot Sizes

  • Dag Haugland
  • Mujahed Eleyat
  • Magnus Lie Hetland
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6971)


In many transportation systems, the shipment quantities are subject to minimum lot sizes in addition to regular capacity constraints. That is, either the quantity must be zero, or it must be between the two bounds. In this work, we consider a directed graph, where a minimum lot size and a flow capacity are defined for each arc, and study the problem of maximizing the flow from a given source to a given terminal. We prove that the problem is NP-hard. Based on a straightforward mixed integer programming formulation, we develop a Lagrangean relaxation technique, and demonstrate how this can provide strong bounds on the maximum flow. For fast computation of near-optimal solutions, we develop a heuristic that departs from the zero solution and gradually augments the set of flow-carrying (open) arcs. The set of open arcs does not necessarily constitute a feasible solution. We point out how feasibility can be checked quickly by solving regular maximum flow problems in an extended network, and how the solutions to these subproblems can be productive in augmenting the set of open arcs. Finally, we present results from preliminary computational experiments with the construction heuristic.


Setup Cost Construction Heuristic Residual Network Residual Graph Maximum Flow Problem 
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  1. 1.
    Bahl, H., Ritzman, L., Gupta, J.: Determining lot sizes and resource requirements: A review. Operations Research 35(3), 329–345 (1988)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2001)zbMATHGoogle Scholar
  3. 3.
    Goldfarb, D., Grigoriadis, M.: A computational comparison of the dinic and network simplex methods for maximum flow. Annals of Operations Research 78, 83–123 (1988)MathSciNetGoogle Scholar
  4. 4.
    Hirsch, W., Dantzig, G.: The fixed charge problem. Naval Research Logistics Quarterly 15, 413–424 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jans, R., Degraeve, Z.: Modeling industrial lot sizing problems: a review. International Journal of Production Research 46(6), 1619–1643 (1968)CrossRefzbMATHGoogle Scholar
  6. 6.
    Jungnickel, D.: Graphs, Networks and Algorithms. Springer, Heidelberg (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    Karimi, B., Ghomi, S.F., Wilson, J.: The capacitated lot sizing problem: a review of models and algorithms. Omega 31, 365–378 (2003)CrossRefGoogle Scholar
  8. 8.
  9. 9.
    Nahapetyan, A., Pardalos, P.: Adaptive dynamic cost updating procedure for solving fixed charge network flow problems. Computational Optimization and Applications 39, 37–50 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Tardos, E.: A strongly polynomial minimum cost circulation algorithm. Combinatorica 5, 247–255 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Voß, S., Woodruff, D.: Connecting mrp, MRP II and ERP supply chain production planning via optimization models. In: Greenberg, H. (ed.) Tutorials on Emerging Methodologies and Applications in Operations Research, p. 8.1–8.30. Springer, Heidelberg (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Dag Haugland
    • 1
  • Mujahed Eleyat
    • 2
  • Magnus Lie Hetland
    • 3
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.Miriam ASHaldenNorway
  3. 3.Department of Computer and Information ScienceNorwegian University of Science and TechnologyTrondheimNorway

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