IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v33y2017i4d10.1007_s10878-016-0031-y.html
   My bibliography  Save this article

Maximum flows in generalized processing networks

Author

Listed:
  • Michael Holzhauser

    (University of Kaiserslautern)

  • Sven O. Krumke

    (University of Kaiserslautern)

  • Clemens Thielen

    (University of Kaiserslautern)

Abstract

Processing networks (cf. Koene in Minimal cost flow in processing networks: a primal approach, 1982) and manufacturing networks (cf. Fang and Qi in Optim Methods Softw 18:143–165, 2003) are well-studied extensions of traditional network flow problems that allow to model the decomposition or distillation of products in a manufacturing process. In these models, so called flow ratios $$\alpha _e \in [0,1]$$ α e ∈ [ 0 , 1 ] are assigned to all outgoing edges of special processing nodes. For each such special node, these flow ratios, which are required to sum up to one, determine the fraction of the total outgoing flow that flows through the respective edges. In this paper, we generalize processing networks to the case that these flow ratios only impose an upper bound on the respective fractions and, in particular, may sum up to more than one at each node. We show that a flow decomposition similar to the one for traditional network flows is possible and can be computed in strongly polynomial time. Moreover, we show that there exists a fully polynomial-time approximation scheme (FPTAS) for the maximum flow problem in these generalized processing networks if the underlying graph is acyclic and we provide two exact algorithms with strongly polynomial running-time for the problem on series–parallel graphs. Finally, we study the case of integral flows and show that the problem becomes $${\mathcal {NP}}$$ NP -hard to solve and approximate in this case.

Suggested Citation

  • Michael Holzhauser & Sven O. Krumke & Clemens Thielen, 2017. "Maximum flows in generalized processing networks," Journal of Combinatorial Optimization, Springer, vol. 33(4), pages 1226-1256, May.
    • Handle: RePEc:spr:jcomop:v:33:y:2017:i:4:d:10.1007_s10878-016-0031-y
    DOI: 10.1007/s10878-016-0031-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-016-0031-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10878-016-0031-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Haiyan Lu & Enyu Yao & Liqun Qi, 2006. "Some further results on minimum distribution cost flow problems," Journal of Combinatorial Optimization, Springer, vol. 11(4), pages 351-371, June.
    2. Chou-Hong J. Chen & Michael Engquist, 1986. "A Primal Simplex Approach to Pure Processing Networks," Management Science, INFORMS, vol. 32(12), pages 1582-1598, December.
    3. Prahalad Venkateshan & Kamlesh Mathur & Ronald H. Ballou, 2008. "An efficient generalized network-simplex-based algorithm for manufacturing network flows," Journal of Combinatorial Optimization, Springer, vol. 15(4), pages 315-341, May.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jan Boeckmann & Clemens Thielen, 2023. "New Ways in Municipal Flood Mitigation: a Mixed-Integer Programming Approach and its Practical Application," SN Operations Research Forum, Springer, vol. 4(4), pages 1-68, December.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. ÇalIskan, Cenk, 2011. "A specialized network simplex algorithm for the constrained maximum flow problem," European Journal of Operational Research, Elsevier, vol. 210(2), pages 137-147, April.
    2. Mohammad Ali Raayatpanah & Salman Khodayifar & Thomas Weise & Panos Pardalos, 2022. "A novel approach to subgraph selection with multiple weights on arcs," Journal of Combinatorial Optimization, Springer, vol. 44(1), pages 242-268, August.
    3. Gengjun Gao & Yuxuan Che & Jian Shen, 0. "Path optimization for joint distribution of medical consumables under hospital SPD supply chain mode," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-18.
    4. Prahalad Venkateshan & Kamlesh Mathur & Ronald H. Ballou, 2008. "An efficient generalized network-simplex-based algorithm for manufacturing network flows," Journal of Combinatorial Optimization, Springer, vol. 15(4), pages 315-341, May.
    5. John W. Chinneck, 1990. "Formulating processing network models: Viability theory," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(2), pages 245-261, April.
    6. Chinneck, J. W. & Moll, R. H. H., 1995. "Processing network models for forest management," Omega, Elsevier, vol. 23(5), pages 499-510, October.
    7. John W. Chinneck, 1992. "Viability analysis: A formulation aid for all classes of network models," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(4), pages 531-543, June.
    8. Gengjun Gao & Yuxuan Che & Jian Shen, 2021. "Path optimization for joint distribution of medical consumables under hospital SPD supply chain mode," Journal of Combinatorial Optimization, Springer, vol. 42(4), pages 866-883, November.
    9. David R. Morrison & Jason J. Sauppe & Sheldon H. Jacobson, 2013. "A Network Simplex Algorithm for the Equal Flow Problem on a Generalized Network," INFORMS Journal on Computing, INFORMS, vol. 25(1), pages 2-12, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jcomop:v:33:y:2017:i:4:d:10.1007_s10878-016-0031-y. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.