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Continuous and discrete flows over time

Author

Listed:
  • Ronald Koch
  • Ebrahim Nasrabadi
  • Martin Skutella

Abstract

Network flows over time form a fascinating area of research. They model the temporal dynamics of network flow problems occurring in a wide variety of applications. Research in this area has been pursued in two different and mainly independent directions with respect to time modeling: discrete and continuous time models. In this paper we deploy measure theory in order to introduce a general model of network flows over time combining both discrete and continuous aspects into a single model. Here, the flow on each arc is modeled as a Borel measure on the real line (time axis) which assigns to each suitable subset a real value, interpreted as the amount of flow entering the arc over the subset. We focus on the maximum flow problem formulated in a network where capacities on arcs are also given as Borel measures and storage might be allowed at the nodes of the network. We generalize the concept of cuts to the case of these Borel Flows and extend the famous MaxFlow-MinCut Theorem. Copyright Springer-Verlag 2011

Suggested Citation

  • Ronald Koch & Ebrahim Nasrabadi & Martin Skutella, 2011. "Continuous and discrete flows over time," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(3), pages 301-337, June.
    • Handle: RePEc:spr:mathme:v:73:y:2011:i:3:p:301-337
    DOI: 10.1007/s00186-011-0357-2
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    References listed on IDEAS

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    1. L. R. Ford & D. R. Fulkerson, 1958. "Constructing Maximal Dynamic Flows from Static Flows," Operations Research, INFORMS, vol. 6(3), pages 419-433, June.
    2. E. J. Anderson & P. Nash & A. B. Philpott, 1982. "A Class of Continuous Network Flow Problems," Mathematics of Operations Research, INFORMS, vol. 7(4), pages 501-514, November.
    3. A. B. Philpott, 1990. "Continuous-Time Flows in Networks," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 640-661, November.
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    Cited by:

    1. Sahar Bsaybes & Alain Quilliot & Annegret K. Wagler, 2019. "Fleet management for autonomous vehicles using flows in time-expanded networks," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(2), pages 288-311, July.
    2. Natashia Boland & Thomas Kalinowski & Simranjit Kaur, 2016. "Scheduling arc shut downs in a network to maximize flow over time with a bounded number of jobs per time period," Journal of Combinatorial Optimization, Springer, vol. 32(3), pages 885-905, October.
    3. S. Khodayifar & M. A. Raayatpanah & P. M. Pardalos, 2019. "A polynomial time algorithm for the minimum flow problem in time-varying networks," Annals of Operations Research, Springer, vol. 272(1), pages 29-39, January.

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