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Modeling and Optimization of Scalar Flows on Networks

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Modelling and Optimisation of Flows on Networks

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2062))

Abstract

Detailed models based on partial differential equations characterizing the dynamics on single arcs of a network (roads, production lines, etc.) are considered. These models are able to describe the dynamical behavior in a network accurately. On the other hand, for large scale networks often strongly simplified dynamics or even static descriptions of the flow have been widely used for traffic flow or supply chain management due to computational reasons. In this paper, a unified presentation highlighting connections between the above approaches are given and furthermore, a hierarchy of dynamical models is developed including models based on partial differential equations and nonlinear algebraic equations or even combinatorial models based on linear equations. Special focus is on optimal control problems and optimization techniques where combinatorial and continuous optimization approaches are discussed and compared.

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Notes

  1. 1.

    Follow Holden and Risebro in [32].

  2. 2.

    See Fig. 1.

  3. 3.

    Theorem 1.1 in [32].

  4. 4.

    Theorem 3.1 in [10].

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Acknowledgements

We wish to thank all our collaborators and co-authors, in particular Michael Herty, Armin Fügenschuh and Alexander Martin. Parts of this work have been taken from the articles [15, 24, 25] as well as [16, 17, 36].

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Correspondence to Simone Göttlich .

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Göttlich, S., Klar, A. (2013). Modeling and Optimization of Scalar Flows on Networks. In: Modelling and Optimisation of Flows on Networks. Lecture Notes in Mathematics(), vol 2062. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32160-3_8

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