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Modeling and Optimization of Scalar Flows on Networks

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Book cover 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].

References

  1. D. Armbruster, D. Marthaler, C. Ringhofer, Kinetic and fluid model hierarchies for supply chains. SIAM J. Multiscale Model. 2, 43–61 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Armbruster, C. de Beer, M. Freitag, T. Jagalski, C. Ringhofer, Autonomous control of production networks using a pheromone approach. Phys. A 363, 104–114 (2006)

    Article  Google Scholar 

  3. D. Armbruster, P. Degond, C. Ringhofer, A model for the dynamics of large queuing networks and supply chains.. SIAM J. Appl. Math. 66, 896–920 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Armbruster, P. Degond, C. Ringhofer, Kinetic and fluid models for supply chains supporting policy attributes. Bull. Inst. Math. Acad. Sinica 2, 433–460 (2007)

    MathSciNet  MATH  Google Scholar 

  5. A. Aw, M. Rascle, Resurrection of second order models of traffic flow. SIAM J. Appl. Math. 60, 916–938 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Aw, A. Klar, T. Materne, M. Rascle, Derivation of continuum flow traffic models from microscopic follow the leader models. SIAM J. Appl. Math. 63, 259–278 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Byrd, J. Nocedal, R. Schnabel, Representations of quasi-newton matrices and their use in limited memory methods. Math. Program. 63, 129–156 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Byrd, P. Lu, J. Nocedal, C. Zhu, A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16, 1190–1208 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. C. Cercignani, The Boltzmann Equation and its Applications (Springer, New York, 1988)

    Book  MATH  Google Scholar 

  10. G. Coclite, M. Garavello, B. Piccoli, Traffic flow on a road network. SIAM J. Math. Anal. 36, 1862–1886 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Colombo, Hyperbolic phase transitions in traffic flow. SIAM J. Appl. Math. 63, 708–721 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Daganzo, A continuum theory of traffic dynamics for freeways with special lanes. Transp. Res. B 31, 83–102 (1997)

    Article  Google Scholar 

  13. C.F. Daganzo, A theory of supply chains. in Lecture Notes in Economics and Mathematical Systems, vol. 526 (Springer, Berlin, 2003), p. viii, 123

    Google Scholar 

  14. C. D’Apice, R. Manzo, A fluid dynamic model for supply chains. Netw. Heterogenous Media 1, 379–389 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Fügenschuh, M. Herty, A. Klar, A. Martin, Combinatorial and continuous models for the optimization of traffic flows on networks. SIOPT 16, 1155–1176 (2006)

    MATH  Google Scholar 

  16. A. Fügenschuh, S. Göttlich, M. Herty, A. Klar, A. Martin, A discrete optimization approach to large scale supply networks based on partial differential equations. SIAM J. Sci. Comput. 30, 1490–1507 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Göttlich, M. Herty, A. Klar, Network models for supply chains. Comm. Math. Sci. 3, 545–559 (2005)

    MATH  Google Scholar 

  18. S. Göttlich, M. Herty, A. Klar, Modelling and optimization of supply chains on complex networks. Comm. Math. Sci. 4, 315–330 (2006)

    MATH  Google Scholar 

  19. J. Greenberg, Extension and amplification of the Aw-Rascle model. SIAM J. Appl. Math. 62, 729–745 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Greenberg, A. Klar, M. Rascle, Congestion on multilane highways. SIAM J. Appl. Math. 63, 818–833 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Günther, A. Klar, T. Materne, R. Wegener, Multivalued fundamental diagrams and stop and go waves for continuum traffic flow equations. SIAM J. Appl. Math. 64, 468–483 (2003)

    MathSciNet  MATH  Google Scholar 

  22. D. Helbing, Improved fluid dynamic model for vehicular traffic. Phys. Rev. E 51, 3164 (1995)

    Article  Google Scholar 

  23. D. Helbing, A. Greiner, Modeling and simulation of multi-lane traffic flow. Phys. Rev. E 55, 5498–5507 (1975)

    Article  Google Scholar 

  24. M. Herty, A. Klar, Modelling, simulation and optimization of traffic flow networks. SIAM J. Sci. Comput. 25, 1066–1087 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Herty, A. Klar, Simplified dynamics and optimization of large scale traffic networks. Math. Models Meth. Appl. Sci. 14, 1–23 (2004)

    Article  MathSciNet  Google Scholar 

  26. M. Herty, M. Rascle, Coupling conditions for a class of second order models for traffic flow. SIAM J. Math. Anal. 38, 595–616 (2006)

    Article  MathSciNet  Google Scholar 

  27. M. Herty, M. Gugat, A. Klar, G. Leugering, Optimal control for traffic flow networks. J. Optim. Theor. Appl. 126, 589–615 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Herty, K. Kirchner, A. Klar, Instantaneous control for traffic flow. Math. Methods Appl. Sci. 30, 153–169 (2006)

    Article  MathSciNet  Google Scholar 

  29. M. Herty, A. Klar, B. Piccoli, Existence of solutions for supply chain models based on partial differential equations.. SIAM J. Math. Anal. 39, 160–173 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Hinze, K. Kunisch, Second order methods for optimal control of time-dependent fluid flow. SIAM J. Contr. Optim. 40, 925–946 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Hinze, R. Pinnau, An optimal control approach to semiconductor design. M3AS 12, 89–107 (2002)

    Google Scholar 

  32. H. Holden, N. Risebro, A mathematical model of traffic flow on a network of unidirectional road. SIAM J. Math. Anal. 4, 999–1017 (1995)

    Article  MathSciNet  Google Scholar 

  33. IBM ILOG CPLEX. IBM Deutschland Gmbh, 71137 Ehningen. Information available at http://www-01.ibm.com/software/integration/optimization

  34. S. Jin, Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48, 235–276 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  35. C. Kelley, Iterative Methods for Linear and Nonlinear Equations (SIAM, Philadelphia, 1995)

    Book  MATH  Google Scholar 

  36. C. Kirchner, M. Herty, S. Göttlich, A. Klar, Optimal control for continuous supply network models. Netw. Heterogenous Media 1, 675–688 (2006)

    Article  MATH  Google Scholar 

  37. A. Klar, R. Wegener, A hierachy of models for multilane vehicular traffic I: Modeling. SIAM J. Appl. Math. 59, 983–1001 (1998)

    Article  MathSciNet  Google Scholar 

  38. A. Klar, R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic. SIAM J. Appl. Math. 60, 1749–1766 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Klar, R. Kuehne, R. Wegener, Mathematical models for vehicular traffic. Surv. Math. Ind. 6, 215 (1996)

    MATH  Google Scholar 

  40. E. Köhler, M. Skutella, Flows over time with load-dependent transit times, in SODA’02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, Society for Industrial and Applied Mathematics Philadelphia, PA, USA, 174–183 (2002)

    Google Scholar 

  41. E. Köhler, M. Skutella, R.H. Möhring, Traffic networks and flows over time, in Book Algorithmics of Large and Complex Networks, Springer-Verlag Berlin, Heidelberg, 166–196 (2009)

    Google Scholar 

  42. S.N. Kruzkov, First order quasi linear equations in several independent variables. Math. USSR Sbornik 10, 217 (1970)

    Article  Google Scholar 

  43. R. Kühne, Macroscopic freeway model for dense traffic, in 9th International Symposium on Transportation and Traffic Theory, ed. by N. Vollmuller (VNU Science Press, Utrecht, 1984), pp. 21–42

    Google Scholar 

  44. J. Lebacque, M. Khoshyaran, First order macroscopic traffic flow models for networks in the context of dynamic assignment, Transportation Planning, ed. by M. Patriksson, K.A.P.M. Labbe, 119–140 (2002)

    Google Scholar 

  45. M. Lighthill, J. Whitham, On kinematic waves. Proc. R. Soc. Edinb. A229, 281–345 (1983)

    MathSciNet  Google Scholar 

  46. S.G. Nash, A. Sofer, Linear and Nonlinear Programming (The McGraw-Hill Companies, New York/St. Louis/San Francisco, 1996)

    Google Scholar 

  47. P. Nelson, A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions. Transp. Theor. Stat. Phys. 24, 383–408 (1995)

    Article  MATH  Google Scholar 

  48. H. Payne, FREFLO: A macroscopic simulation model of freeway traffic. Transp. Res. Rec. 722, 68–75 (1979)

    Google Scholar 

  49. P. Spellucci, Numerische Verfahren der nichtlinearen Optimierung (Birkhäuser, Basel, 1993)

    Book  MATH  Google Scholar 

  50. S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Contr. Optim. 41, 740 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  51. S. Ulbrich, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Syst. Contr. Lett. 3, 309 (2003)

    Google Scholar 

  52. G. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974)

    MATH  Google Scholar 

  53. C. Zhu, R. Byrd, J. Lu, J. Nocedal, L-bfgs-b: Fortran subroutines for large scale bound constrained optimization. Technical Report, NAM-11, EECS Department, Northwestern University, 1994

    Google Scholar 

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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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  1. University of Mannheim, School of Business Informatics and Mathematics, A5, 6, 68131, Mannheim, Germany

    Simone Göttlich & Axel Klar

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  1. Simone Göttlich
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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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