Combinatorial and continuous models for the optimization of traffic flows on networks

SIAM Journal on Optimization

Volumn 16, Issue 4, 2006, Pages 1155-1176

  Fugenschuh A ^a   Herty, M ^b   Klar, A ^b   Martin, A ^a  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

^b Rheinland Pfälzische Technische Universität Kaiserslautern Landau   (Germany)

Author keywords

Combinatorial methods; Partial differential equations; Traffic networks

Indexed keywords

ALGORITHMS; COMBINATORIAL MATHEMATICS; MATHEMATICAL MODELS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS; TELECOMMUNICATION TRAFFIC;

CONTINUOUS MODELS; TRAFFIC FLOW; TRAFFIC NETWORKS;

COMPUTER NETWORKS;

EID: 33750897181     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/040605503     Document Type: Article

References (34)

1. A. AW AND M. RASCLE, Resurrection of second order models of traffic flow, SIAM J. Appl. Math., 60 (2000), pp. 916-938.
2. C. BARDOS, A. Y. LEROUX, AND J. C. NEDELEC, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), pp. 1017-1034.
3. M. BLINKIN, Problem of optimal control of traffic flow on highways, Automat. Remote Control, 37 (1976), pp. 662-667.
4. R. H. BYRD, P. LU, J. NOCEDAL, AND C. ZHU, A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Comput., 16 (1995), pp. 1190-1208.
5. R. BYRD, J. NOCEDAL, AND R. SCHNABEL, Representations of quasi-Newton matrices and their use in limited memory methods, Math. Program., 63 (1994), pp. 129-156.
6. M. CAREY AND E. SUBRAHMANIAN, An approach for modeling time-varying flows on congested networks, Transp. Res. B, (2000), pp. 157-183.
7. G. M. COCLITE AND B. PICCOLI, Traffic flow on a road network, SIAM J. Math. Anal., to appear.
8. R. COLOMBO, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), pp. 708-721.
9. C. M. DAFERMOS, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), pp. 33-41.
10. L. R. FORD AND D. R. FULKERSON, Constructing maximal dynamic flows from static flows, Oper. Res., (1958), pp. 419-433.
11. J. GREENBERG, Extension and amplification of the Aw-Rascle model, SIAM J. Appl. Math., 62 (2001), pp. 729-745.
12. J. GREENBERG, A. KLAR, AND M. RASCLE, Congestion on multilane highways, SIAM J. Appl. Math., 63 (2003), pp. 818-833.
13. M. GUGAT, M. HERTY, A. KLAR, AND G. LEUGERING, Adjoint calculus for traffic flow networks, JOTA, 126 (2005), pp. 589-616.
14. M. GÜNTHER, A. KLAR, T. MATERNE, AND R. WEGENER, Multivalued fundamental diagrams and stop and go waves for continuum traffic flow equations, SIAM J. Appl. Math., 64 (2003), pp. 468-483.
15. D. HELBING, Improved fluid dynamic model for vehicular traffic, Phys. Rev. E, 51 (1995), p. 3164.
16. M. HERTY AND A. KLAR, Modelling, simulation and optimization of traffic flow networks, SIAM J. Sci. Comput., 25 (2003), pp. 1066-1087.
17. M. HERTY AND A. KLAR, Simplified dynamics and optimization of large scale traffic networks, Math. Models Methods Appl. Sci., 14 (2004), pp. 1-23.
18. H. HOLDEN AND N. H. RISEBRO, A mathematical model of traffic flow on a network of unidirectional road, SIAM J. Math. Anal., 4 (1995), pp. 999-1017.
19. H. HOLDEN AND N. H. RISEBRO, Front Tracking for Hyperbolic Conservation Laws, Springer, New York, Berlin, Heidelberg, 2002.
20. ILOG CPLEX DIVISION, Using the CPLEX Callable Library, available online from http://www.cplex.com.
21. O. JAHN, R. H. MÖHRING, AND A. S. SCHULZ, Optimal routing of traffic flows with length restrictions in networks with congestion, in Operations Research Proceedings, K. Inderfurth et al., eds., Springer-Verlag, Berlin, 2000, pp. 437-442.
22. A. KLAR, R. D. KUEHNE, AND R. WEGENER, Mathematical models for vehicular traffic, Surveys Math. Indust., 6 (1996), p. 215.
23. A. KLAR AND R. WEGENER, Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math., 60 (2000), pp. 1749-1766.
24. E. KÖHLER, K. LANGKAU, AND M. SKUTELLA, Time-expanded graphs for flow-dependent transit times, in Algorithms - ESA '02, Lecture Notes in Comput. Sci. 2461, R. Möhring and R. Raman, eds., Springer, Berlin, 2002, pp. 599-611.
25. E. KÖHLER, R. H. MÖHRING, AND M. SKUTELLA, Traffic networks and flows over time, in DFG Research Center: Mathematics for Key Technologies, J. Kramer, ed., Berliner Mathematische Gesellschaft, 2002, pp. 49-70.
26. E. KÖHLER AND M. SKUTELLA, Flows over time with load-dependent transit times, in Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, 2002, pp. 174-183.
27. A. KOTSIALOS, M. PAPAGEORGIOU. M. MANGEAS. AND H. HAJ-SALEM, Coordinated and integrated control of motorway networks via non-linear optimal control, Trans. Res. C, 10 (2002), pp. 65-84.
28. R. D. KÜHNE, Macroscopic freeway model for dense traffic, in 9th International Symposium on Transportation and Traffic Theory, N. Vollmuller, ed., VNU Science Press, Utrecht, the Netherlands, 1984, pp. 21-42.
29. K. LANGKAU, Flows over Time with Flow-Dependent Transit Times, Ph.D. thesis, Fakultät II, Institut für Mathematik, Technische Universität Berlin, 2003.
30. M. J. LIGHTHILL AND J. B. WHITHAM, On kinematic waves, Proc. Roy. Soc. Edinburgh A, 229 (1983), pp. 281-345.
31. P. NELSON, A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions, Transport Theory Statist. Phys., 24 (1995), pp. 383-408.
32. H. J. PAYNE, FREFLO: A macroscopic simulation model of freeway traffic, Transp. Res. Record, 722 (1979), pp. 68-75.
33. G. B. WHITHAM, Linear and Nonlinear Waves, Wiley, New York, 1974.
34. C. ZHU, R. H. BYRD, J. Lu, AND J. NOCEDAL, L-BFGS-B: Fortran Subroutines for Large Scale Bound Constrained Optimization, Tech. Report, NAM-11, EECS Department, Northwestern University, Evanston, IL, 1994.