June  2014, 9(2): 315-334. doi: 10.3934/nhm.2014.9.315

Optimization for a special class of traffic flow models: Combinatorial and continuous approaches

1. 

Department of Mathematics, University of Mannheim, D-68131 Mannheim

2. 

School of Business Informatics and Mathematics, University of Mannheim, D-68131 Mannheim, Germany

3. 

Department of Mathematics, University of Kaiserslautern, D-67663 Kaiserslautern, Germany

Received  November 2013 Revised  March 2014 Published  July 2014

In this article, we discuss the optimization of a linearized traffic flow network model based on conservation laws. We present two solution approaches. One relies on the classical Lagrangian formalism (or adjoint calculus), whereas another one uses a discrete mixed-integer framework. We show how both approaches are related to each other. Numerical experiments are accompanied to show the quality of solutions.
Citation: Simone Göttlich, Oliver Kolb, Sebastian Kühn. Optimization for a special class of traffic flow models: Combinatorial and continuous approaches. Networks & Heterogeneous Media, 2014, 9 (2) : 315-334. doi: 10.3934/nhm.2014.9.315
References:
[1]

A. M. Bayen, R. L. Raffard and C. Tomlin, Adjoint-based control of a new Eulerian network model of air traffic flow, IEEE Transactions on Control Systems Technology, 14 (2006), 804-818. doi: 10.1109/TCST.2006.876904.  Google Scholar

[2]

A. Bressan and K. Han, Optima and equilibria for a model of traffic flow, SIAM Journal on Mathematical Analysis, 43 (2011), 2384-2417. doi: 10.1137/110825145.  Google Scholar

[3]

M. Carey and E. Subrahmanian, An approach to modelling time-varying flows on congested networks, Transportation Research Part B: Methodological, 34 (2000), 157-183. doi: 10.1016/S0191-2615(99)00019-3.  Google Scholar

[4]

A. Cascone, B. Piccoli and L. Rarita, Circulation of car traffic in congested urban areas, Communications in Mathematical Sciences, 6 (2008), 765-784. doi: 10.4310/CMS.2008.v6.n3.a12.  Google Scholar

[5]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM Journal on Mathematical Analysis, 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.  Google Scholar

[6]

C. F. Daganzo, The cell transmission model, part II: Network traffic, Transportation Research Part B: Methodological, 29 (1995), 79-93. doi: 10.1016/0191-2615(94)00022-R.  Google Scholar

[7]

C. F. Daganzo, Fundamentals of Transportation and Traffic Operations, Pergamon-Elsevier, Oxford, U.K., 1997. Google Scholar

[8]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation and Optimization of Supply Chains: A Continuous Approach, SIAM Book Series on Mathematical Modeling and Computation, 2010. doi: 10.1137/1.9780898717600.  Google Scholar

[9]

C. D'Apice, R. Manzo and B. Piccoli, Packet flow on telecommunication networks, SIAM Journal on Mathematical Analysis, 38 (2006), 717-740. doi: 10.1137/050631628.  Google Scholar

[10]

C. D'Apice, R. Manzo and B. Piccoli, A fluid dynamic model for telecommunication networks with sources and destinations, SIAM Journal on Applied Mathematics, 68 (2008), 981-1003. doi: 10.1137/060674132.  Google Scholar

[11]

C. D'Apice, R. Manzo and B. Piccoli, Modelling supply networks with partial differential equations, Quarterly of Applied Mathematics, 67 (2009), 419-440.  Google Scholar

[12]

C. D'Apice, R. Manzo and B. Piccoli, Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks, Journal of Mathematical Analysis and Applications, 362 (2010), 374-386. doi: 10.1016/j.jmaa.2009.07.058.  Google Scholar

[13]

C. D'Apice, R. Manzo and B. Piccoli, Optimal input flows for a PDE-ODE model of supply chains, Communications in Mathematical Sciences, 10 (2012), 1225-1240. doi: 10.4310/CMS.2012.v10.n4.a10.  Google Scholar

[14]

P. Domschke, B. Geißler, O. Kolb, J. Lang, A. Martin and A. Morsi, Combination of nonlinear and linear optimization of transient gas networks, INFORMS Journal on Computing, 23 (2011), 605-617. doi: 10.1287/ijoc.1100.0429.  Google Scholar

[15]

A. Fügenschuh, B. Geißler, A. Martin and A. Morsi, The Transport PDE and Mixed-Integer Linear Programming, in Models and Algorithms for Optimization in Logistics (eds. C. Barnhart, U. Clausen, U. Lauther and R. H. Möhring), Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany, 2009. Google Scholar

[16]

A. Fügenschuh, S. Göttlich, M. Herty, A. Klar and A. Martin, A discrete optimization approach to large scale supply networks based on partial differential equations, SIAM Journal on Scientific Computing, 30 (2008), 1490-1507. doi: 10.1137/060663799.  Google Scholar

[17]

A. Fügenschuh, M. Herty, A. Klar and A. Martin, Combinatorial and continuous models for the optimization of traffic flows on networks, SIAM Journal on Optimization, 16 (2006), 1155-1176. doi: 10.1137/040605503.  Google Scholar

[18]

S. Göttlich, M. Herty, C. Ringhofer and U. Ziegler, Production systems with limited repair capacity, Optimization, 61 (2012), 915-948. doi: 10.1080/02331934.2011.615395.  Google Scholar

[19]

S. Göttlich, M. Herty and U. Ziegler, Numerical discretization of Hamilton - Jacobi equations on networks, Networks and Heterogeneous Networks, 8 (2013), 685-705. doi: 10.3934/nhm.2013.8.685.  Google Scholar

[20]

S. Göttlich, S. Kühn, J.P. Ohst, S. Ruzika and M. Thiemann, Evacuation dynamics influenced by spreading hazardous material, Networks and Heterogeneous Media, 6 (2011), 443-464. doi: 10.3934/nhm.2011.6.443.  Google Scholar

[21]

S. Göttlich and U. Ziegler, Traffic light control: A case study, Discrete and Continuous Dynamical Systems Series S, 7 (2014), 483-501. doi: 10.3934/dcdss.2014.7.483.  Google Scholar

[22]

M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks, J. Optimization Theory Appl., 126 (2005), 589-616. doi: 10.1007/s10957-005-5499-z.  Google Scholar

[23]

M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM Journal on Scientific Computing, 25 (2003), 1066-1087. doi: 10.1137/S106482750241459X.  Google Scholar

[24]

M. Herty and A. Klar, Simplified dynamics and optimization of large scale traffic networks, Mathematical Models and Methods in Applied Sciences (M3AS), 14 (2004), 579-601. doi: 10.1142/S0218202504003362.  Google Scholar

[25]

M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks, Networks and Heterogeneous Media, 2 (2007), 733-750. doi: 10.3934/nhm.2007.2.733.  Google Scholar

[26]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM Journal on Mathematical Analysis, 26 (1995), 999-1017. doi: 10.1137/S0036141093243289.  Google Scholar

[27]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, 2nd edition, Springer, New York, Berlin, Heidelberg, 2002. doi: 10.1007/978-3-642-56139-9.  Google Scholar

[28]

IBM ILOG CPLEX Optimization Studio, Cplex version 12,, 2010., ().   Google Scholar

[29]

G. S. Jiang, D. Levy, C. T. Lin, S. Osher and E. Tadmor, High-Resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 35 (1998), 2147-2168. doi: 10.1137/S0036142997317560.  Google Scholar

[30]

C. T. Kelley, Iterative Methods for Optimization, Society for Industrial and Applied Mathematics, Philadelphia, 1999. doi: 10.1137/1.9781611970920.  Google Scholar

[31]

C. Kirchner, M. Herty, S. Göttlich and A. Klar, Optimal control for continuous supply network models, Networks Heterogenous Media, 1 (2006), 675-688. doi: 10.3934/nhm.2006.1.675.  Google Scholar

[32]

A. Klar, R. D. Kühne and R. Wegener, Mathematical models for vehicular traffic, Surveys on Mathematics for Industry, 6 (1996), 215-239.  Google Scholar

[33]

O. Kolb, Simulation and Optimization of Gas and Water Supply Networks, Ph.D thesis, Technische Universität Darmstadt, 2011. Google Scholar

[34]

O. Kolb and J. Lang, Simulation and continuous optimization, in Mathematical Optimization of Water Networks (eds. A. Martin, K. Klamroth, J. Lang, G. Leugering, A. Morsi, M. Oberlack, M. Ostrowski and R. Rosen), Internat. Ser. Numer. Math., 162, Birkhäuser/Springer Basel AG, Basel, 2012, 17-33. doi: 10.1007/978-3-0348-0436-3.  Google Scholar

[35]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Royal Society of London Proceedings Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar

[36]

R. Manzo, B. Piccoli and L. Rarita, Optimal distribution of traffic flows at junctions in emergency cases, European Journal of Applied Mathematics, 23 (2012), 515-535. doi: 10.1017/S0956792512000071.  Google Scholar

[37]

G. L. Nemhauser and L. A. Wolsey, Integer and Combinatorial Optimization, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, 1988.  Google Scholar

[38]

G. F. Newell, Traffic Flow on Transportation Networks, MIT Press Series in Transportation Studies, MIT Press, Cambridge, MA, USA, 1980. Google Scholar

[39]

P. Spellucci, Numerische Verfahren der Nichtlinearen Optimierung, Birkhäuser-Verlag, Basel, 1993. doi: 10.1007/978-3-0348-7214-0.  Google Scholar

[40]

P. Spellucci, A new technique for inconsistent QP problems in the SQP method, Mathematical Methods of Operations Research, 47 (1998), 355-400. doi: 10.1007/BF01198402.  Google Scholar

[41]

P. Spellucci, An SQP method for general nonlinear programs using only equality constrained subproblems, Mathematical programming, 82 (1998), 413-448. doi: 10.1007/BF01580078.  Google Scholar

[42]

D. Sun, I. S. Strub and A. M. Bayen, Comparison of the performance of four Eulerian network flow models for strategic air traffic management, Networks and Heterogeneous Media, 2 (2007), 569-595. doi: 10.3934/nhm.2007.2.569.  Google Scholar

show all references

References:
[1]

A. M. Bayen, R. L. Raffard and C. Tomlin, Adjoint-based control of a new Eulerian network model of air traffic flow, IEEE Transactions on Control Systems Technology, 14 (2006), 804-818. doi: 10.1109/TCST.2006.876904.  Google Scholar

[2]

A. Bressan and K. Han, Optima and equilibria for a model of traffic flow, SIAM Journal on Mathematical Analysis, 43 (2011), 2384-2417. doi: 10.1137/110825145.  Google Scholar

[3]

M. Carey and E. Subrahmanian, An approach to modelling time-varying flows on congested networks, Transportation Research Part B: Methodological, 34 (2000), 157-183. doi: 10.1016/S0191-2615(99)00019-3.  Google Scholar

[4]

A. Cascone, B. Piccoli and L. Rarita, Circulation of car traffic in congested urban areas, Communications in Mathematical Sciences, 6 (2008), 765-784. doi: 10.4310/CMS.2008.v6.n3.a12.  Google Scholar

[5]

G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM Journal on Mathematical Analysis, 36 (2005), 1862-1886. doi: 10.1137/S0036141004402683.  Google Scholar

[6]

C. F. Daganzo, The cell transmission model, part II: Network traffic, Transportation Research Part B: Methodological, 29 (1995), 79-93. doi: 10.1016/0191-2615(94)00022-R.  Google Scholar

[7]

C. F. Daganzo, Fundamentals of Transportation and Traffic Operations, Pergamon-Elsevier, Oxford, U.K., 1997. Google Scholar

[8]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation and Optimization of Supply Chains: A Continuous Approach, SIAM Book Series on Mathematical Modeling and Computation, 2010. doi: 10.1137/1.9780898717600.  Google Scholar

[9]

C. D'Apice, R. Manzo and B. Piccoli, Packet flow on telecommunication networks, SIAM Journal on Mathematical Analysis, 38 (2006), 717-740. doi: 10.1137/050631628.  Google Scholar

[10]

C. D'Apice, R. Manzo and B. Piccoli, A fluid dynamic model for telecommunication networks with sources and destinations, SIAM Journal on Applied Mathematics, 68 (2008), 981-1003. doi: 10.1137/060674132.  Google Scholar

[11]

C. D'Apice, R. Manzo and B. Piccoli, Modelling supply networks with partial differential equations, Quarterly of Applied Mathematics, 67 (2009), 419-440.  Google Scholar

[12]

C. D'Apice, R. Manzo and B. Piccoli, Existence of solutions to Cauchy problems for a mixed continuum-discrete model for supply chains and networks, Journal of Mathematical Analysis and Applications, 362 (2010), 374-386. doi: 10.1016/j.jmaa.2009.07.058.  Google Scholar

[13]

C. D'Apice, R. Manzo and B. Piccoli, Optimal input flows for a PDE-ODE model of supply chains, Communications in Mathematical Sciences, 10 (2012), 1225-1240. doi: 10.4310/CMS.2012.v10.n4.a10.  Google Scholar

[14]

P. Domschke, B. Geißler, O. Kolb, J. Lang, A. Martin and A. Morsi, Combination of nonlinear and linear optimization of transient gas networks, INFORMS Journal on Computing, 23 (2011), 605-617. doi: 10.1287/ijoc.1100.0429.  Google Scholar

[15]

A. Fügenschuh, B. Geißler, A. Martin and A. Morsi, The Transport PDE and Mixed-Integer Linear Programming, in Models and Algorithms for Optimization in Logistics (eds. C. Barnhart, U. Clausen, U. Lauther and R. H. Möhring), Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany, 2009. Google Scholar

[16]

A. Fügenschuh, S. Göttlich, M. Herty, A. Klar and A. Martin, A discrete optimization approach to large scale supply networks based on partial differential equations, SIAM Journal on Scientific Computing, 30 (2008), 1490-1507. doi: 10.1137/060663799.  Google Scholar

[17]

A. Fügenschuh, M. Herty, A. Klar and A. Martin, Combinatorial and continuous models for the optimization of traffic flows on networks, SIAM Journal on Optimization, 16 (2006), 1155-1176. doi: 10.1137/040605503.  Google Scholar

[18]

S. Göttlich, M. Herty, C. Ringhofer and U. Ziegler, Production systems with limited repair capacity, Optimization, 61 (2012), 915-948. doi: 10.1080/02331934.2011.615395.  Google Scholar

[19]

S. Göttlich, M. Herty and U. Ziegler, Numerical discretization of Hamilton - Jacobi equations on networks, Networks and Heterogeneous Networks, 8 (2013), 685-705. doi: 10.3934/nhm.2013.8.685.  Google Scholar

[20]

S. Göttlich, S. Kühn, J.P. Ohst, S. Ruzika and M. Thiemann, Evacuation dynamics influenced by spreading hazardous material, Networks and Heterogeneous Media, 6 (2011), 443-464. doi: 10.3934/nhm.2011.6.443.  Google Scholar

[21]

S. Göttlich and U. Ziegler, Traffic light control: A case study, Discrete and Continuous Dynamical Systems Series S, 7 (2014), 483-501. doi: 10.3934/dcdss.2014.7.483.  Google Scholar

[22]

M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks, J. Optimization Theory Appl., 126 (2005), 589-616. doi: 10.1007/s10957-005-5499-z.  Google Scholar

[23]

M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM Journal on Scientific Computing, 25 (2003), 1066-1087. doi: 10.1137/S106482750241459X.  Google Scholar

[24]

M. Herty and A. Klar, Simplified dynamics and optimization of large scale traffic networks, Mathematical Models and Methods in Applied Sciences (M3AS), 14 (2004), 579-601. doi: 10.1142/S0218202504003362.  Google Scholar

[25]

M. Herty and V. Sachers, Adjoint calculus for optimization of gas networks, Networks and Heterogeneous Media, 2 (2007), 733-750. doi: 10.3934/nhm.2007.2.733.  Google Scholar

[26]

H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM Journal on Mathematical Analysis, 26 (1995), 999-1017. doi: 10.1137/S0036141093243289.  Google Scholar

[27]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, 2nd edition, Springer, New York, Berlin, Heidelberg, 2002. doi: 10.1007/978-3-642-56139-9.  Google Scholar

[28]

IBM ILOG CPLEX Optimization Studio, Cplex version 12,, 2010., ().   Google Scholar

[29]

G. S. Jiang, D. Levy, C. T. Lin, S. Osher and E. Tadmor, High-Resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 35 (1998), 2147-2168. doi: 10.1137/S0036142997317560.  Google Scholar

[30]

C. T. Kelley, Iterative Methods for Optimization, Society for Industrial and Applied Mathematics, Philadelphia, 1999. doi: 10.1137/1.9781611970920.  Google Scholar

[31]

C. Kirchner, M. Herty, S. Göttlich and A. Klar, Optimal control for continuous supply network models, Networks Heterogenous Media, 1 (2006), 675-688. doi: 10.3934/nhm.2006.1.675.  Google Scholar

[32]

A. Klar, R. D. Kühne and R. Wegener, Mathematical models for vehicular traffic, Surveys on Mathematics for Industry, 6 (1996), 215-239.  Google Scholar

[33]

O. Kolb, Simulation and Optimization of Gas and Water Supply Networks, Ph.D thesis, Technische Universität Darmstadt, 2011. Google Scholar

[34]

O. Kolb and J. Lang, Simulation and continuous optimization, in Mathematical Optimization of Water Networks (eds. A. Martin, K. Klamroth, J. Lang, G. Leugering, A. Morsi, M. Oberlack, M. Ostrowski and R. Rosen), Internat. Ser. Numer. Math., 162, Birkhäuser/Springer Basel AG, Basel, 2012, 17-33. doi: 10.1007/978-3-0348-0436-3.  Google Scholar

[35]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Royal Society of London Proceedings Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar

[36]

R. Manzo, B. Piccoli and L. Rarita, Optimal distribution of traffic flows at junctions in emergency cases, European Journal of Applied Mathematics, 23 (2012), 515-535. doi: 10.1017/S0956792512000071.  Google Scholar

[37]

G. L. Nemhauser and L. A. Wolsey, Integer and Combinatorial Optimization, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, 1988.  Google Scholar

[38]

G. F. Newell, Traffic Flow on Transportation Networks, MIT Press Series in Transportation Studies, MIT Press, Cambridge, MA, USA, 1980. Google Scholar

[39]

P. Spellucci, Numerische Verfahren der Nichtlinearen Optimierung, Birkhäuser-Verlag, Basel, 1993. doi: 10.1007/978-3-0348-7214-0.  Google Scholar

[40]

P. Spellucci, A new technique for inconsistent QP problems in the SQP method, Mathematical Methods of Operations Research, 47 (1998), 355-400. doi: 10.1007/BF01198402.  Google Scholar

[41]

P. Spellucci, An SQP method for general nonlinear programs using only equality constrained subproblems, Mathematical programming, 82 (1998), 413-448. doi: 10.1007/BF01580078.  Google Scholar

[42]

D. Sun, I. S. Strub and A. M. Bayen, Comparison of the performance of four Eulerian network flow models for strategic air traffic management, Networks and Heterogeneous Media, 2 (2007), 569-595. doi: 10.3934/nhm.2007.2.569.  Google Scholar

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