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January  2015, 20(1): 107-127. doi: 10.3934/dcdsb.2015.20.107

Optimal inflow control of production systems with finite buffers

Simone Göttlich 1, and  Patrick Schindler 1,

1. 

University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany, Germany

Received  October 2013 Revised  July 2014 Published  November 2014

We introduce the optimal inflow control problem for buffer restricted production systems involving a conservation law with discontinuous flux. Based on an appropriate numerical method inspired by the wave front tracking algorithm, we present two techniques to solve the optimal control problem efficiently. A numerical study compares the different optimization procedures and comments on their benefits and drawbacks.
Keywords: numerical schemes, conservation law with discontinuous flux, optimal control., Production systems.
Mathematics Subject Classification: Primary: 90B30, 65Mxx; Secondary: 49K2.
Citation: Simone Göttlich, Patrick Schindler. Optimal inflow control of production systems with finite buffers. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 107-127. doi: 10.3934/dcdsb.2015.20.107
References:
[1]

D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Applied Mathematics, 66 (2006), 896-920. doi: 10.1137/040604625.

[2]

D. Armbruster, S. Göttlich and M. Herty, A scalar conservation law with discontinuous flux for supply chains with finite buffers, SIAM J. Applied Mathematics, 71 (2011), 1070-1087. doi: 10.1137/100809374.

[3]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford University Press, 2000.

[4]

CPLEX, Optimization studio version 12, 2010.

[5]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1.

[6]

C. D'Apice, G. Bretti, R. Manzo and B. Piccoli, A continuum-discrete model for supply chains dynamics, Netw. Heterog. Media, 2 (2007), 661-694. doi: 10.3934/nhm.2007.2.661.

[7]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach, SIAM, Society for Industrial and Applied Mathematics, Philadelphia, 2010. doi: 10.1137/1.9780898717600.

[8]

P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Applied Mathematics, 68 (2007), 59-79. doi: 10.1137/060674302.

[9]

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 J. Sci. Comput., 30 (2008), 1490-1507. doi: 10.1137/060663799.

[10]

M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1915-1938. doi: 10.3934/dcds.2012.32.1915.

[11]

T. Gimse, Conservation laws with discontinuous flux functions, SIAM J. Math. Anal., 24 (1993), 279-289. doi: 10.1137/0524018.

[12]

S. Göttlich, M. Herty and A. Klar, Network models for supply chains, Commun. Math. Sci., 3 (2005), 545-559. doi: 10.4310/CMS.2005.v3.n4.a5.

[13]

S. Göttlich, A. Klar and P. Schindler, Discontinuous conservation laws for production networks with finite buffers, SIAM J. Appl. Math., 73 (2013), 1117-1138. doi: 10.1137/120882573.

[14]

S. Göttlich, O. Kolb and S. Kühn, Optimization for a special class of traffic flow models: Combinatorial and continuous approaches, Netw. Heterog. Media, 9 (2014), 315-334. doi: 10.3934/nhm.2014.9.315.

[15]

S. Göttlich, S. Martin and T. Sickenberger, Time-continuous production networks with random breakdowns, Netw. Heterog. Media, 6 (2011), 695-714. doi: 10.3934/nhm.2011.6.695.

[16]

M. Gugat, M. Herty, A. Klar and G. Leugering, Conservation law constrained optimization based upon Front-Tracking, ESAIM, Math. Model. Numer. Anal., 40 (2006), 939-960. doi: 10.1051/m2an:2006037.

[17]

D. Helbing, S. Lämmer, T. Seidel, P. Seba and T. Platkowski, Physics, stability and dynamics of supply networks, Physical Review E, 70 (2004), 066116. doi: 10.1103/PhysRevE.70.066116.

[18]

M. Herty, C. Joerres and B. Piccoli, Existence of solution to supply chain models based on partial differential equation with discontinuous flux function, J. Math. Analysis and Applications, 401 (2013), 510-517. doi: 10.1016/j.jmaa.2012.12.002.

[19]

M. Herty, A. Kurganov and D. Kurochkin, Numerical method for optimal control problems governed by nonlinear hyperbolic systems of PDEs, Commun. Math. Sci., 13 (2015), 15-48. doi: 10.4310/CMS.2015.v13.n1.a2.

[20]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, 2002. doi: 10.1007/978-3-642-56139-9.

[21]

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

[22]

M. La Marca, D. Armbruster, M. Herty and C. Ringhofer, Control of continuum models of production systems, IEEE Trans. Automat. Control, 55 (2010), 2511-2526. doi: 10.1109/TAC.2010.2046925.

[23]

R. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.

[24]

Y. Lu, S. Wong, M. Wang and C.-W. Shu, The entropy solutions for the Lighthill-Whitham-Richards traffic flow model with a discontinuous flow-density relationship, Transportation Science, 43 (2009), 511-530. doi: 10.1287/trsc.1090.0277.

[25]

S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms, SIAM J. Control Optim., 41 (2002), 740-797. doi: 10.1137/S0363012900370764.

[26]

S. Ulbrich, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws, Syst. Control Lett., 48 (2003), 313-328. doi: 10.1016/S0167-6911(02)00275-X.

[27]

J. Wiens, J. Stockie and J. Williams, Riemann solver for a kinematic wave traffic model with discontinuous flux, J. Comput. Phys., 242 (2013), 1-23. doi: 10.1016/j.jcp.2013.02.024.

show all references

References:
[1]

D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Applied Mathematics, 66 (2006), 896-920. doi: 10.1137/040604625.

[2]

D. Armbruster, S. Göttlich and M. Herty, A scalar conservation law with discontinuous flux for supply chains with finite buffers, SIAM J. Applied Mathematics, 71 (2011), 1070-1087. doi: 10.1137/100809374.

[3]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford University Press, 2000.

[4]

CPLEX, Optimization studio version 12, 2010.

[5]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1.

[6]

C. D'Apice, G. Bretti, R. Manzo and B. Piccoli, A continuum-discrete model for supply chains dynamics, Netw. Heterog. Media, 2 (2007), 661-694. doi: 10.3934/nhm.2007.2.661.

[7]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach, SIAM, Society for Industrial and Applied Mathematics, Philadelphia, 2010. doi: 10.1137/1.9780898717600.

[8]

P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Applied Mathematics, 68 (2007), 59-79. doi: 10.1137/060674302.

[9]

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 J. Sci. Comput., 30 (2008), 1490-1507. doi: 10.1137/060663799.

[10]

M. Garavello and P. Goatin, The Cauchy problem at a node with buffer, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1915-1938. doi: 10.3934/dcds.2012.32.1915.

[11]

T. Gimse, Conservation laws with discontinuous flux functions, SIAM J. Math. Anal., 24 (1993), 279-289. doi: 10.1137/0524018.

[12]

S. Göttlich, M. Herty and A. Klar, Network models for supply chains, Commun. Math. Sci., 3 (2005), 545-559. doi: 10.4310/CMS.2005.v3.n4.a5.

[13]

S. Göttlich, A. Klar and P. Schindler, Discontinuous conservation laws for production networks with finite buffers, SIAM J. Appl. Math., 73 (2013), 1117-1138. doi: 10.1137/120882573.

[14]

S. Göttlich, O. Kolb and S. Kühn, Optimization for a special class of traffic flow models: Combinatorial and continuous approaches, Netw. Heterog. Media, 9 (2014), 315-334. doi: 10.3934/nhm.2014.9.315.

[15]

S. Göttlich, S. Martin and T. Sickenberger, Time-continuous production networks with random breakdowns, Netw. Heterog. Media, 6 (2011), 695-714. doi: 10.3934/nhm.2011.6.695.

[16]

M. Gugat, M. Herty, A. Klar and G. Leugering, Conservation law constrained optimization based upon Front-Tracking, ESAIM, Math. Model. Numer. Anal., 40 (2006), 939-960. doi: 10.1051/m2an:2006037.

[17]

D. Helbing, S. Lämmer, T. Seidel, P. Seba and T. Platkowski, Physics, stability and dynamics of supply networks, Physical Review E, 70 (2004), 066116. doi: 10.1103/PhysRevE.70.066116.

[18]

M. Herty, C. Joerres and B. Piccoli, Existence of solution to supply chain models based on partial differential equation with discontinuous flux function, J. Math. Analysis and Applications, 401 (2013), 510-517. doi: 10.1016/j.jmaa.2012.12.002.

[19]

M. Herty, A. Kurganov and D. Kurochkin, Numerical method for optimal control problems governed by nonlinear hyperbolic systems of PDEs, Commun. Math. Sci., 13 (2015), 15-48. doi: 10.4310/CMS.2015.v13.n1.a2.

[20]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, 2002. doi: 10.1007/978-3-642-56139-9.

[21]

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

[22]

M. La Marca, D. Armbruster, M. Herty and C. Ringhofer, Control of continuum models of production systems, IEEE Trans. Automat. Control, 55 (2010), 2511-2526. doi: 10.1109/TAC.2010.2046925.

[23]

R. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.

[24]

Y. Lu, S. Wong, M. Wang and C.-W. Shu, The entropy solutions for the Lighthill-Whitham-Richards traffic flow model with a discontinuous flow-density relationship, Transportation Science, 43 (2009), 511-530. doi: 10.1287/trsc.1090.0277.

[25]

S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms, SIAM J. Control Optim., 41 (2002), 740-797. doi: 10.1137/S0363012900370764.

[26]

S. Ulbrich, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws, Syst. Control Lett., 48 (2003), 313-328. doi: 10.1016/S0167-6911(02)00275-X.

[27]

J. Wiens, J. Stockie and J. Williams, Riemann solver for a kinematic wave traffic model with discontinuous flux, J. Comput. Phys., 242 (2013), 1-23. doi: 10.1016/j.jcp.2013.02.024.

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