# American Institute of Mathematical Sciences

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

## Optimal inflow control of production systems with finite buffers

 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.
Citation: Simone Göttlich, Patrick Schindler. Optimal inflow control of production systems with finite buffers. Discrete & 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.  Google Scholar [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.  Google Scholar [3] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford University Press, 2000.  Google Scholar [4] CPLEX, Optimization studio version 12,, 2010., ().   Google Scholar [5] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] T. Gimse, Conservation laws with discontinuous flux functions, SIAM J. Math. Anal., 24 (1993), 279-289. doi: 10.1137/0524018.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, 2002. doi: 10.1007/978-3-642-56139-9.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] R. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford University Press, 2000.  Google Scholar [4] CPLEX, Optimization studio version 12,, 2010., ().   Google Scholar [5] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd edition, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] T. Gimse, Conservation laws with discontinuous flux functions, SIAM J. Math. Anal., 24 (1993), 279-289. doi: 10.1137/0524018.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer, 2002. doi: 10.1007/978-3-642-56139-9.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] R. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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