# American Institute of Mathematical Sciences

September  2016, 11(3): 415-445. doi: 10.3934/nhm.2016003

## On optimization of a highly re-entrant production system

 1 Department of Information Engineering, Electrical Engineering and Applied Mathematics, University of Salerno, Via Giovanni Paolo II, 132, Fisciano (SA) 2 Department of Differential Equations, Dnipropetrovsk National University, Gagarin av., 72, 49010 Dnipropetrovsk 3 Università degli Studi di Salerno, Dipartimento di Ingegneria dell'Informazione, Ingegneria Elettrica e Matematica Applicata, Via Giovanni Paolo II, 132, 84084 Fisciano (SA)

Received  March 2015 Revised  July 2015 Published  August 2016

We discuss the optimal control problem stated as the minimization in the $L^2$-sense of the mismatch between the actual out-flux and a demand forecast for a hyperbolic conservation law that models a highly re-entrant production system. The output of the factory is described as a function of the work in progress and the position of the switch dispatch point (SDP) where we separate the beginning of the factory employing a push policy from the end of the factory, which uses a quasi-pull policy. The main question we discuss in this paper is about the optimal choice of the input in-flux, push and quasi-pull constituents, and the position of SDP.
Citation: Ciro D'Apice, Peter I. Kogut, Rosanna Manzo. On optimization of a highly re-entrant production system. Networks and Heterogeneous Media, 2016, 11 (3) : 415-445. doi: 10.3934/nhm.2016003
##### References:
 [1] A. Adimurthi, S. S. Ghoshal and G. D. Veerappa Gowda, Exact controllability of scalar conser- vation laws with strict convex flux, Math. Control Relat. Fields, 4 (2014), 401-449. doi: 10.3934/mcrf.2014.4.401. [2] F. Ancona and G. M. Coclite, On the attainable set for Temple class systems with boundary controls, SIAM J. Control Optim., 43 (2005), 2166-2190. doi: 10.1137/S0363012902407776. [3] D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math., 66 (2006), 896-920. doi: 10.1137/040604625. [4] D. Armbruster, D. Marthaler, C. Ringhofer, K. Kempf and T. C. Jo, A continuum model for a re-entrant factory, Oper. Res., 54 (2006), 933-950. doi: 10.1287/opre.1060.0321. [5] D. Armbruster, M. Herty, X. Wang and L. Zhao, Integrating release and dispatch policies in production models, Networks and Heterogeneous Media, 10 (2015), 511-526. doi: 10.3934/nhm.2015.10.511. [6] D. Armbruster and C. Ringhofer, Thermalized kinetic and fluid models for reentrant supply chains, Multiscale Model. Simul., 3 (2005), 782-800. doi: 10.1137/030601636. [7] A. Bressan and G. M. Coclite, On the boundary control of systems of conservation laws, SIAM J. Control Optim., 41 (2002), 607-622. doi: 10.1137/S0363012901392529. [8] P. Baiti, P. LeFloch and B. Piccoli, Uniqueness of classical and nonclassical solutions for nonlinear hyperbolic systems, J. Differential Equations, 172 (2001), 59-82. doi: 10.1006/jdeq.2000.3869. [9] A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, Vol. 20, Oxford University Press, Oxford, 2000. [10] A. Bressan, G. Crasta and B. Piccoli, Well-posedness of the Cauchy problem for n systems of conservation laws, Mem. Amer. Math. Soc., 146 (2000), viii+134pp. doi: 10.1090/memo/0694. [11] G. Bretti, C. D'Apice, R. Manzo and B. Piccoli, A continuum-discrete model for supply chains dynamics, Networks and Heterogeneous Media, 2 (2007), 661-694. doi: 10.3934/nhm.2007.2.661. [12] R. Colombo, M. Herty and M. Mercier, Control of the continuity equation with a non local flow, ESAIM: COCV, 17 (2011), 353-379. doi: 10.1051/cocv/2010007. [13] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, Vol. 136, American Mathematical Society, Providence, RI, 2007. [14] J.-M. Coron, O. Glass and Z. Wang, Exact boundary controllability for 1-D quasilinear hyperbolic systems with a vanishing characteristic speed, SIAM J. Control Optim., 48 (2009), 3105-3122. doi: 10.1137/090749268. [15] J.-M. Coron and Z. Wang, Output feedback stabilization for a scalar conservation law with a nonlocal velocity, SIAM J. Math. Analysis., 45 (2013), 2646-2665. doi: 10.1137/120902203. [16] J.-M. Coron and Z. Wang, Controllability for a scalar conservation law with nonlocal velocity, J. Differential Equations, 252 (2012), 181-201. doi: 10.1016/j.jde.2011.08.042. [17] J.-M. Coron, M. Kawski and Z. Wang, Analysis of a conservation law modelling a highly re-entrant manufacturing system, Discrete and Continuous Dynamical Systems, Series B, 14 (2010), 1337-1359. doi: 10.3934/dcdsb.2010.14.1337. [18] C. D'Apice, S. Goettlich, H. Herty and B. Piccoli, Modeling, Simulation and Optimization of Supply Chains, SIAM, Philadelphia PA, USA, 216 pages, 2010. doi: 10.1137/1.9780898717600. [19] C. D'Apice, P. I. Kogut and R. Manzo, On Approximation of Entropy Solutions for One System of Nonlinear Hyperbolic Conservation Laws with Impulse Source Terms, Journal of Control Science and Engineering, 2010 (2010), Article ID 982369, 10 pp. doi: 10.1155/2010/982369. [20] C. D'Apice, P. I. Kogut and R. Manzo, On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains, Networks and Heterogeneous Media, 9 (2014), 501-518. doi: 10.3934/nhm.2014.9.501. [21] C. D'Apice and R. Manzo, A fluid-dynamic model for supply chains, Networks and Heterogeneous Media, 1 (2006), 379-398. doi: 10.3934/nhm.2006.1.379. [22] C. D'Apice, R. Manzo and B. Piccoli, Modelling supply networks with partial differential equations, Quarterly of Applied Mathematics, 67 (2009), 419-440. doi: 10.1090/S0033-569X-09-01129-1. [23] 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. [24] 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. [25] C. D'Apice, R. Manzo and B. Piccoli, Numerical schemes for the optimal input flow of a supply-chain, SIAM Journal on Numerical Analysis, 51 (2013), 2634-2650. doi: 10.1137/120889721. [26] P. Degond, S. Gottlich, M. Herty and A. Klar, A network model for supply chains with multiple policies, Multiscale Model. Simul., 6 (2007), 820-837. doi: 10.1137/060670316. [27] M. Herty, A. Klar and B. Piccoli, Existence of solutions for supply chain models based on partial differential equations, SIAM J. Math. Anal., 39 (2007), 160-173. doi: 10.1137/060659478. [28] T. Horsin, On the controllability of the Burgers equation, ESAIM Control Optim. Calc. Var., 3 (1998), 83-95. doi: 10.1051/cocv:1998103. [29] P. I. Kogut and R. Manzo, Efficient controls for one class of fluid dynamic models, Far East Journal of Applied Mathematics, 46 (2010), 85-119. [30] P. I. Kogut and R. Manzo, On vector-valued approximation of state constrained optimal control problems for nonlinear hyperbolic conservation laws, Journal of Dynamical and Control Systems, 19 (2013), 381-404. doi: 10.1007/s10883-013-9184-5. [31] 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. [32] T. Li, Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Ser. Appl. Math. 3, American Institute of Mathematical Sciences, Springfield, MO, 2010. [33] V. Perrollaz, Exact controllability of scalar conservation laws with an additional control and in the context of entropy solutions, SIAM J. Control Optim., 50 (2012), 2025-2045. doi: 10.1137/110833129. [34] P. Shang and Z. Wang, Analysis and control of a scalar conservation law modelling a highly re-entrant manufacturing system, J. Differential Equations, 250 (2011), 949-982. doi: 10.1016/j.jde.2010.09.003.

show all references

##### References:
 [1] A. Adimurthi, S. S. Ghoshal and G. D. Veerappa Gowda, Exact controllability of scalar conser- vation laws with strict convex flux, Math. Control Relat. Fields, 4 (2014), 401-449. doi: 10.3934/mcrf.2014.4.401. [2] F. Ancona and G. M. Coclite, On the attainable set for Temple class systems with boundary controls, SIAM J. Control Optim., 43 (2005), 2166-2190. doi: 10.1137/S0363012902407776. [3] D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math., 66 (2006), 896-920. doi: 10.1137/040604625. [4] D. Armbruster, D. Marthaler, C. Ringhofer, K. Kempf and T. C. Jo, A continuum model for a re-entrant factory, Oper. Res., 54 (2006), 933-950. doi: 10.1287/opre.1060.0321. [5] D. Armbruster, M. Herty, X. Wang and L. Zhao, Integrating release and dispatch policies in production models, Networks and Heterogeneous Media, 10 (2015), 511-526. doi: 10.3934/nhm.2015.10.511. [6] D. Armbruster and C. Ringhofer, Thermalized kinetic and fluid models for reentrant supply chains, Multiscale Model. Simul., 3 (2005), 782-800. doi: 10.1137/030601636. [7] A. Bressan and G. M. Coclite, On the boundary control of systems of conservation laws, SIAM J. Control Optim., 41 (2002), 607-622. doi: 10.1137/S0363012901392529. [8] P. Baiti, P. LeFloch and B. Piccoli, Uniqueness of classical and nonclassical solutions for nonlinear hyperbolic systems, J. Differential Equations, 172 (2001), 59-82. doi: 10.1006/jdeq.2000.3869. [9] A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, Vol. 20, Oxford University Press, Oxford, 2000. [10] A. Bressan, G. Crasta and B. Piccoli, Well-posedness of the Cauchy problem for n systems of conservation laws, Mem. Amer. Math. Soc., 146 (2000), viii+134pp. doi: 10.1090/memo/0694. [11] G. Bretti, C. D'Apice, R. Manzo and B. Piccoli, A continuum-discrete model for supply chains dynamics, Networks and Heterogeneous Media, 2 (2007), 661-694. doi: 10.3934/nhm.2007.2.661. [12] R. Colombo, M. Herty and M. Mercier, Control of the continuity equation with a non local flow, ESAIM: COCV, 17 (2011), 353-379. doi: 10.1051/cocv/2010007. [13] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, Vol. 136, American Mathematical Society, Providence, RI, 2007. [14] J.-M. Coron, O. Glass and Z. Wang, Exact boundary controllability for 1-D quasilinear hyperbolic systems with a vanishing characteristic speed, SIAM J. Control Optim., 48 (2009), 3105-3122. doi: 10.1137/090749268. [15] J.-M. Coron and Z. Wang, Output feedback stabilization for a scalar conservation law with a nonlocal velocity, SIAM J. Math. Analysis., 45 (2013), 2646-2665. doi: 10.1137/120902203. [16] J.-M. Coron and Z. Wang, Controllability for a scalar conservation law with nonlocal velocity, J. Differential Equations, 252 (2012), 181-201. doi: 10.1016/j.jde.2011.08.042. [17] J.-M. Coron, M. Kawski and Z. Wang, Analysis of a conservation law modelling a highly re-entrant manufacturing system, Discrete and Continuous Dynamical Systems, Series B, 14 (2010), 1337-1359. doi: 10.3934/dcdsb.2010.14.1337. [18] C. D'Apice, S. Goettlich, H. Herty and B. Piccoli, Modeling, Simulation and Optimization of Supply Chains, SIAM, Philadelphia PA, USA, 216 pages, 2010. doi: 10.1137/1.9780898717600. [19] C. D'Apice, P. I. Kogut and R. Manzo, On Approximation of Entropy Solutions for One System of Nonlinear Hyperbolic Conservation Laws with Impulse Source Terms, Journal of Control Science and Engineering, 2010 (2010), Article ID 982369, 10 pp. doi: 10.1155/2010/982369. [20] C. D'Apice, P. I. Kogut and R. Manzo, On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains, Networks and Heterogeneous Media, 9 (2014), 501-518. doi: 10.3934/nhm.2014.9.501. [21] C. D'Apice and R. Manzo, A fluid-dynamic model for supply chains, Networks and Heterogeneous Media, 1 (2006), 379-398. doi: 10.3934/nhm.2006.1.379. [22] C. D'Apice, R. Manzo and B. Piccoli, Modelling supply networks with partial differential equations, Quarterly of Applied Mathematics, 67 (2009), 419-440. doi: 10.1090/S0033-569X-09-01129-1. [23] 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. [24] 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. [25] C. D'Apice, R. Manzo and B. Piccoli, Numerical schemes for the optimal input flow of a supply-chain, SIAM Journal on Numerical Analysis, 51 (2013), 2634-2650. doi: 10.1137/120889721. [26] P. Degond, S. Gottlich, M. Herty and A. Klar, A network model for supply chains with multiple policies, Multiscale Model. Simul., 6 (2007), 820-837. doi: 10.1137/060670316. [27] M. Herty, A. Klar and B. Piccoli, Existence of solutions for supply chain models based on partial differential equations, SIAM J. Math. Anal., 39 (2007), 160-173. doi: 10.1137/060659478. [28] T. Horsin, On the controllability of the Burgers equation, ESAIM Control Optim. Calc. Var., 3 (1998), 83-95. doi: 10.1051/cocv:1998103. [29] P. I. Kogut and R. Manzo, Efficient controls for one class of fluid dynamic models, Far East Journal of Applied Mathematics, 46 (2010), 85-119. [30] P. I. Kogut and R. Manzo, On vector-valued approximation of state constrained optimal control problems for nonlinear hyperbolic conservation laws, Journal of Dynamical and Control Systems, 19 (2013), 381-404. doi: 10.1007/s10883-013-9184-5. [31] 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. [32] T. Li, Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Ser. Appl. Math. 3, American Institute of Mathematical Sciences, Springfield, MO, 2010. [33] V. Perrollaz, Exact controllability of scalar conservation laws with an additional control and in the context of entropy solutions, SIAM J. Control Optim., 50 (2012), 2025-2045. doi: 10.1137/110833129. [34] P. Shang and Z. Wang, Analysis and control of a scalar conservation law modelling a highly re-entrant manufacturing system, J. Differential Equations, 250 (2011), 949-982. doi: 10.1016/j.jde.2010.09.003.
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