Parameter extraction of complex production systems via a kinetic approach

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June 2016, 9(2): 407-427. doi: 10.3934/krm.2016.9.407

Parameter extraction of complex production systems via a kinetic approach

Ali K. Unver ^1, , Christian Ringhofer ^2, and M. Emir Koksal ^3,

1.	Intel Corporation, Assembly & Test Technology Development, 5000 W. Chandler Blvd, Chandler, AZ 85226, United States
2.	Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804
3.	Ondokuz Mayis University, Department of Mathematics, Samsun, 55139, Turkey

Received May 2015 Revised November 2015 Published March 2016

Abstract
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Continuum models of re-entrant production systems are developed that treat the flow of products in analogy to traffic flow. Specifically, the dynamics of material flow through a re-entrant factory via a parabolic conservation law is modeled describing the product density and flux in the factory. The basic idea underlying the approach is to obtain transport coefficients for fluid dynamic models in a multi-scale setting simultaneously from Monte Carlo simulations and actual observations of the physical system, i.e. the factory. Since partial differential equation (PDE) conservation laws are successfully used for modeling the dynamical behavior of product flow in manufacturing systems, a re-entrant manufacturing system is modeled using a diffusive PDE. The specifics of the production process enter into the velocity and diffusion coefficients of the conservation law. The resulting nonlinear parabolic conservation law model allows fast and accurate simulations. With the traffic flow-like PDE model, the transient behavior of the discrete event simulation (DES) model according to the averaged influx, which is obtained out of discrete event experiments, is predicted. The work brings out an almost universally applicable tool to provide rough estimates of the behavior of complex production systems in non-equilibrium regimes.

Keywords: conservation law, Supply chains, nonlinear parabolic PDE, statistics., re-entrant factory, numerical analysis.

Mathematics Subject Classification: Primary: 65N35, 35K55; Secondary: 62P3.

Citation: Ali K. Unver, Christian Ringhofer, M. Emir Koksal. Parameter extraction of complex production systems via a kinetic approach. Kinetic and Related Models, 2016, 9 (2) : 407-427. doi: 10.3934/krm.2016.9.407

References:

[1]	D. Armbruster, D. Marthaler and C. Ringhofer, A mesoscopic approach to the simulation of semiconductor supply chains, Proceedings of the International Conference on Modeling and Analysis of Semiconductor Manufacturing (MASM2002) (eds. G. Mackulak et al.), 365-369.
[2]	D. Armbruster, D. Marthaler and C. Ringhofer, Kinetic and fluid model hierarchies for supply chains, SIAM J. on Multiscale Modeling, 2 (2003), 43-61. doi: 10.1137/S1540345902419616.
[3]	D. Armbruster and C. Ringhofer, Thermalized kinetic and fluid models for re-entrant supply chains, SIAM J. Multiscale Modeling and Simulation, 3 (2005), 782-800. doi: 10.1137/030601636.
[4]	D. Armbruster, C. Ringhofer and T. Jo, Continuous models for production flows, in Proceedings of the 2004 American Control Conference, Boston, 4589-4594.
[5]	J. Banks, J. Carson and B. Nelson, Discrete Event System Simulation, Prentice Hall, 1999.
[6]	S. Brush, Kinetic Theory, Pergamon Press, 1972.
[7]	C. Cercignani, Boltzmann Equation and its Applications, Applied Mathematical Sciences, 67, 1988. doi: 10.1007/978-1-4612-1039-9.
[8]	G. Fishman, Discrete-Event Simulation, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3552-9.
[9]	D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.
[10]	J. Nocedal and S. Wright, Numerical Optimization, 2nd edition, Springer-Verlag, New York, 2006.
[11]	A. Unver and C. Ringhofer, Estimation of transport coefficients in re-entrant factory models, 15th IFAC Symposium on System Identification, SYSID France, (2009), 705-710.
[12]	A. Unver, C. Ringhofer and D. Armbruster, A hyperbolic relaxation model for product flow in complex production networks, AIMS Procceedings, (2009), 790-799.
[13]	A. Unver, Observation Based PDE Models for Stochastic Production Systems, Thesis (Ph.D.), Arizona State University, 2008.

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References:

[1]	D. Armbruster, D. Marthaler and C. Ringhofer, A mesoscopic approach to the simulation of semiconductor supply chains, Proceedings of the International Conference on Modeling and Analysis of Semiconductor Manufacturing (MASM2002) (eds. G. Mackulak et al.), 365-369.
[2]	D. Armbruster, D. Marthaler and C. Ringhofer, Kinetic and fluid model hierarchies for supply chains, SIAM J. on Multiscale Modeling, 2 (2003), 43-61. doi: 10.1137/S1540345902419616.
[3]	D. Armbruster and C. Ringhofer, Thermalized kinetic and fluid models for re-entrant supply chains, SIAM J. Multiscale Modeling and Simulation, 3 (2005), 782-800. doi: 10.1137/030601636.
[4]	D. Armbruster, C. Ringhofer and T. Jo, Continuous models for production flows, in Proceedings of the 2004 American Control Conference, Boston, 4589-4594.
[5]	J. Banks, J. Carson and B. Nelson, Discrete Event System Simulation, Prentice Hall, 1999.
[6]	S. Brush, Kinetic Theory, Pergamon Press, 1972.
[7]	C. Cercignani, Boltzmann Equation and its Applications, Applied Mathematical Sciences, 67, 1988. doi: 10.1007/978-1-4612-1039-9.
[8]	G. Fishman, Discrete-Event Simulation, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3552-9.
[9]	D. Helbing, Traffic and related self-driven many-particle systems, Reviews of Modern Physics, 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.
[10]	J. Nocedal and S. Wright, Numerical Optimization, 2nd edition, Springer-Verlag, New York, 2006.
[11]	A. Unver and C. Ringhofer, Estimation of transport coefficients in re-entrant factory models, 15th IFAC Symposium on System Identification, SYSID France, (2009), 705-710.
[12]	A. Unver, C. Ringhofer and D. Armbruster, A hyperbolic relaxation model for product flow in complex production networks, AIMS Procceedings, (2009), 790-799.
[13]	A. Unver, Observation Based PDE Models for Stochastic Production Systems, Thesis (Ph.D.), Arizona State University, 2008.

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