American Institute of Mathematical Sciences

Advanced
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

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 & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Armbruster, C. Ringhofer and T. Jo, Continuous models for production flows, in Proceedings of the 2004 American Control Conference, Boston, 4589-4594. Google Scholar

[5]

J. Banks, J. Carson and B. Nelson, Discrete Event System Simulation, Prentice Hall, 1999. Google Scholar

[6]

S. Brush, Kinetic Theory, Pergamon Press, 1972. Google Scholar

[7]

C. Cercignani, Boltzmann Equation and its Applications, Applied Mathematical Sciences, 67, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[8]

G. Fishman, Discrete-Event Simulation, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3552-9.  Google Scholar

[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.  Google Scholar

[10]

J. Nocedal and S. Wright, Numerical Optimization, 2nd edition, Springer-Verlag, New York, 2006.  Google Scholar

[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. Google Scholar

[12]

A. Unver, C. Ringhofer and D. Armbruster, A hyperbolic relaxation model for product flow in complex production networks, AIMS Procceedings, (2009), 790-799.  Google Scholar

[13]

A. Unver, Observation Based PDE Models for Stochastic Production Systems, Thesis (Ph.D.), Arizona State University, 2008.  Google Scholar

show all references

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Armbruster, C. Ringhofer and T. Jo, Continuous models for production flows, in Proceedings of the 2004 American Control Conference, Boston, 4589-4594. Google Scholar

[5]

J. Banks, J. Carson and B. Nelson, Discrete Event System Simulation, Prentice Hall, 1999. Google Scholar

[6]

S. Brush, Kinetic Theory, Pergamon Press, 1972. Google Scholar

[7]

C. Cercignani, Boltzmann Equation and its Applications, Applied Mathematical Sciences, 67, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[8]

G. Fishman, Discrete-Event Simulation, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3552-9.  Google Scholar

[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.  Google Scholar

[10]

J. Nocedal and S. Wright, Numerical Optimization, 2nd edition, Springer-Verlag, New York, 2006.  Google Scholar

[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. Google Scholar

[12]

A. Unver, C. Ringhofer and D. Armbruster, A hyperbolic relaxation model for product flow in complex production networks, AIMS Procceedings, (2009), 790-799.  Google Scholar

[13]

A. Unver, Observation Based PDE Models for Stochastic Production Systems, Thesis (Ph.D.), Arizona State University, 2008.  Google Scholar

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