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A Continuum Model for a Re-entrant Factory

Author

Listed:
  • Dieter Armbruster

    (Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804)

  • Daniel E. Marthaler

    (Northrup Grumman Integrated Systems, Western Region, 17066 Goldentop Road, 9V21/R3-2, San Diego, California 92127-2412)

  • Christian Ringhofer

    (Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804)

  • Karl Kempf

    (Decision Technologies, Intel Corporation, 5000 West Chandler Boulevard, MS CH3-10, Chandler, Arizona 85226)

  • Tae-Chang Jo

    (Mathematics Department, Inha University, 253, Yonghyun-Dong, Nam-Ku, Incheon, 402-751, South Korea)

Abstract

High-volume, multistage continuous production flow through a re-entrant factory is modeled through a conservation law for a continuous-density variable on a continuous-production line augmented by a state equation for the speed of the production along the production line. The resulting nonlinear, nonlocal hyperbolic conservation law allows fast and accurate simulations. Little's law is built into the model. It is argued that the state equation for a re-entrant factory should be nonlinear. Comparisons of simulations of the partial differential equation (PDE) model and discrete-event simulations are presented. A general analysis of the model shows that for any nonlinear state equation there exist two steady states of production below a critical start rate: A high-volume, high-throughput time state and a low-volume, low-throughput time state. The stability of the low-volume state is proved. Output is controlled by adjusting the start rate to a changed demand rate. Two linear factories and a re-entrant factory, each one modeled by a hyperbolic conservation law, are linked to provide proof of concept for efficient supply chain simulations. Instantaneous density and flux through the supply chain as well as work in progress (WIP) and output as a function of time are presented. Extensions to include multiple product flows and preference rules for products and dispatch rules for re-entrant choices are discussed.

Suggested Citation

  • Dieter Armbruster & Daniel E. Marthaler & Christian Ringhofer & Karl Kempf & Tae-Chang Jo, 2006. "A Continuum Model for a Re-entrant Factory," Operations Research, INFORMS, vol. 54(5), pages 933-950, October.
    • Handle: RePEc:inm:oropre:v:54:y:2006:i:5:p:933-950
    DOI: 10.1287/opre.1060.0321
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    References listed on IDEAS

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    1. J. G. Dai & J. H. Vande Vate, 2000. "The Stability of Two-Station Multitype Fluid Networks," Operations Research, INFORMS, vol. 48(5), pages 721-744, October.
    2. Hong Chen & J. Michael Harrison & Avi Mandelbaum & Ann Van Ackere & Lawrence M. Wein, 1988. "Empirical Evaluation of a Queueing Network Model for Semiconductor Wafer Fabrication," Operations Research, INFORMS, vol. 36(2), pages 202-215, April.
    3. J. G. Dai & G. Weiss, 1996. "Stability and Instability of Fluid Models for Reentrant Lines," Mathematics of Operations Research, INFORMS, vol. 21(1), pages 115-134, February.
    4. J. Michael Harrison & Lawrence M. Wein, 1990. "Scheduling Networks of Queues: Heavy Traffic Analysis of a Two-Station Closed Network," Operations Research, INFORMS, vol. 38(6), pages 1052-1064, December.
    5. John D. C. Little, 1961. "A Proof for the Queuing Formula: L = (lambda) W," Operations Research, INFORMS, vol. 9(3), pages 383-387, June.
    6. Lawrence M. Wein, 1990. "Scheduling Networks of Queues: Heavy Traffic Analysis of a Two-Station Network with Controllable Inputs," Operations Research, INFORMS, vol. 38(6), pages 1065-1078, December.
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    Citations

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    Cited by:

    1. Herty, M. & Ringhofer, C., 2007. "Optimization for supply chain models with policies," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 380(C), pages 651-664.
    2. Pihnastyi, Oleh & Yemelianova, Daria & Lysytsia, Dmytro, 2020. "Using PDE-model and system dynamics model for describing multi-operation production lines," MPRA Paper 103975, University Library of Munich, Germany, revised 31 Aug 2020.
    3. Mapundi Banda & Michael Herty, 2012. "Adjoint IMEX-based schemes for control problems governed by hyperbolic conservation laws," Computational Optimization and Applications, Springer, vol. 51(2), pages 909-930, March.
    4. Tanmay Sarkar, 2016. "A numerical study on a nonlinear conservation law model pertaining to manufacturing system," Indian Journal of Pure and Applied Mathematics, Springer, vol. 47(4), pages 655-671, December.
    5. Apostolos Kotsialos, 2010. "A hydrodynamic modelling framework for production networks," Computational Management Science, Springer, vol. 7(1), pages 61-83, January.
    6. Пигнастый, Олег, 2014. "Основы Статистической Теории Построения Континуальных Моделей Производственных Линий [Fundamentals Of The Statistical Theory Of The Construction Of Continuum Models Of Production Lines]," MPRA Paper 95240, University Library of Munich, Germany, revised 20 Aug 2014.
    7. Pihnastyi, Oleh & Korsun, Roman, 2016. "The construction a kinetic equation of the production process," MPRA Paper 92073, University Library of Munich, Germany, revised 19 Mar 2016.
    8. Dong, Ming & He, Fenglan, 2012. "A new continuous model for multiple re-entrant manufacturing systems," European Journal of Operational Research, Elsevier, vol. 223(3), pages 659-668.
    9. Helbing, Dirk & Armbruster, Dieter & Mikhailov, Alexander S. & Lefeber, Erjen, 2006. "Information and material flows in complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 363(1), pages 1-1.
    10. Пигнастый, Олег, 2015. "О Выводе Кинетического Уравнения Производственного Процесса [Derivation of kinetic equations of the production process]," MPRA Paper 93529, University Library of Munich, Germany, revised 07 Aug 2015.
    11. Pihnastyi, Oleh & Khodusov, Valery, 2019. "The optimal control problem for output material flow on a conveyor belt with input accumulating bunker," MPRA Paper 95928, University Library of Munich, Germany, revised 07 Jan 2019.
    12. Yang, Feng & Liu, Jingang, 2012. "Simulation-based transfer function modeling for transient analysis of general queueing systems," European Journal of Operational Research, Elsevier, vol. 223(1), pages 150-166.
    13. Пигнастый, Олег & Ходусов, Валерий, 2017. "Диффузионное Описание Производственного Процесса [Diffusion description of the production process]," MPRA Paper 89250, University Library of Munich, Germany, revised 01 Nov 2017.
    14. Pihnastyi, Oleh & Parachnevych, Oksana, 2018. "On the formulation of the problem of optimal control of production parameters using a two-level model of the production process," MPRA Paper 88761, University Library of Munich, Germany, revised 01 Sep 2018.
    15. Klug, Florian, 2014. "Modelling and analysis of synchronised material flow with fluid dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 393(C), pages 404-417.

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