A Continuum Model for a Re-entrant Factory

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.

[1]  Lawrence M. Wein,et al.  Scheduling Networks of Queues: Heavy Traffic Analysis of a Multistation Network with Controllable Inputs , 2011, Oper. Res..

[2]  Randolph Nelson,et al.  Probability, Stochastic Processes, and Queueing Theory , 1995 .

[3]  Lawrence M. Wein,et al.  Scheduling Networks of Queues: Heavy Traffic Analysis of a Two-Station Network with Controllable Inputs , 1990, Oper. Res..

[4]  Carl M. Harris,et al.  Fundamentals of queueing theory , 1975 .

[5]  Gideon Weiss,et al.  Stability and Instability of Fluid Models for Reentrant Lines , 1996, Math. Oper. Res..

[6]  Skip Thompson,et al.  Delay-differential equations , 2007, Scholarpedia.

[7]  J D Littler,et al.  A PROOF OF THE QUEUING FORMULA , 1961 .

[8]  S.C.H. Lu,et al.  Efficient scheduling policies to reduce mean and variance of cycle-time in semiconductor manufacturing plants , 1994 .

[9]  K. Kempf,et al.  Simulation of emergent behavior in manufacturing systems , 1995, Proceedings of SEMI Advanced Semiconductor Manufacturing Conference and Workshop.

[10]  G. F. Newell.  Approximate Behavior of Tandem Queues , 1979 .

[11]  Christian A. Ringhofer,et al.  Kinetic and Fluid Model Hierarchies for Supply Chains , 2003, Multiscale Model. Simul..

[12]  Hong Chen,et al.  Empirical Evaluation of a Queueing Network Model for Semiconductor Wafer Fabrication , 1988, Oper. Res..

[13]  G. F. Newell.  Scheduling, Location, Transportation, and Continuum Mechanics: Some Simple Approximations to Optimization Problems , 1973 .

[14]  Carl M. Harris,et al.  Fundamentals of queueing theory (2nd ed.). , 1985 .

[15]  Stephen C. Graves,et al.  A Tactical Planning Model for a Job Shop , 1986, Oper. Res..

[16]  K. Mani Chandy,et al.  Open, Closed, and Mixed Networks of Queues with Different Classes of Customers , 1975, JACM.

[17]  D. Helbing.  TRAFFIC MODELING BY MEANS OF PHYSICAL CONCEPTS. , 1996 .

[18]  A.A.J. Lefeber,et al.  Nonlinear models for control of manufacturing systems , 2003 .

[19]  J. Forrester.  Industrial Dynamics , 1997 .

[20]  David L. Woodruff,et al.  CONWIP: a pull alternative to kanban , 1990 .

[21]  Christian A. Ringhofer,et al.  Thermalized Kinetic and Fluid Models for Reentrant Supply Chains , 2005, Multiscale Model. Simul..

[22]  Randolph D. Nelson,et al.  Probability, stochastic processes, and queueing theory - the mathematics of computer performance modeling , 1995 .

[23]  P. R. Kumar,et al.  Re-entrant lines , 1993, Queueing Syst. Theory Appl..

[24]  G. F. Newell.  Approximation Methods for Queues with Application to the Fixed-Cycle Traffic Light , 1965 .

[25]  M J Lighthill,et al.  On kinematic waves II. A theory of traffic flow on long crowded roads , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[26]  Chad D. DeJong,et al.  Simulating the transport and scheduling of priority lots in semiconductor factories , 2002, Proceedings of the Winter Simulation Conference.

[27]  Carlos F. Daganzo,et al.  A theory of supply chains , 2003 .

[28]  John H. Vande Vate,et al.  The Stability of Two-Station Multitype Fluid Networks , 2000, Oper. Res..

[29]  Christian A. Ringhofer,et al.  A Model for the Dynamics of large Queuing Networks and Supply Chains , 2006, SIAM J. Appl. Math..

[30]  D. Armbruster,et al.  Kinetic and fluid models for supply chains supporting policy attributes , 2007 .

[31]  Edward A. MacNair,et al.  Wafer fabrication: 300mm wafer fabrication line simulation model , 2002, WSC '02.

[32]  Edward A. MacNair,et al.  300 mm wafer fabrication line simulation model , 2002, Proceedings of the Winter Simulation Conference.

[33]  Stanley B. Gershwin,et al.  Short term production scheduling of an automated manufacturing facility , 1984, The 23rd IEEE Conference on Decision and Control.

[34]  R. LeVeque.  Finite Difference Methods for Differential Equations , 2005 .

[35]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[36]  J. Little.  A Proof for the Queuing Formula: L = λW , 1961 .

[37]  V. Rich.  Personal communication , 1989, Nature.

[38]  David J. Goodman,et al.  Personal Communications , 1994, Mobile Communications.

[39]  W. Wang,et al.  A model predictive control strategy for supply chain management in semiconductor manufacturing under uncertainty , 2004, Proceedings of the 2004 American Control Conference.

[40]  P. Bárcia,et al.  A revised bound improvement sequence algorithm , 1988 .