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A Continuum Model for a Re-entrant Factory

Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804
Northrup Grumman Integrated Systems, Western Region, 17066 Goldentop Road, 9V21/R3-2, San Diego, California 92127-2412
Department of Mathematics, Arizona State University, Tempe, Arizona 85287-1804
Decision Technologies, Intel Corporation, 5000 West Chandler Boulevard, MS CH3-10, Chandler, Arizona 85226
Mathematics Department, Inha University, 253, Yonghyun-Dong, Nam-Ku, Incheon, 402-751, South Korea
armbruster{at}asu.edu
daniel.marthaler{at}ngc.com
ringhofer{at}asu.edu
karl.g.kempf{at}intel.com
taechang{at}inha.ac.kr

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.

Subject classifications: production/scheduling: approximations; simulations: efficiency; mathematics.
History: Received January 2003; revision received April 2005; accepted September 2005.