A Continuum Model for a Re-entrant Factory

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.

Thermalized Kinetic and Fluid Models for Reentrant Supply Chains

“…Figure 5.7 shows this ratio as a function of time. For the coefficient of variation identically equal to zero (pure convection) the differential equation would reduce to the simple model discussed in [2]. Figure 5.7 shows that the variance of the TPT distribution contributes significantly to the structure of the solution in the transient phase.…”

“…The essential asymptotic tool will be the Chapman-Enskog expansion [4] of the Boltzmann equation for a regime where the average TPT and the time intervals between velocity updates in the random phase are short compared to the overall time scale. Fluid equations for the simulation of TPTs have been studied in [2] and [3], leading to simple convection equations for the density of lots as a function of DOC. We essentially recover these results but obtain as a generalization an additional diffusive term arising from the variance of the TPT distribution T .…”

“…This competition will be governed by a given set of policies used by supplier S m , such as FIFO, PUSH, or PULL [13], which we assume to be given. In particular, it is observed experimentally that for reentrant suppliers there exists a critical load, i.e., a critical value of W m beyond which the TPT τ m n grows rapidly and actually may develop a pole [2]. It is therefore inadequate to compute τ ].…”

Control of Continuum Models of Production Systems

“…ECENTLY Armbruster et al [1] have introduced a continuum model to simulate the average behavior of production systems at an aggregate level. Such a description is appropriate, e.g., for a semiconductor fab which produces a large number of items in a large number of steps.…”

“…We choose the velocity as a function of the total load in the production line describing the equilibrium velocity of the factory as a whole [1]. Note that the velocity function is positive, bounded and monotonically decreasing.…”