Nonlinear Models for Control of Manufacturing Systems

Lefeber, Aaj Erjen

doi:10.1002/3527602585.ch5

Cited by 11 publications

(10 citation statements)

References 34 publications

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“…That is exactly the limit in which we do not expect the PDE to behave very well. It would be interesting to study how the queueing model approaches a PDE model in the limit of many machines and many steps, but this is clearly beyond the current state of our understanding (but see Lefeber 2004). In the meantime, what can be done is to compare the PDE model with discreteevent simulations.…”

Section: A Relatively Small Re-entrant Factorymentioning

confidence: 99%

“…Standard optimal control approaches in the production and inventory modeling context (e.g., Gershwin et al 1985) cannot solve our problem here: They are based on ODEs that cannot take into account the slowdown of the factory, as the load in the factory increases due to the control actions. Lefeber (2004) suggested an approach based on control theory of delay systems. Whether such an approach will work for the re-entrant factory with its large delays remains to be proven.…”

Section: Controlmentioning

confidence: 99%

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A Continuum Model for a Re-entrant Factory

Armbruster

Marthaler²,

Ringhofer

et al. 2006

Operations Research

104

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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.

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Section: A Relatively Small Re-entrant Factorymentioning

confidence: 99%

Section: Controlmentioning

confidence: 99%

A Continuum Model for a Re-entrant Factory

Armbruster

Marthaler²,

Ringhofer

et al. 2006

Operations Research

104

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“…Models like (1) have been used a lot in literature. Examples of these models are the flow model as initiated by Kimemia and Gershwin 14 for modelling failure-prone manufacturing systems, the fluid models or fluid queues as proposed by queueing theorists 10 , or the stochastic fluid model as introduced by Cassandras et al 8 Recently, a new class of models for manufacturing systems has been introduced 2,3, 16 . In these models, the flow of products through a manufacturing system is modelled in a similar way as the flow of cars on a highway.…”

Section: Modelling Manufacturing Systemsmentioning

confidence: 99%

Modelling Manufacturing Systems for Control: A Validation Study

Lefeber

Berg²,

Rooda³

2005

Networks of Interacting Machines

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This contribution deals with the modelling of manufacturing systems for control. First the concept of effective process times is introduced as a means to arrive at relatively simple discrete event models of manufacturing systems based on measured data. Secondly, a control framework is presented. Thirdly, a validation study is presented which shows that the currently available PDE-models for describing manufacturing systems need further improvement. Finally some criteria are specified which a PDE-model should at least meet in order to be considered valid.

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“…An alternative approach has been developed recently [1,2] extending concepts from gas dynamics and traffic theory to production flows (see also [20,14]). This approach not only will allow very fast simulations, it also allows for hierarchical models that, at least on their lowest levels, can be understood qualitatively, leading to insight into the dynamics of supply chains.…”

Section: Dynamics Of Supply Chainsmentioning

confidence: 99%

Dynamical Systems and Production Systems

Armbruster

2004

Nonlinear Dynamics of Production Systems

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Recent applications of nonlinear dynamics to three models of production systems are reviewed: (i) a reorganization procedure for bucket brigade systems is introduced. With such a procedure the bucket brigade self-organizes when a new worker is added whose skill level increases with time due to learning. (ii) Self-organization in chaotic networks of switched arrival systems is presented. (iii) A new continuum model of production flow through re-entrant factories leading to non-local nonlinear partial differential equations is discussed. Such a model may lead to fast simulations of supply chains as well as improved understanding of the dynamics of supply chains.

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Nonlinear Models for Control of Manufacturing Systems

Cited by 11 publications

References 34 publications

A Continuum Model for a Re-entrant Factory

A Continuum Model for a Re-entrant Factory

Modelling Manufacturing Systems for Control: A Validation Study

Dynamical Systems and Production Systems

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