TY - JOUR

T1 - Multiscale analysis of re-entrant production lines

T2 - An equation-free approach

AU - Zou, Y.

AU - Kevrekidis, I. G.

AU - Armbruster, Hans

N1 - Funding Information:
The research of DA was supported by NSF grant DMS-0204543. IGK and YZ gratefully acknowledge support by DOE and by an NSF/ITR grant.

PY - 2006/4/15

Y1 - 2006/4/15

N2 - The computer-assisted modeling of re-entrant production lines, and, in particular, simulation scalability, is attracting a lot of attention due to the importance of such lines in semiconductor manufacturing. Re-entrant flows lead to competition for processing capacity among the items produced, which significantly impacts their throughput time (TPT). Such production models naturally exhibit two time scales: a short one, characteristic of single items processed through individual machines, and a longer one, characteristic of the response time of the entire factory. Coarse-grained partial differential equations for the spatio-temporal evolution of a "phase density" were obtained through a kinetic theory approach in Armbruster and Ringhofer [Thermalized kinetic and fluid models for re-entrant supply chains, SIAM J. Multiscale Modeling Simul. 3(4) (2005) 782-800.] We take advantage of the time scale separation to directly solve such coarse-grained equations, even when we cannot derive them explicitly, through an equation-free computational approach. Short bursts of appropriately initialized stochastic fine-scale simulation are used to perform coarse projective integration on the phase density. The key step in this process is lifting: the construction of fine-scale, discrete realizations consistent with a given coarse-grained phase density field. We achieve this through computational evaluation of conditional distributions of a "phase velocity" at the limit of large item influxes.

AB - The computer-assisted modeling of re-entrant production lines, and, in particular, simulation scalability, is attracting a lot of attention due to the importance of such lines in semiconductor manufacturing. Re-entrant flows lead to competition for processing capacity among the items produced, which significantly impacts their throughput time (TPT). Such production models naturally exhibit two time scales: a short one, characteristic of single items processed through individual machines, and a longer one, characteristic of the response time of the entire factory. Coarse-grained partial differential equations for the spatio-temporal evolution of a "phase density" were obtained through a kinetic theory approach in Armbruster and Ringhofer [Thermalized kinetic and fluid models for re-entrant supply chains, SIAM J. Multiscale Modeling Simul. 3(4) (2005) 782-800.] We take advantage of the time scale separation to directly solve such coarse-grained equations, even when we cannot derive them explicitly, through an equation-free computational approach. Short bursts of appropriately initialized stochastic fine-scale simulation are used to perform coarse projective integration on the phase density. The key step in this process is lifting: the construction of fine-scale, discrete realizations consistent with a given coarse-grained phase density field. We achieve this through computational evaluation of conditional distributions of a "phase velocity" at the limit of large item influxes.

KW - Coarse projective integration

KW - Equation-free

KW - Production line

KW - Re-entrant

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U2 - 10.1016/j.physa.2006.01.043

DO - 10.1016/j.physa.2006.01.043

M3 - Article

AN - SCOPUS:33645133369

VL - 363

SP - 1

EP - 13

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

IS - 1

ER -