# Parameter Extraction of complex production systems via a kinetic approach

**ABSTRACT** Continuum models of re-entrant production systems are developed that treat the flow of products in analogy to traffic flow. Specifically, the dynamics of material flow through a re -entrant factory via a parabolic conservation law is modeled describing the product density and flux in the factory. The basic idea underlying the approach is to obtain transport coefficients for fluid dynamic models in a multi-scale setting simultaneously from Monte Carlo simulations and actual observations of the physical system, i.e. the factory. Since partial differential equation (PDE) -conservation laws are successfully used for modeling the dynamical behavior of product flow in manufacturing systems, a re -entrant manufacturing system is modeled using a diffusive PDE. The specifics of the production process enter into the velocity and diffusion coefficients of the conservation law. The resulting nonlinear parabolic conservation law model allows fast and accurate simulations. With the traffic flow-like PDE model, the transient behavior of the discrete event simulation (DES) model according to the averaged influx, which is obtained out of discrete event experiments, is predicted. The work brings out an almost universally applicable tool to provide rough estimates of the behavior of complex production systems in non -equilibrium regimes.

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**ABSTRACT:**We develop continuum models of re-entrant factory production systems that treat the flow of products in analogy to traffic flow. Specifically, we model the dynamics of material flow through a re-entrant factory via a parabolic conservation law describing the product density and flux in the factory. We first extract the transport coefficients, in particular, velocity and diffusion coefficients of the particles in the production system using discrete event simulation (DES). Since PDE -conservation laws are successfully used for modeling the dynamical behavior of product flow in manufacturing systems, we model the manufacturing system using a diffusive partial differential equation (PDE). The specifics of the production process enter into the velocity and diffusion coefficient of the conservation law. The resulting nonlinear parabolic conservation law model allows fast and accurate simulations.

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Parameter Extraction of Complex Production

Systems via a Kinetic Approach1

A.K. Unvera,∗,

aUniversity of California Los Angeles, Institute for Pure and Applied

Mathematics, 460 Portola Plaza, Los Angeles, CA 90095-7121.

C. Ringhoferb

bArizona State University, Department of Mathematics and Statistics, Tempe, AZ

85287-1804.

Abstract

Continuum models of re-entrant production systems are developed that treat the

flow of products in analogy to traffic flow. Specifically, the dynamics of material flow

through a re - entrant factory via a parabolic conservation law is modeled describing

the product density and flux in the factory. The basic idea underlying the approach

is to obtain transport coefficients for fluid dynamic models in a multi-scale setting

simultaneously from Monte Carlo simulations and actual observations of the physical

system, i.e. the factory. Since partial differential equation (PDE) - conservation

laws are successfully used for modeling the dynamical behavior of product flow

in manufacturing systems, a re - entrant manufacturing system is modeled using

a diffusive PDE. The specifics of the production process enter into the velocity

and diffusion coefficients of the conservation law. The resulting nonlinear parabolic

conservation law model allows fast and accurate simulations. With the traffic flow-

like PDE model, the transient behavior of the discrete event simulation (DES) model

according to the averaged influx, which is obtained out of discrete event experiments,

is predicted. The work brings out an almost universally applicable tool to provide

rough estimates of the behavior of complex production systems in non - equilibrium

regimes.

Key words: supply chains, re-entrant factory, conservation law

∗Corresponding author.

Email addresses: aunver@ipam.ucla.edu (A.K. Unver), ringhofer@asu.edu

(C. Ringhofer).

1This work was supported by NSF grant DMS-0204543.

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1Introduction

In recent years, factories and production systems have become larger and more

complicated. For this reason, a research endeavor has been initiated to find

time and cost efficient ways of production for such supply chains. Moreover,

a wide range of traffic flow theories and models have been developed to find

the most effective means of production. Our goal in this paper is to present a

computationally efficient way to simulate the production systems.

Understanding the behavior of large supply chains under different polices and

scenarios is a major issue for many businesses today. In large factories, no

experiments can be done involving whole supply chains. Therefore, simulation

models are developed, which substitute for the real environment. Especially

in recent years, fast scalable simulations of production flows in a supply chain

have become a very important research topic. The long term goal of a supply

chain simulation is to optimize and control the production across the whole

supply chain. Since most production deals with individual parts and the pro-

cesses that these parts undergo, discrete event simulators would be the regular

method of choice for accurate simulations.

While discrete event simulators have been highly successful to simulate single

factories, they are computationally too expensive to simulate even a moder-

ately complicated supply chain. Also, they are not scalable to a full supply

chain [1]. Alternative models that endow supply chain nodes with fixed pro-

duction capacities and fixed lead times are not accurate enough since they do

not take into account the fact that capacitated system respond nonlinearly to

increases in demand close to the limit of the production capacity.

The approach followed in this paper is to extract TPTs (Throughput Times)

and WIP (Work in Progress) levels from a discrete event simulation, replacing

actual observations of physical system for the purposes of this work, and then

to transform them into transport coefficients of a macroscopic conservation

law. Using discrete event simulations we estimate the transport coefficients;

namely velocity (convection) - ‘C’ and diffusion - ‘D’. Then, we are able to

solve the diffusive partial differential equation (PDE) numerically. The PDE

model that we use in this project is the conservation law given as follows:

∂tρ(x,t)+∂xF(x,t)=0,F(x,t)=Cρ(x,t)−D∂xρ(x,t)(1.1)

where x and t are space and time variables respectively. Here, ρ(x,t) denotes

the density of the particles and F(x,t) denotes the flux, both of which are

used to model the dynamics of material flow through the re-entrant factory.

The velocity and diffusion coefficients are extracted from observations of the

system (replaced by a discrete event simulation model for the purposes of

this paper). Since PDE models are amenable to optimization and control [10],

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we basically merge the randomness and optimization by using both DES and

PDE models.

We base our theory on the assumption that we have the following type of data

available. The production process consists of M stages. am

at which the lot number n arrives at stage m. Thus, the basic input of our

macroscopic model consists of a table of arrival times am

From this table, the throughput times τm

any part of the supply chain can be easily computed. For instance, given the

arrival times am

ndenotes the time

n, n=1:N, m=1:M.

nfor each stage as well as WIP in

nthe WIP in the m-th supplier Wm(t) is computed as

Wm(t)=

?

n

H(t−am

n)−H(t−am+1

n

)

where H denotes the usual Heaviside function. As lots/particles go through

the stages, we compute their velocities and variances depending on observed

parameters. Since each particle has a different processing time at each stage,

it has a varying velocity through out the system, so this causes a variance in

velocity for each particle. According to fluid dynamics theory, variance results

in diffusion. This results in computing TPTs (Throughput Times), means, and

variances depending on a chosen set of state parameters. Assumed data form

the times that lots have passed point in the production process. At the same

time we record some macroscopic quantity (such as the WIP) of the system

at each of the arrival times am

times, parameterized by this macroscopic quantity. We extract the transport

coefficients; namely, velocity (convection) and diffusion coefficients of the par-

ticles in the production system, using discrete event simulation (DES). De-

tailed information on all features of DES can be found in [5], [8]. The velocity

and diffusion coefficients in (1.1) are then related to the mean and variance

of this distribution. Here, the goal is to form a compact model which reorders

itself to optimization. PDE model has this important advantage, it is fast and

optimization can be applied to obtain more accurate results.

n, giving a statistical distribution of throughput

The contents of the rest of the paper are as follows: In Section 2, we present

theoretical background of traffic flow-like PDE models and how to relate TPTs

and WIPs to velocity and diffusion coefficients. In Section 3, we explain the

model hierarchies that we used to derive the flux function and how we compute

the TPT (Throughput Time) distribution and its mean and variance from the

given data. Finally, in Section 5, we apply the theories that we cover in this

paper on a model problem and give numerical results comparing the discrete

event experiments to the PDE model.

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2Traffic Flow-like (PDE) Models

Traffic flow-like (PDE) models are continuum limits of fluid models. These

models have several advantages: They are scalable and, most important of all,

they are amenable to control and optimization [10]. There are two different

types of PDE modeling: First principle modeling and direct modeling from

observations. In the first principle modeling; on average there exists a func-

tional relation between the flux and the WIP such that the flux is given as

a function of WIP, which is known as a clearing function. Whereas in direct

modeling from observation, which we do in our PDE modeling, the clearing

function is replaced by a probability distribution function that is obtained

from the observed data (DES in our case). After we obtain the data (ran-

domly generated times) from DES, we get the TPT’s and then, we extract

the transport coefficients by using mean and variance of TPT’s. We can then

solve the PDE-conservation law given in Equation (1.1). Our ultimate goal is

to predict and optimize the transient behavior of a given system from obser-

vation without knowing details about the internal mechanisms of the system;

i.e. we always want to know how the system behaves even though the features

of the system is changed.

In our traffic flow-like PDE model, we introduce an artificial variable that

represents each product’s percentage of completion, or stage. The stage x

goes from 0 (0%) to 1 (100%). x=0 denotes the start of a product into the

factory, in other words, the product enters the factory, and x=1 denotes the

end of a product that means the product leaves the factory. The PDE model

yields a density of product as a function of time at a given stage according

to the mass conserving PDE. Also, the arrival rates of the products at stages

yield a flux function. Both the density and flux functions are very important

in PDE modeling such that we use them to model the dynamics of material

flow through the whole factory.

The PDE model that we use in this project is a parabolic conservation law. A

conservation law is a relation asserting that a specific quantity is conserved.

Basically, for a quantity, to be conserved means that whatever enters to the

system has to come out of that system after some time.

In our setting, a conservation law can be defined as a partial differential equa-

tion that expresses the fact that some physical quantity is locally conserved

in a fluid or other continuous physical system, such as energy, momentum, or

the quantity of fluid itself. Conservation law in the integral form is given by

∀a,b∈[0,1] ∂t

?b

aρ(x,t)dx=influx - outflux(2.2)

where ρ(x,t) is the conserved variable. It is the density of products with units

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[parts/space] in the system. The flux is given by F(x,t) with units [parts/time].

The integral in Equation (2.2) gives the number of products per time between

x=a and x=b.

The total number of products in the system can be found by taking the integral

of density of products ρ(x,t) over the stage variable x from 0 to 1. We get the

total WIP W(t) as a function of time as follows

W(t)=

?1

0ρ(x,t)dx.(2.3)

3 Model Hierarchies for Supply Chains

We present a model hierarchy to derive the flux function F given in Equation

(1.1) for a given supply chain. The model hierarchy consists of three parts,

namely; a particle model for the trajectories of individual parts, a kinetic

model for the phase space density, and a diffusion convection equation.

3.1 Particle Models

In particle models, the evolution of a large ensemble of lots (particles) is

modeled by describing the trajectories of each individual lot in phase space

by a set of Newton equations. We introduce the trajectories as x=ξ(t) and

v=∂ξ

which we get through the TPT’s obtained from DES (detailed information

on all features of DES can be found in [5], [8]), so that we obtain a random

variable v from

dP{1

τm

n

Here, n and m denotes the numbers of lots and stages, respectively; and am

represents the times for each of these lots at each stage.

∂t. The velocity v is updated periodically from a velocity distribution V,

=v}=Vm(v,am

n)dv. (3.4)

n

Now, the position ξ(t) after a small time interval dt is given by:

ξ(t+dt)=ξ(t)+dt v(3.5)

where v is the random velocity variable.

Similarly, we have a rule for updating the velocity v according to a random

variable κ, which we calculate by flipping a coin. The probability of κ=1 equals

wdt and the probability of κ=0 is 1−wdt. Here w is the update frequency. In

Section 3.2 (Kinetic Models), we explain how to choose the update frequency

w. Now, according to this rule, if κ=0, we maintain the current velocity, and

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