1

Quality-Reliability Chain Modeling for System-Reliability Analysis of

Complex Manufacturing Processes

Yong Chen and Jionghua Jin

Abstract: System reliability of a manufacturing process should address effects of both the

manufacturing system (MS) component reliability and the product quality. In a multi-station

manufacturing process (MMP), the degradation of MS components at an upstream station can

cause the deterioration of the downstream product quality. At the same time, the system

component reliability can be affected by the deterioration of the incoming product quality of

upstream stations. This kind of quality and reliability interaction characteristics can be observed

in many manufacturing processes such as machining, assembly, and stamping. However, there is

no available model to describe this complex relationship between product quality and MS

component reliability. This paper, considering the unique complex characteristics of MMP,

proposes a new concept of quality and reliability chain (QR-Chain) effect to describe the

complex propagation relationship of the interaction between MS component reliability and

product quality across all stations. Based on this, a general QR-chain model for MMP is

proposed to integrate the product quality and MS component reliability information for system

reliability analysis. For evaluation of system reliability, both the exact analytic solution and a

simpler upper bound solution are provided. The upper bound is proved equal to the exact

solution if the product quality does not have self-improvement, which is generally true in many

MMP. Therefore, the developed QR-chain model and its upper bound solution can be applied to

many MMP.

Index Terms doubly stochastic process, multi-station manufacturing process, quality and

reliability dependency, quality and reliability chain, system reliability evaluation.

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Acronyms

MS manufacturing system

MMP multi-station manufacturing process

QR-Chain quality and reliability chain

TE tooling element

PQC product quality characteristic

DOE design of experiments

NHPP non-homogeneous Poisson process

r.v. random variable

pdf probability distribution function

Notation

n number of process-variables in an MS

m number of product quality characteristics

p number of MS components

l number of noise-variables in the system

g number of s-independent random increments in the component

degradation model

t number of operation cycles, which is an operating time index

Kt production mission time (reliability evaluation lifetime)

)(tq j the jth quality index

)(tiλ catastrophic failure rate of component i at operation t

i0λ initial fixed failure rate

),( baunif the distribution of equal probability of 0.5 at a and b

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nRt ∈)(X vector of process-variables at time t

)(tY Tm tYtYtY )](...)()([ 21 , vector of PQC at time t

[ ]ib the ith element of vector b

a design specification vector for product quality

XF cdf of a random variable X

BA ⊗ Kronecker product of A and B

yx o Hadamard product of two vectors x and y ( [ ] [ ] [ ] iii yxzyxz ⋅=⇒= o )

)(kµ mean vector of )( ktX

0µ initial mean of process-variable at t0

)(kΣ covariance matrix of )( ktX

0Σ initial covariance of process-variable at t0

1. Introduction

1.1 Problem Statement

An MMP is referred as a process to produce a product through a series of stations, which

are used in various industry sectors. Examples of MMP include multi-assembly stations in

automotive body assembly, transfer or progressive dies in stamping processes, and multiple

pattern lithography operations in semiconductor manufacturing. In these processes,

unanticipated machine/tool failures or degradations can cause unanticipated process downtime or

nonconforming products. Therefore, the development of a system reliability model for a

manufacturing process needs to consider both effects of product quality and MS component

reliability.

System reliability is generally defined as the probability that a system performs its

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intended function under operating conditions for a specified period of time [1]. The intended

function of a manufacturing process should consider not only the MS uptime at each station, but

also the produced product quality. In the literature, many research efforts were conducted to

study various reliability models and evaluation methods [2], [3], [4], [5], where system failures

are determined only by MS component failures due to their malfunctions or their degradations

beyond the maximum acceptable amount of tool wear. The dependency of the MS reliability on

the product quality has been overlooked on papers about system reliability modeling of a

manufacturing process.

For a single-station manufacturing process, the concept of the interaction between

product quality and MS component reliability, the QR-Co-Effect, was proposed in [6] for

manufacturing process reliability modeling. Figure 1 shows that the MS component degradation

affects the outgoing product quality. Meanwhile the incoming raw material/part quality can

affect the MS component reliability such as degradation rate and catastrophic failure rate. Based

on this model, [6] proves that there is an overestimation of the manufacturing process reliability

if the QR-Co-Effect is not considered in the system reliability modeling.

In-Coming Product quality

Product Quality

QR Co-Effect

QR Co-Effect

Component Reliability

Tool/Machine Component Reliability

Nonconforming Product Quality

System Reliability & Down-Time Analysis

Component Degradation

Component Catastrophic Failure

Outgoing Product Quality

Figure 1. QR-Co-Effect between product quality and MS component reliability

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In the single-station situation [6], it is reasonable to assume that incoming product quality

is independent of time. Moreover, the component catastrophic failure rate, although affected by

incoming product quality, is also a time invariant variable. Therefore, the event of

nonconforming product quality can be reasonably considered as s-independent of component

catastrophic failure. However, this assumption does not hold in an MMP due to the dependent

propagation of product quality across stations. Therefore, the development of an effective model

to describe the quality & reliability dependency, and its propagation throughout all stations, is

one of the major concerns for system reliability evaluation in an MMP.

1.2 QR-Chain Effect in an MMP

Definitions:

• Manufacturing system a system consisting of several tools and equipments to perform

the designed operations in an MMP and produce products as its end item;

• Component/MS component a physical part of a MS, such as a tool, an equipment, or a

machine. In this paper, component means the component of an MS, rather than the

component of a product;

• Product Both the incoming & outgoing workpieces at each station of an MMP, which

includes the intermediate parts and the final product;

• Component reliability information includes component catastrophic failure

information such as catastrophic failure rate and component degradation information such

as wear rate;

• System catastrophic failure System failure due to component catastrophic failures;

• System failure due to nonconforming products an event that the produced products are

out of specifications. For MMP, the specifications can be generally assigned to the

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intermediate or the final products.

• System reliability of an MMP—the probability that neither the system catastrophic

failure nor the failure due to nonconforming products occurs during a specific period of

time.

In an MMP, each station consists of multiple components. To simplify the problem, all

components in an MMP are assumed to connect in series, that is, the catastrophic failure of any

component will lead to the system catastrophic failure. All the developed models and

methodologies in this paper can be easily adapted to more general hybrid series/parallel systems.

The product variation is propagated in an MMP [7], [8]. The final product quality is

affected by the accumulation or stack up of all variations generated at previous stations.

Considering the QR-Co-Effect at each station, the variation propagation in product quality leads

to the propagation of the interaction between the MS component reliability and the product

quality (as shown in Figure 2), which is called as the QR-Chain effect. The QR-Chain effect is a

unique characteristic of an MMP and will be studied in this paper.

Product Quality

Product Quality

Product Quality

Product Quality

Station 2 Station 3

QR-Co-Effect Variation Propagation

Component Reliability

Component Reliability

Component Reliability

Quality Quality Quality Quality

Station 1

Variation Propagation

Figure 2. General concepts of the QR-Chain in MMPs

The QR-Chain effect can be observed in many MMPs. A few examples are given below

to illustrate the common characteristics of the QR-Chain effect in various manufacturing

processes:

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1. Machining processes: The QR-Chain effect can be observed in a cylinder head

machining process. For an example from [9], as shown in Figure 3, there are two stations

consisting of drilling a hole in a cylinder head (station I) and then tapping a thread on it

afterwards (station II). In station I, the material properties of the incoming workpiece (taken as

the product quality) have an important impact on the wear and breakage rate of the drill (taken as

the component reliability). The drill condition further impacts the quality of the hole drilled in

this station in terms of size, straightness, orientation, etc. In the next tapping station (station II),

those drilled hole quality characteristics of station I are essential factors affecting the thread

quality of station II and the wear and breakage rate of the taper tool. Therefore, the QR-Co-

Effect at station I has propagated to station II.

Y1, Y2, Z, M1, A1, B, and C are datum

Y1, Y2

M1 Z

A1

B C

Station I: Drill bolt hole Station II: Tapping threads in the bolt

Figure 3. QR-Chain in a machining process

2. Transfer or progressive die stamping processes: Similar (to example 1) QR-Chain

effects can be observed in a stamping process with transfer or progressive dies, where multiple

stations are used to form a part. Figure 4 shows a doorknob process consists of 6 stations in a

sequence of i. notch and cutoff, ii. blanking, iii. the 1st draw, iv. the 2nd draw, v. the 3rd draw, and

vi. bulging [10]. In general, the product quality in previous stations impacts the current die or

tool degradation, which further impacts the quality of products produced in the current station.

As an example, the blanking die worn-out at station 2 (taken as component reliability) generates

burr on the edge of the part (taken as the outgoing product quality of station 2). In the following

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draw stations from stations 3 to 5, the size of the burr (taken as incoming product quality) has an

important impact on the draw die wear (taken as component reliability), and a further impact on

the dimension and surface quality of the formed part. In the last bulging station, the combined

effects of the formed part dimension and the size of burr on the part surface have impacts on the

rubber tool wear rate (rubber functions as a die) at the bulging station. Thus, a clear QR-Chain

effect can be observed in this process.

Draw Redraw 2nd-RedrawCutoff Blanking

Doorknob Product

TonnageSensors

Stamping Press

Multiple operations

Bulging

Slide

Blank

Notch Draw Redraw 2nd-RedrawCutoff Blanking

Doorknob Product

TonnageSensors

Stamping Press

Multiple operations

Bulging

Slide

Blank

Notch

Figure 4. QR-Chain in a stamping process with a transfer die

3. Autobody assembly processes: An automotive body assembly consists of more than

200 sheet metal parts and 80 assembly stations. The quality of the assembled part is determined

by the locating-hole on the part and the locating pins on the fixture. The locating error due to the

fixtures’ tolerance or degradation (taken as component reliability) at previous stations is reflected

as the incoming locating-hole deviation/variation of the part at the current assembly station.

Figure 5a shows the hole-center deviation of the raw stamping part (taken as incoming

workpiece quality of the current station) impacts the locating pin degradation or broken rate

(taken as component reliability) of the current station. The combined effects of the locating-hole

deviation of the stamping part and the locating pin degradation lead to outgoing part deviation of

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the current assembly station. Again, the assembled subassembly is used as the input of the next

assembly station. As a result, there is a QR-Co-Effect at each station, and this QR-Co-effect

propagates from one station to the next station in an assembly process. Figure 5b is an example

of a side frame assembly process represented with a hierarchical structure in [11]. The output of

the stations at the lower layer of the hierarchical structure becomes the input of the stations at the

upper layer. And the QR-Co-Effect propagates from the bottom to the top of the hierarchical

structure of an autobody assembly process.

Part 1

Part 2

a. Single station

A-Pillar Inner Panel

Aperture Complete

Rear Quarter Inner Panel

Aperture Outer Complete

Front Aperture Panel

Rear Quarter Outer Panel Panel Assembly Taillamp

Reinf. Qtr. Outer Upper Corner

B-Pillar Inner Panel

Rail Roof Side Panel

Aperture Inner Complete

b. Multiple stations

Figure 5. QR-Chain in the side frame assembly process

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From these MMP examples, the common characteristics of the QR-Chain effect are

summarized as:

a. The degradation of MS components can be modeled by a stochastic process.

b. The quality of the outgoing products at a station depends on the reliability of system

components at the current station as well as the quality of incoming products produced by

the previous stations.

c. The reliability of the MS components at the current station can be affected by incoming

product quality from the previous stations.

d. The stochastic deterioration of the outgoing product quality is caused by the stochastic

degradation of the MS components. The dependent propagation of product quality across

stations causes the s-dependency of the failures of MS components and the system failure

due to nonconforming products. This is a special characteristic for an MMP compared

with a single station process.

1.3 Literature Review of System Reliability Modeling

Fault tree and Markov chain are popular models to evaluate system reliability [12], [13],

[14], [15], [16], [17]. However, these traditional models have the following disadvantages in

studying the QR-Chain effect:

1. They are unable to model the continuous changes of component wear states;

2. They are unable to capture the complex dependency between MS component reliability

and product quality, which is typically a continuous functional relationship;

3. With the QR-Chain effect, strong and complex interactions exist among product quality,

component degradation, and component catastrophic failures. The fault tree model is not

suitable to represent all these interactions;

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4. For an MMP with the QR-Chain effect, there are tens to hundreds of components. The

interactions among these components are very complex. Therefore discretizing the

component degradation state and modeling all component state transitions by a Markov

chain is generally not feasible. Moreover, the resulted Markov chain is also too difficult

to solve.

Because of these 4 reasons, neither the fault tree model nor the Markov chain model is a proper

tool to evaluate system reliability for MMP with the QR-Chain effect.

The ineffectiveness of using the fault tree and the Markov chain to model MMP with the

QR-Chain effect is because they are not designed to capture the failure-generating mechanisms

of components. Over the past few years, some literature devoted to the evolution of a relatively

new class of failure models is based on physics-of-failure and the characteristics of operating

environment. A comprehensive review of this kind of models is in [18]. Among these models,

[19] and [20] studied the item degradation with diffusion processes. The item is considered as

failed when its degradation state reaches a threshold level. Another type of model studies the

covariate induced failure rate processes [21], [22]. In these models, the failure rate of an item is

influenced by a covariate whose evolvement is also a stochastic process. These two types of

models can be used to model the component degradation and the component catastrophic failure

of a manufacturing process, by treating the product quality deterioration as a covariate process

influencing the component catastrophic failures. However, none of these models can be used to

model MMPs with the QR-Chain effect because:

1. Most of these models are developed for the single component failure rather than the

failure of a multi-component and multi-station system.

2. None of these models captures the product quality information. However, as mentioned

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before the failure of an MMP should be defined based on product quality rather than the

degradation state of a single component.

Thus, a model describing the relationship between component performance and product quality,

and a scheme to assess the product quality are the most essential elements in modeling of MMP

with the QR-Chain effect.

This paper introduces a QR-Chain model for MMP to integrate the MS component

degradation and catastrophic failure information with the product quality information. Based on

this model, a general methodology is presented for system reliability analysis in MMP. Section

2 proposes a new system reliability model, i.e., the QR-Chain model, for general MMP. Section

3 performs the system reliability evaluation to obtain an analytic solution for the proposed QR-

Chain model; an upper bound of the system reliability is derived. Section 4 uses an example to

illustrate the analysis procedures and the effectiveness of the proposed methodology.

2. QR-Chain Modeling

Based on the characteristics of the QR-Chain effect summarized in Section 1, this section

discusses the modeling procedures describing the relationship between component performance

and product quality, the degradation process of components, the assessment criteria of product

quality, and the impact of the product quality on the component catastrophic failures.

Assumptions:

1. Discrete part manufacturing processes are considered, where the number of

operation cycles is treated as the time index.

2. The component degradation can be modeled by a discrete approximation to a

diffusion process with mean of the wear linearly dependent on the component

degradation state and its variance as constant.

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3. The wear within a time interval follows Normal distribution.

4. Without loss of generality, the component is in the ideal state if X(t)=0.

5. All components have non-decreasing wear.

6. The product quality can be assessed by the s-expected squared deviation of the

PQC from the target.

7. Pra system component fails during the next operation time, given that it still

works at the current operation time is assumed to be proportional to the linear

combination of the squared deviations of certain PQC.

8. The effects of different PQC on component catastrophic failure are s-independent.

2.1 Relationship between Component Performance and Product Quality

The product quality in an MMP is generally affected by the state of multiple MS

components. The important impact of the MS component performance on the product quality

has been demonstrated in [23]. In [23], the component state is described by process-variables.

Another type of variables affecting the product quality is the random noise that is not determined

by the component state, called the noise-variables. Examples of noise-variables include random

variations of raw material quality and random environmental variations. In general, noise-

variables randomly change from one operation time to the next. Considering further the

interaction between the process-variables and the noise-variables, the following general linear

model is assumed for the PQC Yj(t), j=1,2, …, m, which is called as the process model in this

paper,

mjttηtY tjT

tTj

Tjjj ,...,2,1 ,)()()( =××+×+×+= zΓXzβXα (1)

where lTltttt Rzzz ∈= ],...,,[ 21z is the vector of noise-variables, with mean E(zt) and covariance

matrix Cov(zt) independent of the time index, mjj ,...,2,1, =η are constants, jα and jβ are

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vectors characterizing the effects of X(t) and zt, and jΓ is a matrix characterizing the effects of

the interactions between X(t) and zt.

Remarks: The proposed process model (1) is called the response model in robust parameter

design [24], [25]. In parameter design, the process-variables in (1) are often called the main

effect of the control factors and the noise-variables are called the main effect of the noise factors.

Based on the effect ordering principle in parameter design, the main effect of the control factors,

the main effect of the noise factors, and the control by noise interactions are the most significant

effects on product quality. Thus the linear combination of these three effects, as in (1), is widely

used in robust parameter design to model the PQC. When the physical process knowledge is

available, the process model (1) can be obtained based on the specific physical process models.

Otherwise, it can be generally obtained by using DOE since the process model (1) has the same

structure as the response model in robust parameter design. The term ‘control factor’ is not used

in this paper because the process-variables are exactly determined by their inherent component

degradation states, rather than by externally controlled process parameter setups.

2.2 System Component Degradation

In general, X(t) in (1) change over time due to system component degradation. Let the

time axis be divided into contiguous and uniform intervals of length h, and the successive

endpoints of the intervals be denoted by h, 2h, 3h, …, kh, …. The process-variable vector

X(tk)∈ Rn represents the degradation state of system components at time tk≡kh, k=1,2,…. The

production mission time is hKtK ⋅= . A well-known single item degradation model [19] is

kkkkk tXtXtXtX εσµ ⋅⋅+⋅=−+ )()()()( 1 (2)

where εk, k≥1 is a sequence of independently and identically distributed random variables.

When εk, k≥1 is s-normally distributed, (2) becomes a discrete approximation to a diffusion

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process.

In [4], the attention is restricted to a simpler class of the diffusion processes where µ(⋅)

and σ(⋅) are independent of X and k. For a multivariate version of (2), the following model can

be obtained based on assumption 2:

... ,2 ,1 ,0 ,)()( 1 =×+×=+ ktt kkkkk εGXAX (3)

where gk R∈ε ; Ak and Gk are possibly time-varying, known matrices of appropriate dimension;

1, ≥kkε are i.i.d. and s-normal, with ),(~ Qµε εNk ; ),(~)( 000 ΣµX Nt . Eq (3) generally

describes a Gauss-Markov process. The assumption of the s-normal distribution of kε here is

due to its mathematical tractability and the enormous literature on diffusion processes and its

applications in degradation modeling. For a large scale MMP with a large number of operation

cycles during each time interval, the s-normal distribution assumption can also be justified by the

central limit theorem if the wear increments of each operation cycle during a time interval are

independent. From assumptions 4 and 5, for any 1≤j≤n, Pr[X(t0)]j<0 and

[ ] 0)()(Pr 1 <−+ jkk tt XX can be ignored. In fact, decreasing (or negative) wear is not

meaningful in many applications [18].

2.3 Product Quality Assessment and System Failure due to Nonconforming Products

Based on the degradation model assumed in the previous section, (1) can be rewritten as

mjtttttηtY kktjT

ktTjk

Tjjj ,...,2,1, ,)()()( 1 =<≤××+×+×+= +zΓXzβXα (4)

Under a given manufacturing system component degradation state X(tk), Yj(t) is still a random

variable due to randomness of the noise-variable zt. The product quality, as a system

performance index, is defined by the mean and variance of Yj(t) under a given component state

X(tk). Due to the term )( kTj tXα × in (4), the change of X(tk) over time leads to a mean shift of

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Yj(t). Due to the term tjT

kt zΓX ××)( in (4), the change of )( ktX can also lead to the change of

the variability of Yj(t). Since variability is a very important quality index, it is important to

include the interaction between the process-variable and the noise-variable in our model.

Quality is generally defined as the closeness of a quality characteristic to the target. In

order to capture both the mean shift and the variability information, the product quality can be

assessed as in assumption 6. Following this concept, under given component degradation state

X(tk), qj(t) can be defined as

122 )),(|))((())(|)(())(|))((())(|( +<≤−+=−≡ kkkjjkjkjjkj tttttYEttYVarttYEttq XXXX γγ (5)

where jγ is the target value for the jth PQC. Based on assumption 4, the mean of the PQC

achieves the target value when X(tk)=0. Thus from (4),

)())(|)(( tTjjkjj EttYE zβ0X ⋅+=== ηγ ; (6)

))(()cov())(()cov())(|)(( kTjtj

Tkjt

Tjkj ttttYVar XΓzΓXβzβX ××××+××= . (7)

From (4) and (6),

)())())((()(

))()()(())(|))((( 22

kT

tjjtjjT

k

tjT

kkTjkjj

tEEt

EttttYE

XzΓαzΓαX

zΓXXαX

××+×+×=

××+×=−γ (8)

From (5), (7), and (8), the )(tq j is a quadratic function of X(tk); thus

1 ,,...,2,1 ,)()())(|( +<≤=+××= kkjkjT

kkj tttmjdttttq XBXX (9)

Ttjjtjj

Tjtjj EE ))(())(()cov( zΓαzΓαΓzΓB ×+××++××≡

jtTjjd βzβ ××≡ )cov( .

Bj in (9) is positive semidefinite, and 0≥jd .

For each quality index, there is a threshold value based on the product design

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specifications. Let the threshold for the jth quality index be aj; then define qtE as the event that

all quality indexes are within the specification by time t:

( )Im

jjj

qt taqE

1

0,)(=

≤≤∀≤≡ ττ

Based on this definition, there is no system failure due to nonconforming products by time t if

qtE holds.

2.4 Component Catastrophic Failure and Its Induced System Catastrophic Failure

From assumption 7,

pitt

tttiTii 1,2,..., )),)(())(((

)(,operation at it works|1operation at fails component Pr

0 =−−×+=+

γYγYs

Y

oλ (10)

where mi R∈s , called the QR-coefficient in this paper, has only nonnegative elements;

Tm ]...[ 21 γγγ≡γ . With the failure rate being minimum at γY =)(t (quality

characteristics right on the target), and with assumption 8, the quadratic relationship in (10) can

be considered as a second order approximation of a general functional relationship based on the

Taylor series. By taking s-expectation on the noise-variables in Y(t), and using the definition of

the failure rate, )(tiλ can be written as

))(|(

))(|))(())((())(|))(())(((E(

)(,at it works|)1,( during fails component Pr)(

0

00i

kTii

kTiik

Ti

ki

tt

tttEttt

ttttit

Xqs

XγYγYsXγYγYs

X

×+=

−−×+=−−×+=

+=

λλλ

λoo (11)

where Tkmkkk ttqttqttqtt ))](|())(|())(|([))(|( 21 XXXXq K= .

Define ctE as the event that catastrophic failures never occurred at any of the p system

components by time t. Then there is no system catastrophic failure by time t if ctE holds.

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3. System Reliability Evaluation

Based on the definition of system reliability of an MMP with the QR-Chain effect, the

system reliability at the production mission time, tK is Pr)( ct

qtK KK

EEtR ∩= .

3.1 Challenges in System Reliability Evaluation

The complex interactions among various elements of the QR-Chain model lead to

following challenges in system reliability evaluation:

• s-dependency between qtE and c

tE : Both qtE and c

tE depend on the system component

degradation in an MMP. So they are not s-independent to each other in general:

ctqt

ct

qt EEEEtR PrPrPr)( ⋅≠∩= .

• s-dependency among system component degradation: Generally the degradation of a

system component depends not only on its current state, but also on that of other system

components. So the degradation processes of system components are not s-independent.

• s-dependency among catastrophic failures of system components: The catastrophic

failures of system components at each station of an MMP can depend on the incoming

product quality. The same quality characteristic of the incoming product can affect

multiple system components. Different quality characteristics are generally not s-

independent of each other, because they all depend on the system component degradation

processes in previous stations. As a result, the catastrophic failures of system

components are not s-independent of each other.

• Doubly stochastic property: From (9) & (11), the )(tiλ depends on another stochastic

process ,...2,1),( =ktkX . In the literature, the failure process with its failure rate

depending on another stochastic process is called “doubly stochastic Poisson process”

[26]. Thus the failure process of each manufacturing system component in the QR-Chain

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model is a doubly stochastic Poisson process.

3.2 System Reliability Evaluation of MMPs

Three steps are proposed to evaluate the QR-Chain model:

Step 1. Conditioning on the degradation path of each component, the component

catastrophic failure process, which is a doubly stochastic Poisson process, becomes a NHPP.

And the dependency of qtE and c

tE can also be removed.

Step 2. Uncondition on the degradation paths by implementing s-expectation to the

conditional system reliability calculated in Step 1.

Step 3. Reorganize the integrand in (17) obtained in Step 2 into the form of the pdf of a

Gaussian r. v.

3.2.1 Step 1

A conditional system reliability is evaluated in this step by conditioning on the

degradation path ,...2,1),( =ktkX , or TK

TTTK ttt ])()()([ 10 XXXX K≡ . Conditioning on

KX , qtK

E and ctK

E are s-independent. So the conditional system reliability

KqtK

ctKK KK

EEtR XXX |Pr|Pr)|( ⋅= . (12)

KctK

E X|Pr and KqtK

E X|Pr can be calculated by Results 1 & 2.

Result 1. Let Tmddd ][ 21 K≡d and ( )∑

=

×+≡p

i

Tiic

10 dsλ . There exists a positive

semidefinite matrix UK, such that

( )( )KKTKKKK

ct tcEK

xUxxX ××+⋅−== exp|Pr .

The proof of Result 1 is in Appendix A1.

Result 2. Ω is a domain in nKR ×+ )1( s.t. IIK

k

m

jjjkjk

TK datt

0 1

)()(= =

−≤××⇔∈ xBxΩx . Let

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∈

≡otherwise,0

if,1 ΩxKKI

KKKqt IEK

== xX|Pr .

This result is obvious from the definition of the system failure due to nonconforming products

and (9).

From Result 1, Result 2, and (12),

( )KKKtR xX =| ( )( ) K exp Itc KKTKK ⋅××+⋅−= xUx (13)

3.2.2 Step 2

Unconditioning on XK by implementing s-expectation to (13), then

( )[ ] ( ) ( )KKKKKKKK KK

dFtRtRtR xxXxX XX ∫ ==== ||E)( .

The ∫ in this equation denotes a multi-fold integral. Further from (13),

( )( ) ( )KKKKTKKK K

dFItctR xxUx X⋅××+⋅−= ∫ exp)(

( )( )∫∈

××+⋅=Ωx

X xxUxK

K KKKTKK dFtc )(-exp (14)

From (3), X(tk) is Gaussian. Suppose ))(),((~)( kkNtk ΣµX , then )(kµ and )(kΣ can be

obtained recursively by

011 )0( ,,...,2,1,)1()( µµµGµAµ ==×+−×= −− Kkkk kk ε ,

01111 )0( , ..., ,2 ,1 ,)1()( ΣΣGQGAΣAΣ ==××+×−×= −−−− Kkkk Tkk

Tkk .

There exist constant matrices H(i), i≥1, so that

,...2,1 ,,...,2,1 ,)()()()( ,1 ==×+×××= −++ iKkitt ik

kkikik εHXAAX K (15)

where [ ]TTik

Tk

Tk

ik11

,−++= εεεε K . From (15),

21

)()())(),(cov(),( 1 kttkik kikkik ΣAAXXΣ ×××=≡+ −++ K and ),(),( kikikk T +=+ ΣΣ .

Let [ ]TTTTK K )()1()0( µµµµ K≡ and

≡

)()1,()0,(

),1()1()0,1(

),0()1,0()0(

KKK

K

K

K

ΣΣΣ

ΣΣΣΣΣΣ

Σ

K

MOMM

K

K

.

Since X(t0), X(t1), …, X(tK) are jointly Gaussian,

),(~ KKK N ΣµX (16)

Substitute )( KKF xX in (14) based on (16), then

( )( )( )

( ) ( )∫∈

−+⋅

−××−−

⋅××+⋅−=

Ωx

xµxΣµxΣ

xUxK

KKKKT

KK

K

KnKKTKKK dtctR 1

2

1

2

)1( 2

1exp

2

1exp)(

π

( )∫∈

−+⋅

−××−+××−

⋅⋅−=

Ωx

xµxΣµxxUxΣK

KKKKT

KKKKTK

K

KnK dtc )()(2

1exp

2

1)exp( 1

2

1

2

)1(

π (17)

3.2.3 Step 3

We transform the integrand in (17) to the form of the pdf of another multivariate

Gaussian r.v., KX~

, based on Lemma 1.

Lemma 1. There exists a positive definite matrix KΣ~

, a vector Kµ~ , and a scalar sK>0 such that

KKKKT

KKKKKT

KKKKTK s+−××−=−××−+×× −− )~(

~)~(

2

1)()(

2

1 11 µxΣµxµxΣµxxUx (18)

Lemma 1 is proved in appendix A2. From Lemma 1, the integrand in (17) can be

transformed into the form of a multivariate Gaussian pdf as in Result 3.

Result 3. Let X~

be a )1( +Kn dimension Gaussian r. v. whose pdf is

( )( )

( ) ( )

−××−⋅−

⋅= −

+ KKKT

KK

K

KnK

Kf µxΣµx

Σx

X~~~

2

1exp

~2

1 1

2

1

2

)1(~

π

Then the system reliability can be written as

22

( )∫∈

⋅⋅−⋅⋅−=Ωx

XxΣ

Σ K

KKK

K

KKK dF

stctR ~2

1

2

1

~)exp()exp()( (19)

Eq (19) can be proved by reorganizing the exponential part of the integrand in (17) based on

Lemma 1.

The multi-fold integral in (19) can be calculated by the simulation: Let Ns Gaussian r.v.

be generated with mean Kµ~ and variance KΣ~

, among which N0 generated r.v. fall in the quality

constraint ΩΩΩΩ. Then N0/Ns is an estimate of the integral ∫∈ Ωx

Xx

K

KKdF )(~ .

3.2.4 Self-Improvement of Product Quality and the Upper Bound of System Reliability

The dimension of the integral in (19) is n⋅(K+1), which is generally very large.

Especially the integral dimension depends on the production mission time tK. It generally takes

enormous computation resources to generate r.v. with such a large dimension. However, taking

advantages of the properties of Gaussian random variables, we can appreciably save the

computation resources for the situation that the product quality does not have self-improvement;

this is discussed as follows.

In (9), although X(tk) is non-decreasing and Bj is positive semidefinite, qj(t) is not

monotone non-decreasing over time t in general. If qj(t) is decreasing at t, we say that the

product quality is self-improved at that time. If the product of an MMP does not have self-

improvement, i.e., qj(t) is monotone non-decreasing, the evaluation of the system reliability can

be much easier by taking advantages of the properties of Gaussian r.v.. First, examine (Lemma

2) under what condition the product quality of an MMP does not have self-improvement.

Lemma 2. If all elements of , ..., ,2 ,1 , mjj =B in (9) are nonnegative, then the product quality of

the MMP does not have self-improvement. Particularly, if , ..., ,2 ,1 , mjj =B are diagonal, the

23

product quality of the MMP does not have self-improvement.

If jB is a diagonal matrix, then it has only nonnegative elements, because jB is positive

semidefinite. The proof of Lemma 2 is obvious from (9) and assumption 4 and 5.

If the product quality of an MMP does not have self-improvement, the event that qj(t) is

within the specifications at any time by tK is equivalent to the event that it is within the

specifications at time tK.

∈

≡otherwise,0

)(,1 KKt

tI

K

Ωx

where Im

jjjKjK

TKK dattt

1

)()()(=

−≤××⇔∈ xBxΩx . If the product quality of an MMP does

not have self-improvement, then

KtK II = .

Thus, (19) can be rewritten as

( )∫∈

⋅⋅−⋅−=KK

K

t

KK

K

KKK dF

stctR

ΩxX

xΣΣ )(

~2

1

2

1

~)exp()exp()( (20)

The integral domain of (20) depends only on )( Ktx , not on )( 1−Ktx , )( 2−Ktx , … , )( 0tx .

So the integral can be calculated based on the marginal distribution of )(~

KtX , which is the last n

elements of KX~

. Because KX~

is multivariate Gaussian, the marginal distribution for )(~

KtX can

be directly obtained from the distribution of KX~

. Following this, the n(K+1) fold integral can be

reduced to an n-fold integral as in Result 4.

Result 4.

( )∫∈

⋅⋅−⋅⋅−=KK

K

t

KtK

K

KKK tdF

stctR

ΩxX

xΣΣ )(

)(~2

1

2

1)(

~)exp()exp()( (21)

24

The proof of Result 4 is in Appendix A3. Obtaining the distribution of )(~

KtX in (21) is

discussed in Appendix A4.

In Result 4, the dimension of )(~

KtX is n, which is independent of the mission time index

K . If the distribution of )(~

KtX is known, evaluation of (21) requires the numerical calculation

of an integral of n fold rather than n(K+1) fold. Thus, (21) gives a simplified analytic solution of

the system reliability for the case when the product quality has no self-improvement. This result

is meaningful because the product quality of the majority of MMPs does not have self-

improvement.

In general cases when the product quality can have self-improvement, it is easy to see

that KKK tI ΩX ∈⇒= )(1 , but the converse ( 1)( =⇒∈ KKK It ΩX ) is not true in general. So

(21) tends to overestimate the system reliability in general cases. However, (21) can be treated

as an upper bound estimation of the system reliability, which is much easier to be evaluated than

the exact system reliability in (19).

4. Numerical Analysis

A manufacturing process with 3 stations, in Figure 6, is considered in the numerical

analysis. In this process, there are 3 TE treated as the manufacturing system components at each

station. Let TE1, TE2, and TE3 denote the tooling elements at station 1, station 2, and station 3,

respectively. The performance of these three TEs is described by the process-variables

[ ]TtXtXtXt )()()()( 321≡X . The noise-variables itz , i=1,2,3,4,5,6 (Figure 6), interact with

the process-variables and impact the product quality. There are 4 product quality characteristics

4 ,3 ,2 ,1 ),( =itYi . )(1 tY and )(2 tY are measurements on the outgoing products of station 1 and

station 2, respectively. )(3 tY and )(4 tY are measurements on the final product.

25

Y1(t) Y2(t) Y3(t)

Y4(t)

X1(t) X2(t)

Station 1 Station 2 Station 3

X3(t)

z2t~N(0,0.001) z3t~unif(-1,1) z5t~unif(-1,1) z1t~unif(-1,1) z4t~N(0,0.001) z6t~N(0,0.001)

Figure 6. An example of a three station manufacturing process

4.1 The process model

The process model, (22)-(25), describes the effects of the noise-variables and the process-

variables on the product quality characteristics

tt ztXztXtY 11211 )(483.0)(274.0)( ⋅⋅−+⋅= (22)

tt ztXztXtYtY 324212 )(564.0)(365.0)(372.0)( ⋅+⋅+⋅+⋅= tttt ztXztXzztXtX 32114221 )()(180.0564.0372.0)(365.0)(102.0 ⋅+⋅⋅−⋅+⋅+⋅+⋅= (23)

)(303.0)(229.0)(064.0)(775.0722.0)(303.0)(628.0)( 321536323 tXtXtXztXztXtYtY tt ⋅+⋅+⋅=⋅⋅+⋅−⋅+⋅=

tttttt ztXztXztXzzz 533211642 )(775.0)(628.0)(113.0722.0354.0234.0 ⋅⋅+⋅⋅+⋅⋅−⋅−⋅+⋅+ (24)

tztXtXtY 5334 )(703.0)(426.0)( ⋅⋅−⋅= (25)

The distributions of 6 5, ,4 ,3 ,2 ,1 , =izit are shown in Figure 6. Refer to the general process

model (1), for each )(tYj in this example, 0=jη . And jα , jβ , and jΓ can be obtained by

comparing (22)-(25) with (1) as

−=

000000

000000

00000483.0

1Γ ;

−=

000000

000100

00000180.0

2Γ ;

−=

0775.00000

000628.000

00000113.0

3Γ ;

−=

0703.00000

000000

000000

4Γ ;

26

[ ]T00274.01 =α ; [ ]T0365.0102.02 =α ;

[ ]T303.0229.0064.03 =α ; [ ]T426.0004 =α ;

[ ]T0000101 =β ; [ ]T00564.00372.002 =β ;

[ ]T722.00354.00234.003 −=β ; [ ]T0000004 =β .

4.2 Manufacturing system component degradation model

The following component degradation model is used for this MMP

kkk tt εXX +=+ )()( 1 , k=0, 1, 2, …

where ),(~ Qµε εNk , [ ]T22210 3 ⋅= −εµ ,

⋅= −

5.200

05.20

005.2

10 7Q , and h=1000. The

initial value of the process-variables ),(~)0( 00 ΣµX N , where [ ]T1.01.01.00 =µ and

40 10

5.200

05.20

005.2−⋅

=Σ .

Compared with (3), kkk ∀== , , IGIA .

4.3 Product quality assessment

Based on (9),

1 4, ,3 ,2 ,1 ,)()())(|( +<≤=+××= kkjkjT

kkj tttjdttttq XBXX

jB and jd can be obtained from (9) and the process model (22)-(25) as

=

000

000

00308.0

1B ,

=

000

0133.1037.0

0037.0043.0

2B ,

27

=

692.0069.0019.0

069.0447.0015.0

019.0015.0017.0

3B ,

=

676.000

000

000

4B ,

31 10−=d , 3

2 10457.0 −×=d , 33 10701.0 −×=d , and 04 =d . The thresholds for ))(|(1 kttq X and

))(|(2 kttq X are ∞; the thresholds for ))(|(3 kttq X and ))(|(4 kttq X are both 0.04. That is, the

quality specifications are only assigned for the product quality characteristics on the final

product.

4.4 Component catastrophic failure

Due to the QR-Chain effect, the failure rates of TE2 and TE3 are affected by )(1 tq and

)(2 tq , respectively. In this system,

71 106)( −×=tλ ; )(103106)( 1

472 tqt ⋅×+×= −−λ ; )(103106)( 2

473 tqt ⋅×+×= −−λ .

Comparing this equation with (11),

7030201 106 −×=== λλλ ; [ ]T00001 =s ; [ ]Ts 0002 =s ; [ ]Ts 0003 =s ,

s= 4103 −× for this MMP.

4.5 System reliability evaluation

Since all elements of 4 ,3 ,2 ,1 , =jjB , are nonnegative, from Lemma 2 the product

quality of the manufacturing process in Figure 6 does not have self-improvement. Thus, Result 4

can be used to evaluate the system reliability. Figure 7 shows the reliability plots calculated with

Result 4. Three cases as follows are compared in Figure 7a:

Case i: 0 ,0 == skε , i.e., the component degradation and the impact of product quality on MS

component catastrophic failure rate are ignored.

Case ii: 0 ,0 =≠ skε , i.e., the component degradation is considered; but the impact of product

28

quality on MS component catastrophic failure rate is ignored.

Case iii: 0 ,0 ≠≠ skε , i.e., both component degradation and the impact of product quality on MS

component catastrophic failure rate are considered. Therefore, case iii considers the

QR-Chain effect studied in this paper.

Compared with case iii, both case i and case ii tend to overestimate the system reliability because

they incorrectly neglect the QR-Chain effect existing in an MMP. If a scheduled maintenance

policy is designed based on the results from case i or case ii, many unanticipated failures of the

manufacturing process will be resulted in due to the nonconforming product quality or the impact

of the incoming product quality of each station. A sensitivity analysis for various values of the

QR-coefficient s is in Figure 7b.

0 5 10 15 20 25 30 35 40 45 50 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35 40 45 50 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Reliability

Time Interval

(i) εεεεk=0; s=0

(ii) εεεεk≠0; s=0 (iii) εεεεk≠0; s≠0

Reliability

Time Interval

s=0

s=10-4

s=3×10-4

s=3×10-3

s=10-3

s=5×10-4

a. System reliability in three different cases b. Sensitivity analysis for different s

Figure 7. System reliability and sensitivity analysis

Acknowledgement

In this research, Mr. Y. Chen and Dr. J. Shi are partially supported by the National

Science Foundation (NSF) Grant: NSF-DMI 9713654; Dr. J. Jin is partially supported by the

Faculty Small Grant from the University of Arizona Foundation: UA 4-59617.

29

Appendix

A1. Proof of Result 1

Conditioning on KX , the catastrophic failures of each component are s-independent

following an NHPP. Furthermore, all components are connected in series. Thus,

⋅−= ∑ ∑=

−

=

p

i

K

kkiK

ct htEK

1

1

0

)(exp|Pr λX (A1)

From (9) and (11),

[ ] Kkpitttt k

m

jjji

Tk

Tiik

Tiiki ..., ,1 ,0 , ..., 2, 1, ),()()()(

100 ==×

⋅×+×+=×+= ∑

=XBsXdsqs λλλ

.

So, ( ) [ ]∑ ∑ ∑∑ ∑∑ ∑=

−

= ==

−

==

−

=

⋅×

⋅×+

⋅×+=

⋅p

i

K

k

m

jjji

Tp

i

K

k

Tii

p

i

K

kki khkhht

1

1

0 11

1

00

1

1

0

)()()( XBsXdsλλ

Let [ ]∑∑= =

⋅⋅≡p

i

m

jjjih

1 1

' BsU . Because the elements of si are nonnegative and Bj, j=1,2,…, m are

positive semidefinite, 'U is positive semidefinite. Also

( ) K

p

i

K

k

Tii tchKch ⋅=⋅⋅=⋅×+∑∑

=

−

=1

1

00 dsλ . Thus

( )∑∑ ∑−

==

−

=

××+⋅=

⋅1

0

'

1

1

0

)()(K

kk

TkK

p

i

K

kki tttcht XUXλ , (A2)

Finally, Let

⊗≡

××

××

)()(

)('

)(

nnnKn

nnKKKK 00

0UIU , where I(K×K) is the K×K identity matrix. UK is positive

semidefinite. The n×n zero matrix in UK is for X(tK), which does not contribute to the

catastrophic failure rates of components.

From (A1) and (A2),

( )( )KKTKKK

ct tcEK

XUXX ××+⋅−= exp|Pr .

30

A2. Proof of Lemma 1

1−KΣ is positive definite and UK is positive semidefinite (from Result 1)

⇒ 11 2~ −− +⋅≡ KKK ΣUΣ is positive definite.

Let ( ) ( ) 2~

,2/2/~ 11111 −−−−− +⋅≡××+≡ KKKKKKKK ΣUΣµΣΣUµ and

( ) ( ) ( )( )( ) ( )( )( ) ( )[ ] 02/2/

2/2/2/2/

11

11111

>××+××=

×

×+××−××≡

−−−

−−−−−

KK

T

KKTK

TK

KK

T

KK

T

KTKKK

TKKs

µΣΣUUµ

µΣΣUΣµµΣµ.

Substitute KKK s and~

,~ 1−Σµ , the equation

KKKKT

KK s+−××− − )~(~

)~(2

1 1 µxΣµx )()(2

1 1KKK

TKKKK

TK µxΣµxxUx −××−+××= −

is resulted.

A3. Proof of Result 4

From (20),

( )∫∈

⋅⋅−⋅⋅−=KK

K

t

KK

K

KKK dF

stctR

ΩxX

xΣΣ )(

~2

1

2

1

~)exp()exp()(

( )∫ =∈⋅⋅−⋅⋅−= )(

~|)(

~Pr

~exp)exp( ~2

1

2

1 KKKKKK

K

KK

KdFt

stc xxXΩXΣ

ΣX

( ) ~|)(

~PrE

~exp)exp( ~

2

1

2

1 KKKK

K

KK t

stc

K

XΩXΣΣ

X∈⋅⋅−⋅⋅−=

( ))(

~Pr

~exp)exp( 2

1

2

1 KKK

K

KK t

stc ΩXΣ

Σ∈⋅⋅−⋅⋅−=

( )( )∫∈

⋅⋅−⋅⋅−=KK

K

t

KtK

K

KK tdF

stc

ΩxX

xΣΣ )(

)(~2

1

2

1

~)exp()exp(

31

A4. Distribution of )(~

KtX

Based on the property of multivariate Gaussian r.v., )(~

KtX is Gaussian with

))(~

),(~(~)(~

KKNtK ΣµX , where ))(~

()(~KtEK Xµ ≡ and ))(

~cov()(

~KtK XΣ ≡ . Partition KK µΣ ~ ,

~ as

following,

( ) ( )

( ) ( )

( )

( )

=

=

×

×

××

××

1

1~ 2221

1211~

n

nKK

nnnKn

nnKnKnKK µ2

µ1µΣΣ

ΣΣΣ .

It is easy to see that 22)(~

and )(~ ΣΣµ2µ == KK . So,

( )( )( )

( ) ( ) )()(~)()(~

)(~)(2

1exp

)(~

2

1 1

2

1

2

)(~ KK

TK

nKt

tdKtKKt

K

tdFK

xµxΣµx

Σx

X

−××−−

⋅= −

π

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Yong Chen is currently an assistant professor in the Department of Mechanical and Industrial Engineering at the University of Iowa. He received the B. E. degree in computer science from Tsinghua University, China in 1998, the Master degree in Statistics, and the Ph. D. degree in Industrial & Operations Engineering from the University of Michigan in 2003. His research interests include the fusion of advanced statistics, applied operations research, and engineering knowledge to develop in-process quality and productivity improvement methodologies achieving automatic fault diagnosis, proactive maintenance, and integration of quality and reliability in complex manufacturing systems. Mail: Department of Mechanical and Industrial Engineering, The University of Iowa, Iowa City, IA 52242-1527. Email: yongchen@engineering.uiowa.edu. Jionghua (Judy) Jin received her B.S. and M.S. degree in Mechanical Engineering, both from Southeast University in 1984 and 1987 respectively, and her Ph.D. degree in Industrial and Operations Engineering at the University of Michigan in 1999.

Dr. Jin is currently an assistant professor in the Department of Systems and Industrial Engineering at the University of Arizona. Her research expertise is in the areas of systematic process modeling for variation analysis, automatic feature extraction for monitoring and fault diagnosis, and optimal maintenance decision with the integration of the quality and reliability interaction.

Dr. Jin currently serves on the Editorial Board of IIE Transactions on Quality and Reliability. She is a member of INFORMS, IIE, ASQ, ASME, and SME. She was the recipient of the CAREER Award from National Science Foundation in 2002, and the Best Paper Award from ASME, Manufacturing Engineering Division in 2000.

Mail: Department of Systems & Industrial Engineering, The University of Arizona, P.O. Box 210020, Tucson, AZ 85721-0020. Email: jhjin@sie.arizona.edu.