quality-reliability chain modeling for system-reliability...
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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: [email protected]. 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: [email protected].