# A Unified Approach for Predicting Long- and Short-Term  Capability Indices with Dependence on Manufacturing Target Bias

Authors: Nikhil T. Satyala; R. J. Pieper
Journal: International Journal of Quality, Statistics, and Reliability
 (2008)

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## Abstract

It is shown that the exact solution for the capability index (CPI) for Gaussian-distributed process with target bias can be expressed in terms of an unbiased CPI and a normalized target bias. The principal advantage of this specific formulation is that it facilitates evaluation of the degradation of the capability of the process due to bias between process mean and the process target. It is shown how this formalism, initially developed for the short-term process, is readily extended to long-term process for which the distribution is Gaussian. Readily isolated in the latter case are the two long-term CPI degrading effects, namely, process instability and target bias. Sufficient conditions to guarantee that long-term processes are distributed as Gaussian are discussed. Within the context of these assumed conditions, a new paradigm for a long-term locator ‘‘k’’ is proposed. For a three sigma process the results indicate that the exact CPI model is a less pessimistic predictor than both of the industry CPI models tested.


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## Body

## 1. Introduction

In 1979, Taguchi and Wu [1] introduced a
viewpoint on estimating the loss (in monetary units, i.e., cost) associated  with lack of precision and accuracy in a manufacturing process. The preeminent manufacturing  precursor to the viewpoint introduced by Taguchi is the classical
“goal-post model” in which the only consideration to production cost is whether
the product parameters fall within the process specification limits. Consistent
with the goal-post philosophy, the level of process control is typically
characterized in terms of what are known as capability indices (CPIs) [2].
Capability indices provide a numerical assessment of the ability of a process
in attaining the predefined specifications [3, 4].A manufacturing process would commonly be
described in terms of three parameters: the finite target value (τ), an upper specification limit (USL), and a
lower specification limit (LSL). All the parts for which the measured value “x”
for a certain specification exceeds the USL or falls below the LSL are
rejected. If the process target value for the product characteristic is
centered between the USL and LSL, then the tolerances are said to be symmetric.
The capability index is of interest to the manufacturing community because it
consolidates the details in a complicated multifaceted manufacturing process
down to one quantity which can be used to predict the fraction of parts
rejected. Typical capability index values can range from 0.7 to 2.0. In the jargon of the community, three sigma process
would correspond to a capability index of 1.0 while a much improved six sigma
process suggested originally by Motorola would correspond to a capability index
of 2.0 [2]. The standard deviation of the process is indicative of the level of
precision. The absolute value of the difference between the distribution mean
and the process target (i.e., target bias) is indicative of the process
accuracy. According to the Taguchi guide for improved manufacturing quality, it
is much easier to adjust the manufacturing process to improve accuracy than to
adjust the process to improve the precision [2].  The most commonly assumed probabilistic
distribution for a product characteristic with measured value “x” is the normal
distribution which can be defined in terms of mean μ, and a standard deviation σ.The primary situation for the application
of asymmetric tolerances [5] occurs when the product parameter of interest
exhibits a skewed distribution [6]. Historically, capability indices were first
applied under the assumption that the mean of the process is on the target [2].
Target bias is at best approximately zero. In some practical cases, it maybe
necessary to consider the impact of the distribution mean of the product
parameter being off target. There have been a variety of target-bias-dependent
capability index models introduced. A noncomprehensive but high profile list of
such models has been assembled for purposes of this paper.What follows first is a brief qualitative
review of capability indices that are assuming zero target bias. The short-term
capability index,Cp, is gauged within a relatively narrow window of
time. The long-term capability index [2], Cpk can be found in the
literature to be applied in two ways. It could be applied to extend the short-term
capability concept (e.g., measured over days or even hours) to long-term (e.g., measured
over weeks or months).  It is assumed, in
this case, that process mean shifts around the target but on the average is “on
target.” The concept is that a time-wise shifting around in the short-term
process is accounted for with a probability density function (PDF) averaging
leading to a higher standard deviation. The long-term precision in the
manufacturing process is degraded relative to short-term and, therefore, the long-term
capability index is lower than the short-term capability index.On the other hand,Cpk has also
found utility as a capability index that can include the impact of target bias
[2, 7]. However, as pointed out in [6], this type of usage of Cpk to account for the target bias is questionable.Lastly, a third-measured
paradigm for a capability index, Cpm paradigm [2, 7, 8] is also
commonly invoked in the community to account for target bias. Because Cpm can be related to Taguchi loss functions [2], it is sometimes referred to as
the Taguchi index. One advantage of the Cpm approach is that it is
nonparametric, that is, makes no a priori assumptions on the underlying
distribution of the specified product parameter distribution.It has been shown [5] that a
probabilistic description of the manufacturing process can be used to predict
the exact dependence for the fraction of rejected components and related to a
CPI. This has been done under assumptions of a normal
(Gaussian) distribution for the process
product and symmetric specification limits. The derived CPI with target bias
was shown to be expressed in terms of four parameters, process mean, process
target, upper specification limit, and lower specification limit. Additionally,
it has been demonstrated that this exact solution is equivalent to a reparameterized
solution expressed in terms of appropriately defined upper and lower capability
indices [5].A target bias-dependent
capability index (CPI) for the symmetric-specification limit
Gaussian-distributed process is proposed and tested. It is shown that various
exact expressions reported in the literature are equivalent to a proposed short-term CPI model dependent on only two parameters, unbiased
short-term CPI and a normalized target bias. One advantage of this particular
formulation is that it facilitates the evaluation of the degradation of the
capability of the process due to an offset between the mean and target of a process
parameter. The second advantage of this parameterization is that it allows for
a convenient comparison of the exact CPI model with two other commonly used
industry models which also estimate the CPI with target offset. A third
advantage is that the proposed formalization facilitates setting up a CPI model
for the Gaussian-distributed long-term process with a methodology unified in
approach with that of the proposed short-term CPI model. Readily isolated are the
two long-term CPI degrading effects, namely, long-term process instability and
target bias. Sufficient conditions to guarantee that the long-term processes
are distributed as Gaussian are discussed. Within the context of these assumed
conditions, a new paradigm for a long-term locator “k” is proposed.Two
implementation schemes for the proposed model are discussed. One method is
based on the availability of numerical built-in mathematical routines for the
error function and its inverse. The second scheme supplants the built-in
functions used in the first scheme with recently reported analytical approximations
[9, 10]. For a three sigma process, the results indicate that the exact CPI model
is a less pessimistic predictor than both of the industry CPI models
tested.


## 2. Background

### 2.1. Background on CPI Model

In general,
the measurements for the process parameter to meet desired specifications are
characterized by a distribution having a meanμ and a standard deviation σ [2].  The process has an upper specification limit
(USL) and a lower specification limit (LSL).
The distance between the USL and LSL, as represented on Figure 1, is 2Δ.
The specifications are considered to be symmetric if the target satisfies the
condition τ=(USL+LSL)/2 [2]. The defining recipe for the
capability index intended for situations for which the measured parameter
distribution is normal and has symmetric limits is (1)Cpo=USL−LSL6σ=Δ3σ.  The subscript
“o” in (1) indicates that it does not account for any target bias. Generalizations
of (1) to cover asymmetry in tolerances and nonnormal distributed parameters can
be found in the literature [6].  The
capability index is a direct measure of the process control and relates Cpo to a fraction of rejection, (2)po=2Φ(−3Cpo), where Φ(z) is the standard normal cumulative density
function (CDF). However, (2) can be applied to a process with normal
distribution, no target bias, and symmetric limits.Figure 1
PDF showing manufacturing specifications and rejected fraction of parts.


### 2.2. Distribution Independent Observation

Independent of whether the distribution
is normal with symmetric limits or not, the fraction rejected, also known as
the “component of nonconformity” [6] can be computationally predicted by evaluating
the CDF of the process distribution at selected points. As suggested by Figure 1, this prediction rule is given by [5] (3)p=F(LSL)+(1−F(USL)). This combines
the parts that do not meet the specifications, that is, the parts that have a
measured product characteristic “x” which is either lower than the LSL or higher
than the USL. In the spirit of (2), the generalized process capability index
should be consistent with the rule (4)Cp=−13Φ−1(p2),  where for the normal distribution
with symmetric limits,  po=p, and (4) reduces to special case described by (2).
From Figure 1, it follows that (5)USL=τ+Δ,LSL=τ+Δ.    Consistent
with (3) and (4), it can be shown that [5] (6)p=Φ(LSL−μσ)+Φ(−(USL−μσ)). However, (6)
conjunction with (4) produces an exact short-term capability index model which
agrees with the Boyles [5] yield index model. Demonstration details are
provided in Appendix 1.


## 2.1. Background on CPI Model

In general,
the measurements for the process parameter to meet desired specifications are
characterized by a distribution having a meanμ and a standard deviation σ [2].  The process has an upper specification limit
(USL) and a lower specification limit (LSL).
The distance between the USL and LSL, as represented on Figure 1, is 2Δ.
The specifications are considered to be symmetric if the target satisfies the
condition τ=(USL+LSL)/2 [2]. The defining recipe for the
capability index intended for situations for which the measured parameter
distribution is normal and has symmetric limits is (1)Cpo=USL−LSL6σ=Δ3σ.  The subscript
“o” in (1) indicates that it does not account for any target bias. Generalizations
of (1) to cover asymmetry in tolerances and nonnormal distributed parameters can
be found in the literature [6].  The
capability index is a direct measure of the process control and relates Cpo to a fraction of rejection, (2)po=2Φ(−3Cpo), where Φ(z) is the standard normal cumulative density
function (CDF). However, (2) can be applied to a process with normal
distribution, no target bias, and symmetric limits.Figure 1
PDF showing manufacturing specifications and rejected fraction of parts.


## 2.2. Distribution Independent Observation

Independent of whether the distribution
is normal with symmetric limits or not, the fraction rejected, also known as
the “component of nonconformity” [6] can be computationally predicted by evaluating
the CDF of the process distribution at selected points. As suggested by Figure 1, this prediction rule is given by [5] (3)p=F(LSL)+(1−F(USL)). This combines
the parts that do not meet the specifications, that is, the parts that have a
measured product characteristic “x” which is either lower than the LSL or higher
than the USL. In the spirit of (2), the generalized process capability index
should be consistent with the rule (4)Cp=−13Φ−1(p2),  where for the normal distribution
with symmetric limits,  po=p, and (4) reduces to special case described by (2).
From Figure 1, it follows that (5)USL=τ+Δ,LSL=τ+Δ.    Consistent
with (3) and (4), it can be shown that [5] (6)p=Φ(LSL−μσ)+Φ(−(USL−μσ)). However, (6)
conjunction with (4) produces an exact short-term capability index model which
agrees with the Boyles [5] yield index model. Demonstration details are
provided in Appendix 1.


## 3. Model-A: Standard Normal Version and Computer Implementation

### 3.1. Model-A Analysis

The PDF represented
in Figure1 can be transformed to the standard normal version as shown in
Figure 2 [11]. From (5), it can be stated that (7)LSL−μσ=(τ−μ)−Δσ=−Δσ+τ−μσ,USL−μσ=(τ+Δ)−μσ=Δσ+τ−μσ.  After
defining a normalized target bias, (8)δb=(τ−μσ). Therefore,
for normal process PDFs with symmetric limits and target bias, it follows that
the proposed exact Model-A [11] has the following rejection fraction after
substituting from (1), (7), and (8) in (6), (9)pA=Φ(−3Cpo+δb)+Φ(−3Cpo−δb). Consistent
with the general approach (4), (10)CpA=−13Φ−1(pA2). A check with δb=0 from (9) yields pA=po,
and consistent with (10), CpA=Cpo.Figure 2
Transformed PDF showing the specification limits rejected
fractions of parts.


### 3.2. Model-A Implementation Using Built-In Error Function

Noting that
the standard normal arguments “z” needed in (9) are expected to be negative for
reasonably limited target bias, the following conversion rule valid for z≤0 is useful with MATLAB [12], (11)Φ(z)=0.5−0.5erf(|z|2), and for inspection of (11), it
follows that (12)z=2erf−1(0.5−Φ(z)0.5). Appendix 2 describes the
definitions and the approximations considered for the error function (erf) and
the inverse error function (erf−1) in (11) and (12), respectively.
However, (11) and (12) can make use of built-in error and inverse error
functions of MATLAB. An alternative to using built-in routines for the error
function and its inverse is to employ approximate analytic expressions described
in the following subsection.


### 3.3. Model-A Implementation Using Analytic Approximation for Error Function

The built-in error function routines, available in MATLAB
[12], can be replaced with analytic approximations in predicting the bias-dependent
capability indices. After defining a=0.14, the
error function can be approximated as [9, 10] (13)erf(x)≅1−exp(−x2((4/π)+ax21+ax2)). Moreover, after taking r=2/(πa) and t(x)=ln(1−x2), the inverse of the error function can be
approximated as [9, 10] (14)erf−1(x)≈(−r−t(x)2+(r+t(x)2)2−t(x)a)0.5 A comparison of (13) and
(14) with the MATLAB built-in routines showed a maximum percentage error of
0.58% for the error function and 0.004% for its inverse. Hence, (13) and (14)
are applicable to (11) and (12), respectively.To demonstrate this approach, the
built-in error function-based Model-A predictions are compared with the
analytic Model-A predictions forCpo=1.0,Cpo=1.4,Cpo=1.6, and Cpo=2.0. It can be seen from Figure 3 that both the
numerical and the analytic approaches are serving as approximately equivalent
predictors. The accuracy of the analytic
approximation approach compared to that of the built-in numerical
implementation proved to be very good with the maximum percentage error of
0.91% atCpo=1.Figure 3
Model-A comparisons for numerical and analytic versions.Figure3 also shows the comparison of
Model-A with a well-established Boyles model [5] from the community. The
predictions at various target bias values have been considered to establish a
concise comparison of the behavioral pattern of the proposed model with the
already existing industry model.The comparison of Model A with
Boyle’s exact model [5] requires specification of the USL and the LSL values (e.g.,
USL  =  58 and LSL  =  26) [5]. For symmetrical specification limits, this implicitly
determines the target value. The USL and LSL values when taken in combination
with the selected values, for the unbiased short-term CPI (1) and the
normalized target bias (8), lead to the process mean and standard deviation
target values. With the four quantities{USL, LSL, μ,
and σ}, numerically determined application of (6)
and (4) will predict the exact value for the target-bias-dependent CPI which
for comparison purposes has been included with the datasets plotted on Figure 3.


## 3.1. Model-A Analysis

The PDF represented
in Figure1 can be transformed to the standard normal version as shown in
Figure 2 [11]. From (5), it can be stated that (7)LSL−μσ=(τ−μ)−Δσ=−Δσ+τ−μσ,USL−μσ=(τ+Δ)−μσ=Δσ+τ−μσ.  After
defining a normalized target bias, (8)δb=(τ−μσ). Therefore,
for normal process PDFs with symmetric limits and target bias, it follows that
the proposed exact Model-A [11] has the following rejection fraction after
substituting from (1), (7), and (8) in (6), (9)pA=Φ(−3Cpo+δb)+Φ(−3Cpo−δb). Consistent
with the general approach (4), (10)CpA=−13Φ−1(pA2). A check with δb=0 from (9) yields pA=po,
and consistent with (10), CpA=Cpo.Figure 2
Transformed PDF showing the specification limits rejected
fractions of parts.


## 3.2. Model-A Implementation Using Built-In Error Function

Noting that
the standard normal arguments “z” needed in (9) are expected to be negative for
reasonably limited target bias, the following conversion rule valid for z≤0 is useful with MATLAB [12], (11)Φ(z)=0.5−0.5erf(|z|2), and for inspection of (11), it
follows that (12)z=2erf−1(0.5−Φ(z)0.5). Appendix 2 describes the
definitions and the approximations considered for the error function (erf) and
the inverse error function (erf−1) in (11) and (12), respectively.
However, (11) and (12) can make use of built-in error and inverse error
functions of MATLAB. An alternative to using built-in routines for the error
function and its inverse is to employ approximate analytic expressions described
in the following subsection.


## 3.3. Model-A Implementation Using Analytic Approximation for Error Function

The built-in error function routines, available in MATLAB
[12], can be replaced with analytic approximations in predicting the bias-dependent
capability indices. After defining a=0.14, the
error function can be approximated as [9, 10] (13)erf(x)≅1−exp(−x2((4/π)+ax21+ax2)). Moreover, after taking r=2/(πa) and t(x)=ln(1−x2), the inverse of the error function can be
approximated as [9, 10] (14)erf−1(x)≈(−r−t(x)2+(r+t(x)2)2−t(x)a)0.5 A comparison of (13) and
(14) with the MATLAB built-in routines showed a maximum percentage error of
0.58% for the error function and 0.004% for its inverse. Hence, (13) and (14)
are applicable to (11) and (12), respectively.To demonstrate this approach, the
built-in error function-based Model-A predictions are compared with the
analytic Model-A predictions forCpo=1.0,Cpo=1.4,Cpo=1.6, and Cpo=2.0. It can be seen from Figure 3 that both the
numerical and the analytic approaches are serving as approximately equivalent
predictors. The accuracy of the analytic
approximation approach compared to that of the built-in numerical
implementation proved to be very good with the maximum percentage error of
0.91% atCpo=1.Figure 3
Model-A comparisons for numerical and analytic versions.Figure3 also shows the comparison of
Model-A with a well-established Boyles model [5] from the community. The
predictions at various target bias values have been considered to establish a
concise comparison of the behavioral pattern of the proposed model with the
already existing industry model.The comparison of Model A with
Boyle’s exact model [5] requires specification of the USL and the LSL values (e.g.,
USL  =  58 and LSL  =  26) [5]. For symmetrical specification limits, this implicitly
determines the target value. The USL and LSL values when taken in combination
with the selected values, for the unbiased short-term CPI (1) and the
normalized target bias (8), lead to the process mean and standard deviation
target values. With the four quantities{USL, LSL, μ,
and σ}, numerically determined application of (6)
and (4) will predict the exact value for the target-bias-dependent CPI which
for comparison purposes has been included with the datasets plotted on Figure 3.


## 4. Alternative Popular Methods

### 4.1. Model-X: The AMT Model

Model-X is
based on incorporating the target bias with capability index by first defining
a location index [2, 7, 8], (15)kb=|μ−τ|Δ.  The
subscript b indicates that this model includes the target bias. However, (15)
can be combined with the short-term capability index Cpo to define the Model-X capability index rule as (16)CpXb≡Cpo(1−kb).  From (15), it follows that (17)CpXb=Cpo(1−(μ−τσ)σΔ). This can be simplified via (1) and
(8) [11]: (18)CpXb=Cp−|δb|3, and hence the corresponding
fraction of rejection is (19)px=2Φ(−3Cpxb).


### 4.2. Model-Y:Cpm Model

TheCpm model was modeled to include the impact of the bias of the mean from the target
and the variance of the process parameter. As in the similarly defined Taguchi
loss function, it is not assumed that the PDF is normal [2, 11]. The
capability index in this model is defined using the variance of the process as (20)Cpm≡Δ3σm. In (20), the variance is given by σm=σ2+(μ−τ)2.
Hence, the Model-Y [11] with bias can be defined as [2, 7] (21)CpYb≡Δ3σm=(Δ3σ)11+δb2=Cpo1+δb2, and the
corresponding fraction of rejection predicted by (22)pY=2Φ(−3CpYb).


### 4.3. Computer Tests for Various Capability Index Models

The normalized fraction rejected for
models A, X, and Y are defined in terms of the fraction rejected under zero
bias conditions given in (2) as (23)(pAN,pXN,pYN)≡pA,pX,pYpo. In Figure 4, the fraction rejected
obtained from (23) is plotted versus normalized target bias (8) as derived from
(9), (19), and (22) for models A, X, and Y, respectively. It should be noted
that Model-X and Model-Y are both more pessimistic (i.e., higher fraction is rejected)
than that predicted from Model-A.Figure 4
Plot showing fraction rejected for models A, X and Y (Cp=1) versus normalized bias.The dependence of
bias-inclusive capability indices on short-term capability index and normalized
target bias for the models A, X, and Y were taken from (10), (18), and (21),
respectively. These were plotted versus the normalized target bias as shown in
Figure 5.Figure 5
Plot showing capability indices for models A,
X, and Y (Cpo=1) versus normalized target bias.These final results indicate that
the Model-X and Model-Y capability index rules are consistently overpessimistic
(i.e., lower in value). Of the two industry standard models Model-Y, usingCpm,
should be a better choice than Model-X
in that it is closer in prediction to Model-A.


## 4.1. Model-X: The AMT Model

Model-X is
based on incorporating the target bias with capability index by first defining
a location index [2, 7, 8], (15)kb=|μ−τ|Δ.  The
subscript b indicates that this model includes the target bias. However, (15)
can be combined with the short-term capability index Cpo to define the Model-X capability index rule as (16)CpXb≡Cpo(1−kb).  From (15), it follows that (17)CpXb=Cpo(1−(μ−τσ)σΔ). This can be simplified via (1) and
(8) [11]: (18)CpXb=Cp−|δb|3, and hence the corresponding
fraction of rejection is (19)px=2Φ(−3Cpxb).


## 4.2. Model-Y:Cpm Model

TheCpm model was modeled to include the impact of the bias of the mean from the target
and the variance of the process parameter. As in the similarly defined Taguchi
loss function, it is not assumed that the PDF is normal [2, 11]. The
capability index in this model is defined using the variance of the process as (20)Cpm≡Δ3σm. In (20), the variance is given by σm=σ2+(μ−τ)2.
Hence, the Model-Y [11] with bias can be defined as [2, 7] (21)CpYb≡Δ3σm=(Δ3σ)11+δb2=Cpo1+δb2, and the
corresponding fraction of rejection predicted by (22)pY=2Φ(−3CpYb).


## 4.3. Computer Tests for Various Capability Index Models

The normalized fraction rejected for
models A, X, and Y are defined in terms of the fraction rejected under zero
bias conditions given in (2) as (23)(pAN,pXN,pYN)≡pA,pX,pYpo. In Figure 4, the fraction rejected
obtained from (23) is plotted versus normalized target bias (8) as derived from
(9), (19), and (22) for models A, X, and Y, respectively. It should be noted
that Model-X and Model-Y are both more pessimistic (i.e., higher fraction is rejected)
than that predicted from Model-A.Figure 4
Plot showing fraction rejected for models A, X and Y (Cp=1) versus normalized bias.The dependence of
bias-inclusive capability indices on short-term capability index and normalized
target bias for the models A, X, and Y were taken from (10), (18), and (21),
respectively. These were plotted versus the normalized target bias as shown in
Figure 5.Figure 5
Plot showing capability indices for models A,
X, and Y (Cpo=1) versus normalized target bias.These final results indicate that
the Model-X and Model-Y capability index rules are consistently overpessimistic
(i.e., lower in value). Of the two industry standard models Model-Y, usingCpm,
should be a better choice than Model-X
in that it is closer in prediction to Model-A.


## 5. Extension to Long-Term Process

The standard approach [2, 13] for
predicting the long-term capability index is given by (24)Cpko≡Δ3σk. The suffix “o” in (24) indicates that there is no target bias
and is now expressed in terms of a long-term standard deviation σk. Consistent with the qualitative description in
Section 1 for the long-term capability index as required by (24) (25)σk≥σ.  This is consistent with a long-term
process being less precise, that is, higher standard deviation than that associated
with the corresponding short-term processes. By assumption, the short-term
processes have the same standard deviation. The claim here is that under
certain restrictions, to be described, the long-term process will be Gaussian,
with standard deviation σk, and mean μ¯,
the latter is given by (26)μ¯=avg{μi},i=1,2,…,M, where avg{} is the average operator and i indexes the M short-term
processes to be averaged. If there is a target bias, μ¯≠τ,
it can be accounted for with Model-A type analysis by defining a normalized
target bias [11]: (27)δbk=τ−μ¯σk. The long-term capability index representation equivalent to
(24) is (28)Cpko=Cpo(1−k)≤Cpo,0≤k<1. Unlike the locator index defined by (15), this representation
for CPI in (28) assumes a location index k which only accounts for the long-term
effective spread. The target bias is accounted for via (27). A revised form for
the locator index which excludes target bias effect is [11] (29)krevised≡avg{|μ¯−μi|}Δ,1>k≥0.   It should be noted that in this revised form of locator (29),
the average of the short-term process means μ¯ supplants τ in (15). If the long-term process is “on
target” (i.e., if μ¯=τ), then the definition (29) reduces to the
commonly used (15) [2]. As discussed in what follows, the restrictive
mathematical conditions for reproducing an exact long-term Gaussian process
from the superposition of short-term Gaussian processes will lead to a different
locator model than commonly seen model in (15) or (29).The long-term (LT) process distribution PDF can be viewed as
being constructed from the mathematical average of multiple short-term PDFs as represented
in the classic Harris and Lawson text on six sigma methods [13]. In Appendix 3,
it is formally shown that a long-term Gaussian PDF can be constructed from an
average of short-term PDFs, (30)fLT(y)=1M∑i=1MfST(y,μi). In this case, the short-term process PDF is given by (31)fST(y,μi)=12πσe−(y−μi)2/2σ2,−∞≤y≤∞,i=1,2,…,M. The often-cited mean-finding rule [2], taking discrete-term expectation
operator E{} [2], with respect to long-term PDF for y on both sides of (30),
would lead to (32)μ¯=E{y}=1M∑i=1ME{yfST(y,μi)}=1M∑i=1Mμi, which confirms (26). No claim is made that fLT(y) is guaranteed to be Gaussian.  For example, if two short-term Gaussians with
same standard deviation σ but with different means were averaged via (30),
the resultant process would not be a Gaussian. Nonetheless, the assumption which
is often made and is implicit in the applicability of (24) is that the
overriding long-term distribution in (30) is approximately normal with mean μ¯ and standard deviation σk [2, 13]. This assumption is shown to be
justified in Appendix 4 with the restriction that the random variable
associated with the (short-term) mean, μ, is approximately distributed Gaussian: (33)fμ(μ)=12πσμe−(μ−μ¯)2/2σμ2,−∞≤μ≤∞, with an average mean μ¯ and a standard deviation σμ.
Consider first a special case that the short-term distributions are identical (i.e.,
same constant mean μ=μ¯).
Then, the mean μ ceases to be a random variable in the usual
sense. This is accounted for in (33) by taking a limit σμ→0 and in that limit (33) reduces to the Dirac
delta function [2]: (34)limσμ→0fμ(μ)→δ(μ−μ¯). For large-enough sampling of the short-term PDF (see (30)) (i.e., M→∞), discrete averaging with μi can be switched to averaging over the
continuous random variable, μ,
with an integration rule, (35)fLT(y)=E{fST(y,μ)}=∫−∞∞fST(y,μ)fμ(μ)dμ. In the special case where all the short-term Gaussian
distributions are the same (i.e., same mean), the PDF for random variable μ reduces to a Dirac delta function (34) and
subsequent substitution into (35) yields, as expected, a long-term PDF that is
identical to the time-wise stable short-term processes. It is demonstrated in
Appendix 4 that the integrated (35) is distributed Gaussian with mean μ¯ and standard deviation (36)σk=σ2+σμ2=σ1+(σμσ)2. In Appendix 4, it is shown that (37)σμ=π2avg{|μ¯−μi|}. It follows from (24) and (36) that the capability index is
given by (38)Cpko=Cpo1+(σμ/σ)2=Cpo1+(π/2(avg{|μ¯-μi|}/σ))2, and as expected Cpko is reduced with increasing temporal
instability in the short-term process as gauged by σμ. Consistent with (28), the effective locator,
k, would be given by (39)keff=1−11+(σμ/σ)2=1−11+(π/2(avg{|μ¯−μi|}/σ))2. For relatively small ratios of x=σμ/σ, a two-term expansion can be approximated as (40)11+x2≈1(1+x2/2)≈(1−x22)|x|≪1. Hence, it facilitates for setting up an approximation for (39), (41)keff≈12(σμσ)2=12(π2avg{|μ¯−μi|}σ)2,σμσ≪1. As expected, keff=0 for stable short-term processes (i.e., when σμ=0); and the locator keff increases with increasing instability in the short-term
process. If there is no target bias, then μ¯=τ and according to (27), δbk=0.Under the
restriction (33), a unified approach, which includes impact of target bias, is
possible with the identical mathematical thread-of-logic (Section 3). A long-term
Model-A-type capability index with nonzero target bias is then given by (42)pkA=Φ(−3Cpko+δbk)+Φ(−3Cpko−δbk). Following (8), (9), and (10) after the substitutions, (43)Cpo→Cpko,δb→δbk,CpA→CpkA,pA→pkA,σ→σk.


## 6. Conclusion

It is has been shown that a short-term CPI model dependent on only two
parameters, unbiased short-term CPI and a normalized target bias is equivalent
to various exact CPI expressions reported in the literature. The demonstrated
principal advantage of this specific formulation is that it facilitates the
evaluation of the degradation of the capability of the process due to an offset
between the mean and the target of a process parameter. The unified methodology
for predicting short-term CPI is applicable to the long-term CPI, pending a condition
that the long term process is distributed Gaussian. Sufficient conditions to
guarantee that the long-term processes are distributed as Gaussian were
discussed. Within the context of these assumed conditions, a new paradigm for a
long-term locator “k” is proposed.Two implementation schemes for the
proposed reformulation for the exact solution were discussed. One method is
dependent on the availability of a built-in error function and its inverse while
the other method uses an analytic approximation for the error function and its
inverse.The second scheme supplants the built-in
functions used in the first scheme with recently reported analytical
approximations. For a three sigma process, the results indicate that the exact
CPI model is a less pessimistic predictor than both of the industry CPI models
tested.  Our results indicate that theCpm model (Model-Y) and the AMT model (Model-X) were more pessimistic than the
exact model (Model-A) in estimating manufacturing loss.In the literature,
methods have been reported to account for nonnormality in process distribution.
For future work, it would be interesting to demonstrate that any distribution
can be converted to an equivalent Gaussian. In such case, the Model-A approach would again be
applicable as long as a combination of equivalent process specifications and
equivalent Gaussian parameters are appropriately defined.



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*Source: 594753-2009-03-04.xml*