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International Journal of Production
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Quality-yield measure for production
processes with very low fraction
defective
W. L. Pearn , Y. C. Chang & Chien-Wei Wu a
Department of Industrial Engineering & Management , National Chiao Tung University , 1001 Ta Hsueh Road, Hsinchu, Taiwan
b
Department of Industrial Engineering & Management , National Chiao Tung University , 1001 Ta Hsueh Road, Hsinchu, Taiwan E-mail:
Published online: 22 Feb 2007.
To cite this article: W. L. Pearn , Y. C. Chang & Chien-Wei Wu (2004) Quality-yield measure
for production processes with very low fraction defective, International Journal of Production Research, 42:23, 4909-4925, DOI: 10.1080/00207540410001699417
To link to this article: http://dx.doi.org/10.1080/00207540410001699417
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vol. 42, no. 23, 4909–4925
Quality-yield measure for production processes with very low
fraction defective
W. L. PEARN*, Y. C. CHANG and CHIEN-WEI WU
Process yield is the most common criterion used in manufacturing industry for measuring process performance. A more advanced measurement formula, called the quality yield index, Yq, is proposed to calculate the quality yield for arbitrary
processes by taking customer loss into consideration. Yqpenalizes yield for the
variation of the product characteristics from its target, which presents a measure of the average product loss. Quality yield could be expressed as the traditional yield minus the truncated expected relative loss within the specifications to quan-tify how well a process can reproduce product items satisfactory to the customers. The paper proposes a reliable approach for measuring quality yield by converting the estimate into a lower confidence bound for processes with a very low fraction of defectives. The lower confidence bound not only provides information about actual process performance that is tightly related to both the fraction of defec-tive units and customer quality loss, but also is useful in making decisions for capability testing.
1. Introduction
During the last decade, numerous process capability indices, including Cp, Cpk,
Cpmand Cpmk(Kane 1986, Chan et al. 1988, Pearn et al. 1992), have been proposed
in manufacturing industries to provide numerical measures on process performance. Those indices are effective tools for process capability analysis and quality assurance. Two process characteristics including the process location in relation to its target value and the process spread (overall process variation) are used to establish the formula of those capability indices. The closer the process output is to the target value and the smaller the process spread, the more capable is the process. That is, the larger the process capability index, the more capable is the process. Because Cpand
Cpkare independent of the target T, they can fail to account for process loss incurred
by the departure from the target. For this reason, two more advanced indices, Cpm
and Cpmk, were developed. Those indices have been defined explicitly as follows:
Cp¼ USL LSL 6 , Cpk¼min USL 3 , LSL 3 , Cpm¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ ð T Þ2 q and Cpmk¼min USL 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ ð T Þ2 q , LSL 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ ð T Þ2 q 8 > < > : 9 > = > ;,
Revision received November 2003.
Department of Industrial Engineering & Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan.
*To whom correspondence should be addressed. E-mail: [email protected]
International Journal of Production ResearchISSN 0020–7543 print/ISSN 1366–588X online # 2004 Taylor & Francis Ltd http://www.tandf.co.uk/journals
DOI: 10.1080/00207540410001699417
where is the process mean, is the process standard deviation, and USL and LSL are the upper and the lower specification limits, respectively. The indices are designed to monitor the performance for normal and near-normal processes with symmetric tolerances. It has been assumed that the target T ¼ M ¼ (USL þ LSL)/2 (which is quite common in practical situations) for the simplicity of the present discussions. It is essential that process capability indices must be applied under the condition that the process is in statistical control (stable).
The index Cpconsiders the overall process variability relative to the
manufactur-ing tolerance, reflectmanufactur-ing product quality consistency. Due to the simplicity of the design, Cpcannot reflect the tendency of process centring (targeting). The index Cpk
takes the process mean into consideration but can fail to distinguish between on-target processes from off-target processes. The index Cpm takes the proximity
of process mean from the target value into account, which is more sensitive to process departure than Cpk. Because Cpmis based on the average process loss relative
to the manufacturing tolerance, it has been alternatively called the Taguchi index. The index Cpmkis constructed from combining the modifications to Cpthat produced
Cpkand Cpm, which inherits the merits of both indices.
In the literature, several authors have promoted the use of various process capability indices and examined with differing degrees of completeness. Examples include Chou and Owen (1989), Chou et al. (1990), Franklin and Wasserman (1992), Kushler and Hurley (1992), Kotz et al. (1993), Va¨nnman and Kotz (1995), Va¨nnman (1997), Kotz and Lovelace (1998), Hoffman (2001), Pearn and Shu (2003), and references therein. Kotz and Johnson (2002) presented a thorough review for the development of process capability indices in the past 10 years, and Spiring et al. (2003) consolidated the research papers in process capability analysis for 1990–2002. Applications of those indices include the manufacturing of semi-conductor products (Hoskins et al. 1988), head/gimbals assembly for memory storage systems (Rado 1989), flip-chips and chip-on-board (Noguera and Nielsen 1992), rubber edge (Pearn and Kotz 1994–95), aluminium electrolytic capacitors (Pearn and Chen 1997), and couplers and wavelength division multiplexers (Wu and Pearn 2003). Other applications include performance measures on process with tool-wear problem (Spiring 1989), supplier selections (Tseng and Wu 1991, Chou 1994), capability measures for multiple manufacturing streams (Bothe 1999) and many others.
1.1. Process yield
An important measure for interpreting process capability is yield, defined as: Y ¼
ZUSL LSL
dF ðxÞ,
where F(x) is the cumulative distribution function of the measured characteristic X. The disadvantage of the yield measure is that it does not distinguish among the products that fall inside of the specification limits.
1.2. Process loss
To rectify this disadvantage, the quadratic loss function is considered to distin-guish the products by increasing the penalty as the departure from the target increases. However, the quadratic loss function itself does not provide comparison
with the specification limits and depends on the unit of the characteristic. To address these issues, Johnson (1992) developed the relative expected loss Lefor a symmetric
case as follows: Le¼ Z 1 1 ðx T Þ2 d2 " # dF ðxÞ ¼ 2þ ð T Þ2 d2 ,
where d ¼ (USL LSL)/2 is the half specification width. This measure has a direct relationship with Cpm because Le¼(3Cpm)2. The advantage of Leover Cpmis that
the estimator of the former has better statistical properties than that of the latter, as the former does not involve a reciprocal transformation of process mean and variance.
1.3. Quality yield
To incorporate the proportion conforming measure Y with loss function-based index Le, Ng and Tsui (1992) proposed the quality yield (Q-yield) index, Yq. In
contrast to the yield index Y, Q-yield emphasizes the ability of the process clustering around the target, which therefore reflects the degree of the process targeting (cen-tring) by considering only the relative loss within the specifications. By only taking the relative expected loss Le within the specifications into account, Ng and Tsui
defined the standardized quality as one minus the relative loss, and so the Q-yield, Yqis defined as the expected value of the standardized quality within the
specifica-tion: Yq¼ Z USL LSL 1 ðx T Þ 2 d2 " # dF ðxÞ:
This Q-yield index differs from the expected relative worth index defined in Johnson (1992) by truncating the deviation outside the specifications. With this truncation, the Q-yield index will be between 0 and 1 and thus provides a standard-ized measure. In addition, by relating to the yield measure widely accepted in the manufacturing industry, it will be understood and accepted as a capability measure. Similar to the yield measure Y, an ideal Yqis 1, which provides the user a clear guide
about the standard. Similar to the yield Y, Yq requires no normality assumption.
While yield is the proportion of conforming products, Q-yield can be interpreted as the average degree of products reaching ‘perfect’ or ‘on target’.
The present paper first rewrites the Q-yield as a representation of process yield and expected relative loss, focusing on production processes with a very low fraction of defectives. It then obtains a lower confidence bound on process capability index Cpk and an upper confidence bound on the expected relative loss to convert the
estimated Q-yield into a reliable lower confidence bound, which is the main contri-bution of the present work. The paper is organized as follows. Section 2 presents the comparisons of yield and Q-yield, with some illustrative examples. Section 3 inves-tigates the estimator of Q-yield. Since Y Leprovides a lower bound on the Q-yield,
estimations of process yield Y and process loss Le are also explored. Section 4
proposes a reliable method to obtain a lower confidence bound on Q-yield. Section 5 presents an application example of the amplified pressure sensor (APS). Section 6 demonstrates the proposed methodology by calculating the Q-yield for pressure sensor product. Conclusions are made in Section 7.
2. Comparisons of yield and Q-yield
Process yield is currently defined as the percentage of the processed product units passing the inspections. Units are inspected according to specification limits placed on various key product characteristics and sorted into two categories: passed (conforming) and rejected (defectives). Use of yield as a quality measure implies that each rejected unit costs the factory an additional amount (scrap or repair), while each passed unit costs the factory nothing additional. By inference, all passed units are equally acceptable to the next-in-line customer. A customer in this sense refers to any user of goods such as materials, components, subassemblies, assemblies or systems.
However, customers do notice unit-to-unit difference in these characteristics, especially when the variance is large and/or the mean is offset from the target. A more customer-oriented measure Yq is then proposed to account for both the
fraction of defectives and variation from target for the passed units. Penalty to the yield increases as the departure from the target T increases. When all conforming products are on target, then Yq¼Y. Figures 1a and b show two normally
distri-buted processes, N( ¼ T, ¼ d/3) and N( ¼ T þ d/3, ¼ d/6), respectively, with the quadratic loss function. The latter process has a higher yield but with a lower Q-yield since it has larger departure from the target value than the former. Furthermore, if the process characteristic X follows uniform distribution, U(LSL, USL), then the yield is Y ¼ 1.00 (100% conforming) and Q-yield is Yq¼0.665 (66.5% perfect),
Figure 1. (a) Plots of process N(T, d/3) with loss function, (b) Plots of process N(T þ d/3, d/6) with loss function, (c) Plots of process U (LSL, USL) with loss function, (d) Plots of process
w2(3) with loss function.
respectively. Obviously, this is a low-quality process. On the other hand, if X follows the chi-square distribution with three degrees of freedom, the yield would be Y ¼ 0.888 (88.8% conforming) and Q-yield would be Yq¼0.62 (62% perfect)
(figures 1c, d).
To extend the applicability of the plot for normally distributed processes, we rewrite the definition of Y and Yq as a function of Cd¼( T )/d and Cv¼/d.
Note that the subindex Cdmeasures the departure ratio, and the subindex Cv
mea-sures the process variation relative to the specification tolerances. The value of Cd
(abscissa) considered is from 2 to 2 and hence is from T 2d to T þ 2d. Moreover, Cv (ordinate) is from 0 to 1 to cover a wide range of . Therefore,
using Cdas the x-axis and Cvas the y-axis, one can plot the surface of Y and Yq
with various 2 Cd2 and 0 Cv1 (figure 2a and b, respectively). Figure 2c
and d displays the cross-section plots of Y and Yqversus 2 Cd2 for various
Cv¼1/6, 1/4, 1/3, 1/2, 1 (top to bottom in plot). Note that the plots of Y and
Yq are invariable irrespective of the specification limits. Processes with multiple
characteristics with different characteristic specification limits can thus be plotted simultaneously on a single chart.
Therefore, high Q-yields are desirable and can be viewed as improved product quality from the customer’s viewpoint. Q-yield is more flexible because it compares
(a) (b)
(c) (d)
Figure 2. (a) Surface plot of Y versus 2 Cd2 and 0 Cv1, (b) Surface plot of Yq
versus 2 Cd2 and 0 Cv1, (c) Plots of Y versus 2 Cd2 for various Cv¼1/6, 1/4,
1/3, 1/2, 1 (top to bottom), (d) Plots of Yqversus 2 Cd2 for various Cv¼1/6, 1/4, 1/3,
1/2, 1 (top to bottom).
the quality of different characteristics of a product on a single percentage scale and indicates how close a product comes to meeting 100% customer satisfaction. Comparing with the existing capability indices, note that those capability indices rely on the underlying assumption of normal distribution. Although new capability indices have been developed for non-normal distributions (e.g. the Clements 1989 and Johnson et al. 1994 methods). Those indices are more complicated to analyse and harder to interpret, and are sensitive to data peculiarities such as bimodality or truncation. Second, these indices do not explicitly account for the manufactur-ing cost or customer’s loss. Capability indices are generally defined with respect to the specification limits rather than to the customer’s functional limits. Table 1 summarizes values of those indices for some cases to illustrate the differences among Y, Yqand Cp, Cpk, Cpm, Cpmk.
3. Estimation of Q-yield, Yq
Ng and Tsui (1992) proposed a sample estimator based on a finite population of products. Suppose X1, X2, . . . , Xn denote the sample measurements of product
characteristics. It follows that yield and Q-yield are estimated by collected sample data and can be defined as follows:
^ Y Y ¼ X LSLXiUSL 1 n, YY^q¼ X LSLXiUSL 1 ðXiT Þ2=d2 n " # :
The sampling distribution and sampling errors are investigated. The decision-maker would be interested in a lower bound on the Q-yield rather than just the sample point estimate. Further, as the rapid advancement of manufacturing technol-ogy and customers demand, when the fraction of defectives is very low, such as in parts per million (ppm), products almost all fall between LSL and USL, one cannot even observe a defective item on inspection for a reasonable sample size. Thus, such an approach is not applicable for the low defective processes (since the sample point estimate is almost certain to be zero). The Q-yield index Yq can be rewritten as
follows: Yq¼Y ZUSL LSL ðx T Þ2 d2 " # dFðxÞ Y Le:
Thus, the measure Y Leprovides a lower bound on the Q-yield Yq. For processes
with very low fraction of defectives, the approximation of Yqusing Y Le would
Case Y(%) Yq(%) Cp Cpk Cpm Cpmk N(T, d ) 68.27 48.39 0.33 0.33 0.33 0.33 N(T, d/2) 95.45 76.99 0.67 0.67 0.67 0.67 N(T, d/3) 99.73 88.94 1.00 1.00 1.00 1.00 N(T, d/4) 99.99 93.75 1.33 1.33 1.33 1.33 N(T d/3, d/2) 90.50 69.13 0.67 0.44 0.55 0.37 N(T d/3, d/3) 97.72 78.41 1.00 0.67 0.71 0.47 N(T d/3, d/4) 99.62 82.70 1.33 0.89 0.80 0.53 N(T d/3, d/6) 99.997 86.11 2.00 1.33 0.89 0.60
Table 1. Comparisons of yield, Q-yield, and Cp, Cpk, Cpm, and Cpmk.
be quite accurate. Subsequently, we discuss the estimators of process yield Y and process loss Le.
3.1. Estimation of process yield, Y
The index Cpk is yield-based, which provides a lower bound on the process
yield, i.e. 2(3Cpk) 1 yield (3Cpk) (Boyles 1991). Table 2 shows some indexes
with two-sided specifications and the corresponding maximal non-conforming units (ppm) for a normally distributed process.
When Cpk¼C, b ¼ d/ can be expressed as b ¼ 3C þ ||. Thus, the index Cpk
can be expressed as a function of the characteristic parameter : Cpk¼d j Mj
3 ¼
d= jj
3 ,
where ¼ ( M )/.
Construction of the exact lower confidence bounds on Cpkis complicated since
the distribution of ^CCpkinvolves the joint distribution of two non-central t-distributed
random variables, or alternatively, the joint distribution of the folded-normal and the chi-square random variables, with an unknown process parameter even when the samples are given (Pearn et al. 1992). Numerous methods for obtaining approx-imate confidence bounds of Cpkhave been proposed (e.g. Bissell 1990, Chou et al.
1990, Zhang et al. 1990, Porter and Oakland 1991, Kushler and Hurley 1992, Rodridguez 1992, Nagata and Nagahata 1994, Tang et al. 1997).
Using the integration technique similar to that presented in Va¨nnman (1997), Pearn and Lin (2003) obtained an exactly explicit form of the cumulative distribution function of the natural estimator ^CCpk under the normal assumption, which is expressed in terms of a mixture of the chi-square distribution and the normal distribution, for x > 0, where G() is the cumulative distribution function of the chi-square distribution with degrees of freedom n 1, w2n1and () is the probability density function of the standard normal distribution:
FCC^pkðxÞ ¼1 Zbpffiffin 0 G ðn 1Þðb ffiffiffi n p tÞ2 9nx2 ! ðt þ pffiffiffinÞ þðt pffiffiffinÞ dt: (A brief derivation of the cumulative distribution function of ^CCpk is included in
the appendix.) Hence, given the sample of size n, the confidence level , the estimated value ^CCpkand the parameter , using numerical integration technique with iterations,
the 100 % lower confidence bounds for Cpkand CL, where bL¼3CLþ||, can be
obtained by solving the following equation: Z bL ffiffin p 0 G ðn 1ÞðbL ffiffiffi n p tÞ2 9n ^CC2 pk ! ðt þ pffiffiffinÞ þðt pffiffiffinÞ dt ¼ 1 : Cpk 0.7 0.8 0.9 1 1.1 1.2 1.3 1.33 ppm 35729 16395 6934 2700 967 318 96 66 Cpk 1.4 1.5 1.6 1.67 1.7 1.8 1.9 2.0 ppm 27 6.795 1.587 0.544 0.34 0.067 0.012 0.002 Table 2. Some Cpkindex values with the corresponding defective units (in ppm) for
a normally distributed process.
A 100 % lower confidence bound on the process yield Y can then be expressed as 2(3CL) 1.
However, since the process parameters and are unknown, then the distri-bution characteristic parameter, is also unknown, which has to be estimated in real applications, naturally done by substituting and with the sample mean XX and the sample standard deviation S. Such approach (and most existing methods) intro-duces additional sampling errors from estimating in finding the lower confidence bounds, which certainly would make the decisions less reliable and provide less quality assurance to the customers. To eliminate the need for further estimating the distribution characteristic parameter , we examine the behaviour of the lower confidence bound values CL against the parameter . The results indicate that the
lower confidence bound is decreasing in and reaches its minimum at ¼ 1.00 in all cases and stays at the same value for 1.00 with accuracy up to 104. Figure 3a to d plots the curves of the lower confidence bound, CL, versus the parameter
¼0(0.05)3.00, n ¼ 25, 50, 75, 100, 150 and 200 with confidence level ¼ 0.95, for ^
C
Cpk¼1.00, 1.33, 1.67 and 2.00, respectively. Hence, for practical purpose, we may
solve the above equation with ^ ¼ ¼1:00 to calculate the required lower confidence bounds for given ^CCpk, n and , without having to estimate further the parameter .
Thus, based on such an approach, the confidence level can be ensured and the decisions made are indeed more reliable.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.6 0.5 0.7 0.8 0.9 1.0 1.2 1.1 1.3 1.4 1.5 1.6 1.5 1.4 1.6 1.7 1.8 1.9 2.0 0.9 0.8 1.0 1.1 1.2 1.3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (a) (b) (d) (c)
Figure 3. (a) Plots of CLversus || for ^CCpk¼1.00, n ¼ 25, 50, 75, 100, 150, 200 (bottom
to top), (b) Plots of CLversus || for ^CCpk¼1.33, n ¼ 25, 50, 75, 100, 150, 200 (bottom to top),
(c) Plots of CLversus || for ^CCpk¼1.67, n ¼ 25, 50, 75, 100, 150, 200 (bottom to top),
(d) Plots of CLversus || for ^CCpk¼2.00, n ¼ 25, 50, 75, 100, 150, 200 (bottom to top).
3.2. Estimation of process loss, Le
Johnson (1992) proposed the relative expected squared error loss Le by
approaching capability from the point of view of the loss function. However, here the opposite concept of worth was used. It was assumed that a characteristic achieves its maximum worth WT, when X ¼ T, with decreasing values of worth as
X moves away from the target value T (eventually the worth becomes zero, then negative). The worth function can be described by WTk(X T )2, for
WTk(X T )2, and it will become zero when |X T | ¼ (WT/k)1/2. Johnson
viewed the pair (T þ (WT/k )1/2, T (WT/k )1/2) in the role of specification limits
for Cpm and defined ¼ (WT/k)1/2. The ratio of worth to maximum worth is
called the relative worth and can be defined as: W ðX Þ ¼1 kðX T Þ 2 WT ¼1 ðX T Þ 2 2 ,
where (X T )/2is the relative loss. The expected relative loss, Le¼E[(X T )]/2,
is used to quantify capability and is effectively equivalent to Cpm, since:
Le¼
d 3 2
=C2pm:
Suppose the product has zero worth outside the specifications by setting ¼ d, the relationship between Le and Cpm becomes Le¼(3Cpm)2. A natural unbiased
estimator of Leis: ^ L Le¼ 1 nd2 Xn i¼1 ðXiT Þ2:
4. Lower confidence bounds on Q-yield, Yq
Now we deal with the lower confidence limit on the Q-yield. Given a sample of size n, confidence level , estimated value ^CCpk and the estimated relative loss ^LLe,
the lower confidence bounds of Yq can be easily obtained by some mathematical
manipulations. The 100 % lower confidence bound of Yqcan be expressed as:
LYq¼LYULe ¼2ð3CLÞ 1 n þ ^ll w02 nð1 2; ^llÞ " # ^ L Le,
where LY is a lower 100 1% confidence bound on Y, ULe is an upper 100 2%
confidence bound on Le and ¼ 12. All derivations are shown below.
A lower 100 % confidence bound for Yqand Y simultaneously can be derived as:
PðY LY, YqLYqÞ ¼PðY LYÞ PðYqLYqjY LYÞ
¼PðY LYÞ PðLeY LYqjY LYÞ PðY LYÞ
PðLeLYLYqÞ
¼12:
As noted above, the yield-based index Cpkgives a lower bound on the process yield.
Hence, the probability P(Y LY) is equivalent to the probability P(CpkCL).
Solve P(Y LY) ¼ 1 for LY, one obtains LY¼2(3CL) 1, where CL is the
100 1% lower confidence bound on Cpk. Next, we proceed with the expression
PðLe LYLYqÞ ¼2. Under normal assumptions, ðn þ lÞ ^LLe=Le is distributed
as w0n2ðlÞ, a non-central chi-squared distribution with n degrees of freedom and
non-centrality parameter l ¼ n( T )2/2. Let ULe ¼ULeðX1, X2, . . . , XnÞ be a
statistic calculated from the sample data satisfying PðLeULeÞ ¼2, where the
confidence level 2does not depend on Le. Then, ULeis an 100 2% upper confidence
bound for Le. Note that:
PðLeULeÞ ¼Pððn þ lÞ ^LLe=Le ðn þlÞ ^LLe=ULeÞ ¼Pðw0n2ðlÞ ðn þ lÞ ^LLe=ULeÞ ¼2: Thus, ðn þ lÞ ^LLe=ULe ¼w 0 2 nð1 2; lÞ, where w 0 2 nð1 2; lÞ is the (lower) (1 2)th
percentile of the w0n2ðlÞ distribution. An 1002% upper confidence limit on Le can
be expressed, in terms of ^LLe, as:
ULe¼ n þl w02 nð1 2; lÞ ^ L Le:
l can be estimated by ^ll ¼ n½ð XX T Þ=Sn2, where XX ¼
Pn
i¼1Xi=n and Sn¼
½Pni¼1ðXi XX Þ2=n1=2. Substitute the results of LY and LYq back to the equation,
an 100 % lower confidence bound for Yqand Y simultaneously can be expressed as:
P Y 2ð3CLÞ 1, Yq2ð3CLÞ 1 n þ ^ll w02 nð1 2; ^llÞ " # ^ L Le ! :
5. Application to amplified pressure sensor (APS)
Consider the following case taken from a manufacturing factory making a series of original equipment manufacturer (OEM) pressure sensors, which combines state-of-the-art pressure sensor technology with signal conditioning to produce a fully signal conditioned, amplified, temperature-compensated sensor in a dual in-line package (DIP) configuration. Combining the sensor and signal conditioning circuitry in a single package simplifies the use of advanced silicon micromachined pressure sensors. The pressure sensor can be directly mounted onto a standard printed circuit board and no additional components are required to obtain an amplified high-level, calibrated pressure measurement. The pressure sensors are based on highly stable, piezoresistive pressure sensor chips mounted on a ceramic substrate. Two different pin configurations of the APS part, one for classical through-hole printed circuit board applications the other for surface mount applications, are available (figure 4a, b). Note that the only difference between the two is the pins. The ceramic housing, cap and ports are identical between the two configurations.
An electronically programmable application-specific integrated circuit (ASIC) is contained in the same package to provide calibration and temperature compensa-tion. The model is designed for operating pressure ranges from 0–5 to 0–100 psi. In addition, the sensor output is ratio metric with the supply voltage. Some features of the model are as follows: wide selection of full-scale ranges to 100 psi; low pressure (0–0.15 psi (full scale) FS) based on unique low-pressure die; amplified, calibrated, fully signal conditioned amplified output of 4.0 (volts direct current) VDC FS span (0.5–4.5 V signal); output ratio metric with supply voltage; temperature compensa-tion for span and offset; gage, differential and absolute version; DIP package for convenient personal computer board mounting; and small, lightweight package.
Some typical applications are barometric measurement; medical instrumentation; pneumatic control; gas flow; respirators and ventilators and ventilating and air-conditioning.
5.1. Amplified pressure product capabilities
The series pressure product provides a significant advantage to the user due to a number of improvements associated with the technology used in fabricating this part. These advantages include integrated amplification, electronic-trim for more precise control of gain and offset, and fewer external support components. The following notes are meant as an aid to the user to document some of these improvements. The amplified configuration has some key advantages, but these also must be considered in designing the systems into which pressure sensor parts are used. For instance, a fairly standard ‘trick’ when using an unamplified part in such absolute applications as barometric measurements is to use a 5 PSIA part. This allows it to operate in the 15 PSIA range with effectively 10 psi overpressure to get three times more unamplified output from the part. The addition of amplification at the mea-surement site has several key advantages. One is the required support circuitry. The pressure sensor has been designed to eliminate the need for external components. It requires no external components. The pressure sensor model with the gain of the part testing is shown in figure 5.
One of the key features of the pressure sensor is that it is electronically trimmed. As such, the part can be tested and verified before the final trim parameters are programmed. With the conventional laser-trimmed components, the final
Figure 5. Application schematic with 1.5 mA drive at 25C after a 10 s warm up. Figure 4. (a) Standard through hole pin configuration for the APS, (b) Surface-mount
pin configuration for the APS.
performance is set by how well the test system can measure millivolt level signals and resistances ranging from<50 Ohms to >5 MOhms. All of this is done at the end of long test cables and this further makes measurements more uncertain. There are several alternative measures on the manufacturability of a part. Yield from a manu-facturer’s viewpoint is critical, but so to is the distribution of parts as manufactured. The tighter the distribution on key parameters, the higher the quality of the part and the lower is the probability that the end-customer will get a part that will not meet the published specification.
6. Q-yield calculation for pressure sensor product
To illustrate how the proposed Q-yield lower confidence bound could be estab-lished and applied to actual data collected from the factories, we consider the follow-ing example taken from a company located in the Science-Based Industrial Park, Taiwan, manufacturing and designing a pressure sensor product. For a particular model of amplified pressure sensor process, capability analysis with focus on two key characteristics, Span and Zero, are taken. Span limits are 100 mV about a 2.000 V target (USL ¼ 2.100, LSL ¼ 1.900, T ¼ 2.000) and the Zero limits are set to 80 mV about a 2.500 V target (USL ¼ 2.580, LSL ¼ 2.420, T ¼ 2.500). Tight control of Zero and Span during testing will make the part more capable. We test 100 parts in each key characteristic. The collected data are shown in table 3. Figure 6a and b shows the histogram with density of the 100 APS data measurements for the Zero and Span, respectively. Proceeding with the calculations with a 95% level of
Zero (V) Span (V) 2.5445 2.5310 2.5204 2.5406 2.0512 2.0532 2.0396 2.0035 2.5455 2.5305 2.5418 2.5390 2.0594 2.0507 2.0382 2.0512 2.5338 2.5721 2.5430 2.5570 2.0517 2.0050 2.0276 1.9956 2.5482 2.5573 2.5403 2.5539 2.0038 2.0300 2.0719 2.0038 2.5306 2.5329 2.5391 2.5493 2.0532 2.0318 1.9957 2.0629 2.5471 2.5495 2.5202 2.5452 2.0235 2.0308 2.0226 2.0409 2.5482 2.5355 2.5470 2.5528 2.0373 1.9684 2.0113 2.0092 2.5474 2.5611 2.5434 2.5335 2.0501 2.0037 2.0295 2.0524 2.5532 2.5419 2.5327 2.5416 2.0575 2.0557 2.0333 2.0584 2.5511 2.5455 2.5618 2.5506 2.0070 2.0374 2.0563 2.0094 2.5490 2.5476 2.5490 2.5382 1.9716 2.0152 2.0392 2.0113 2.5543 2.5375 2.5454 2.5225 2.0390 2.0504 2.0529 2.0463 2.5454 2.5466 2.5253 2.5405 2.0316 1.9912 2.0824 2.0307 2.5279 2.5333 2.5586 2.5432 2.0000 2.0243 2.0825 2.0180 2.5381 2.5364 2.5563 2.5521 2.0102 1.9842 2.0300 2.0433 2.5453 2.5396 2.5493 2.5402 2.0112 2.0482 2.0440 1.9793 2.5379 2.5486 2.5382 2.5432 2.0024 2.0277 2.0199 2.0255 2.5270 2.5484 2.5461 2.5409 2.0638 2.0252 2.0006 2.0227 2.5367 2.5289 2.5335 2.5429 2.0518 2.0668 2.0142 2.0239 2.5518 2.5346 2.5265 2.5409 2.0105 2.0254 1.9966 2.0359 2.5462 2.5432 2.5390 2.5358 2.0536 2.0377 2.0162 1.9897 2.5542 2.5583 2.5361 2.5454 2.0275 2.0231 2.0636 2.0289 2.5398 2.5331 2.5440 2.5424 1.9993 1.9831 2.0533 2.0238 2.5203 2.5464 2.5270 2.5607 1.9985 2.0519 2.0041 2.0499 2.5291 2.5445 2.5360 2.5502 2.0254 2.0709 2.0162 2.0156
Table 3. APS data of 100 measurements for the Zero and Span.
confidence, we obtain the calculated sample mean, sample derivation, estimated Cpk
index values, Cpklower confidence bounds, estimated Levalues, Leupper confidence
bounds, estimated yield, yield lower confidence bounds, estimated yield and Q-yield lower confidence bounds (table 4).
Table 5 shows the manufacturing capabilities for the pressure sensor processes using the estimated yield, estimated Q-yield and their corresponding lower confidence bounds. The plot of Q-yield versus yield is shown in figure 7. These two dimensions of product quality are useful because one dimension represents customer satisfaction while the other represents factory fulfilment. The triangle with vertices (0, 0), (1, 0) and (1, 1) contains the set of all (Y, Yq). The objective of quality improvement is to
move towards the point (1, 1). The engineers can effectively monitor and get the most priority of all process characteristics simultaneously.
2.420 2.436 2.452 2.468 2.484 2.500 2.516 2.532 2.548 2.564 2.580 0 10 20 30 40 (a) (b) 1.900 1.920 1.940 1.960 1.980 2.000 2.020 2.040 2.060 2.080 2.100 0 5 10 15 20
Figure 6. (a) Histogram of the APS data measurements for the Zero, (b) Histogram of the APS data measurements for the Span.
7. Conclusions
Process capability indices, which establish the relationship between the actual process performance and the manufacturing specifications, have been the focus of recent research in quality assurance and capability analysis. The Q-yield, Yq, has
been proposed to calculate the process capability by taking customer loss into consideration. It penalizes yield for variation of the product characteristics from its target, combining the proportion of conformities and the average process loss. The present paper develops a reliable approach to obtain a lower confidence bound for Yq, which can be applied to production processes with very low fraction of
defectives where existing method cannot be applied. The lower confidence bound provides information about actual process performance for both the fraction of defective units and customer quality loss. The results obtained allow one to perform capability testing based on yield and customer satisfactions. A real-world applica-tion to the amplified pressure sensor manufacturing process is also presented for illustrative purposes.
Appendix: Derivation of the cumulative distribution function of ^CCpk
Let X1, X2, . . . , Xnbe a random sample of size n drawn from a normal
distribu-tion with mean and variance 2measuring the characteristic under investigation.
Figure 7. Plot of Q-yield versus yield.
Estimated yield Yield LCB Estimated Q-yield Q-yield LCB Zero 1.0000 0.9999 0.7041 0.6016 Span 1.0000 0.9962 0.8582 0.8054
Table 5. Comparison of estimated yield and Q-yield, associated with LCB for the APS products.
X
X S CC^pk CL LL^e ULe YY^ LY YY^q LYq
Zero 2.5424 0.0099 1.2705 1.0821 0.2959 0.3983 1.0000 0.9999 0.7041 0.6016 Span 2.0286 0.0246 0.9660 0.8165 0.1418 0.1908 1.0000 0.9962 0.8582 0.8054 Table 4. Calculated statistics, estimated process capability measures and corresponding
lower confidence bound of the APS products.
The natural estimator ^CCpk is obtained by replacing the process mean and
pro-cess standard deviation by their conventional estimators X and S, respectively. The following expression occurs:
^ C Cpk¼
d XX M
3S :
For the sake of deriving the cumulative distribution function of ^CCpk, the
follow-ing notations are introduced:
. K ¼ (n 1)S2/2, which is distributed as w2n1.
. Z0¼pffiffiffinð XX M Þ=, which is distributed as Nðpffiffiffin, 1Þ with ¼ ( M )/. . H ¼ |Z0|, which is distributed as a folded-normal distribution with probability
density function fHðhÞ ¼ ðh þ
ffiffiffi n p
Þ þðh pffiffiffinÞfor h 0, where () is the probability density function of the standard normal distribution.
For x >0 , the cumulative distribution function of ^CCpkcan be derived as follows:
FCC^pkðxÞ ¼ P ^CCpkx ¼P ffiffiffiffiffiffiffiffiffiffiffi n 1 p ðbpffiffiffinH Þ 3pffiffiffiffiffiffiffinK x ! ¼1 P pffiffiffiffiffiffiffinK < ffiffiffiffiffiffiffiffiffiffiffi n 1 p ðbpffiffiffinH Þ 3x ! ¼1 Z1 0 P pffiffiffiffiffiffiffinK < ffiffiffiffiffiffiffiffiffiffiffi n 1 p ðbpffiffiffinH Þ 3x jH ¼ h ! fHðhÞdh ¼1 Z1 0 P pffiffiffiffiffiffiffinK < ffiffiffiffiffiffiffiffiffiffiffi n 1 p ðbpffiffiffinhÞ 3x ! fHðhÞdh;
where b ¼ d/. Since K is distributed as w2n1: P pffiffiffiffiffiffiffinK < ffiffiffiffiffiffiffiffiffiffiffi n 1 p ðbpffiffiffinhÞ 3x ! ¼0 for h > bpffiffiffinand x > 0: Therefore, FCC^pkðxÞ ¼1 Zbpffiffin 0 P pffiffiffiffiffiffiffinK < ffiffiffiffiffiffiffiffiffiffiffi n 1 p ðbpffiffiffinhÞ 3x ! fHðhÞdh ¼1 Zbpffiffin 0 G ðn 1Þðb ffiffiffi n p hÞ2 9nx2 ! fHðhÞdh, for x > 0,
where G() is the cumulative distribution function of w2n1. Substituting fH(t ) leads
to the result: FCC^pkðxÞ ¼1 Zbpffiffin 0 G ðn 1Þðb ffiffiffi n p tÞ2 9nx2 ! ðt þ pffiffiffinÞ þðt pffiffiffinÞ dt, for x > 0: References
BISSELL, A. F., 1990, How reliable is your capability index? Applied Statistics, 39, 331–340.
BOTHE, D. R., 1999, A capability index for multiple process streams. Quality Engineering, 11,
613–618.
BOYLES, R. A., 1991, The Taguchi capability index. Journal of Quality Technology, 23, 17–26.
CHAN, L. K., CHENG, S. W. and SPIRING, F. A., 1988, A new measure of process capability:
Cpm. Journal of Quality Technology, 20, 162–175.
CHOU, Y. M. and OWEN, D. B., 1989, On the distributions of the estimated process capability
indices. Communications in Statistics: Theory and Methods, 18, 4549–4560.
CHOU, Y. M., OWEN, D. B. and BORREGO, A. S., 1990, Lower confidence limits on process
capability indices. Journal of Quality Technology, 22, 223–229.
CHOU, Y. M., 1994, Selecting a better supplier by testing process capability indices. Quality
Engineering, 6, 427–438.
CLEMENTS, J. A., 1989, Process capability calculations for non-normal distributions. Quality
Progress, 22, 95–100.
FRANKLIN, L. A. and WASSERMAN, G. S., 1992, Bootstrap lower confidence limits for capability
indices. Journal of Quality Technology, 24, 196–210.
HOFFMAN, L. L., 2001, Obtaining confidence intervals for Cpkusing percentiles of the
distri-bution of Cp. Quality and Reliability Engineering International, 17, 113–118.
HOSKINS, J., STUART, B. and TAYLOR,J., 1988, A Motorola Commitment: A Six-Sigma Mandate
(Motorola).
JOHNSON, T., 1992, The relationship of Cpm to squared error loss. Journal of Quality
Technology, 24, 211–215.
JOHNSON, N. L., KOTZ, S. and PEARN, W. L., 1994, Flexible process capability indices. Pakistan
Journal of Statistics, 10, 23–31.
KANE, V. E., 1986, Process capability indices. Journal of Quality Technology, 18, 41–52.
KOTZ, S. and LOVELACE, C., 1998, Process Capability Indices in Theory and Practice (London:
Arnold).
KOTZ,S. and JOHNSON, N. L., 2002, Process capability indices — a review, 1992–2000. Journal
of Quality Technology, 34, 1–19.
KOTZ, S., PEARN, W. L. and JOHNSON, N. L., 1993, Some process capability indices are
more reliable than one might think. Journal of the Royal Statistical Society, Series C, 42, 55–62.
KUSHLER, R. and HURLEY, P., 1992, Confidence bounds for capability indices. Journal of
Quality Technology, 24, 188–195.
NAGATA, Y. and NAGAHATA, H., 1994, Approximation formulas for the lower confidence limits
of process capability indices. Okayama Economic Review, 25, 301–314.
NG, K. K. and TSUI, K. L., 1992, Expressing variability and yield with focus on the customer.
Quality Engineering, 5, 255–267.
NOGUERA, J. and NIELSEN, T., 1992, Implementing six sigma for interconnect technology.
ASQC Quality Congress Transactions, Nashville, 538–544.
PEARN, W. L. and CHEN, K. S., 1997, Capability indices for non-normal distributions with
an application in electrolytic capacitor manufacturing. Microelectronics and Reliability, 37, 1853–1858.
PEARN, W. L. and KOTZ, S., 1994–95, Application of Clements’ method for calculation second
and third generation process capability indices for non-normal Pearsonian populations. Quality Engineering, 7, 139–145.
PEARN,W. L. and LIN, P. C., 2003, Testing process performance based on the capability index
Cpkwith critical values. Computers and Industrial Engineering (in press).
PEARN, W. L. and SHU, M. H., 2003, Lower confidence bounds with sample size information
for Cp applied to production yield assurance. International Journal of Production
Research, 41, 3581–3599.
PEARN, W. L., KOTZ, S. and JOHNSON, N. L., 1992, Distributional and inferential properties
of process capability indices. Journal of Quality Technology, 24, 216–231.
PORTER, L. J. and OAKLAND, S., 1991, Process capability indices — an overview of theory
and practice. Quality and Reliability Engineering International, 7, 437–448.
RADO, L. G., 1989, Enhance product development by using capability indexes. Quality
Progress, 22, 38–41.
RODRIGUEZ, R. N., 1992, Recent developments in process capability analysis. Journal of
Quality Technology, 24, 176–187.
SPIRING, F., LEUNG, B., CHENG, S. and YEUNG, A., 2003, A bibliography of process capability
papers. Quality and Reliability Engineering International, 19, 445–460.
SPIRING, F. A., 1989, An application of Cpmto the tool-wear problem. ASQC Quality Congress
Transactions, Toronto, 123–128.
TANG, L. C., THAN, S. E. and ANG, B. W., 1997, A graphical approach to obtaining confidence limits of Cpk. Quality and Reliability Engineering International, 13, 337–346.
TSENG, S. T. and WU, T. Y., 1991, Selecting the best manufacturing process. Journal of Quality Technology, 23, 53–62.
VA¨NNMAN, K., 1997, Distribution and moments in simplified form for a general class of
capability indices. Communications in Statistics: Theory and Methods, 26, 159–179.
VA¨NNMAN, K. and KOTZ, S., 1995, A superstructure of capability indices distributional
properties and implications. Scandinavian Journal of Statistics, 22, 477–491.
WU, C. W. and PEARN, W. L., 2003, Measuring manufacturing capability for couplers
and wavelength division multiplexers (WDM). International Journal of Advanced Manufacturing Technology(in press).
ZHANG, N. F., STENBACK, G. A. and WARDROP, D. M., 1990, Interval estimation of process
capability index Cpk. Communications in Statistics: Theory and Methods, 19, 4455–4470.
