DOI 10.1007/s00170-003-1879-4 O R I G I N A L A R T I C L E
W.L. Pearn · Y.C. Chang · Chien-Wei Wu
Bootstrap approach for estimating process quality yield
with application to light emitting diodes
Received: 06 May 2003 / Accepted: 21 July 2003 / Published online: 24 November 2004 Springer-Verlag London Limited 2004
Abstract Process capability indices have been widely used by quality professionals for measuring process performance. Al-though process yield is the most common criterion used in the manufacturing industry for measuring process performance, a more advanced measurement formula Yq, called quality yield
index, has been proposed as an alternative measure of process performance. Quality yield can be viewed as the classical pro-cess yield minus the truncated expected relative propro-cess loss, within the specifications, which focuses on customer satisfac-tion. By taking customer loss into consideration, the advantage of using the quality-yield measure as process performance is that the formula can be applied to processes with arbitrary distribu-tions. Unfortunately, statistical properties of the estimated Yqare
mathematically intractable. Therefore, capability testing cannot be performed. In this paper, a nonparametric but computer inten-sive method called bootstrap is used to obtain a lower confidence bound on quality yield for capability testing purposes. Simula-tion studies are conducted to examine the sampling distribuSimula-tion of the estimated Yq. An application using the index Yq for the
light emitting diode manufacturing process is presented for illus-tration purposes.
Keywords Bootstrap methods· Lower confidence bound · Process capability indices· Quality yield · Simulation
1 Introduction
Process capability indices are convenient and powerful tools for measuring process performance. In recent years, process capa-bility indices have received substantial research attention in the W.L. Pearn (u) · Y.C. Chang
Department of Industrial Engineering & Management, National Chiao Tung University,
Taiwan
E-mail: [email protected] C.-W. Wu
Department of Business Administration, Feng Chia University,
Taiwan
quality assurance and statistical literature. Those indices quan-tify process performance by taking into consideration process location, process variation, and manufacturing specifications, which reflect process consistency, process accuracy, process yield, and process loss. The process indices Cp, Cpk, Cpm and
Cpmk [1–3] have become popular as unitless measures, which
combine natural process tolerance, manufacturing specifications, process centering, and the target value of the process. Those in-dices convey critical information regarding whether a process is capable of reproducing items satisfying the customer’s require-ment. In practice, a minimal capability requirement would be preset by the customers/engineers. If the prescribed minimum capability fails to be met, one would conclude that the process is incapable. Four basic well-known capability indices are:
Cp= USL− LSL 6σ , (1) Cpk= min USL− µ 3σ , µ − LSL 3σ , (2) Cpm= USL− LSL 6σ2+ (µ − T )2, (3) Cpmk= min USL− µ 3σ2+ (µ − T )2, µ − LSL 3σ2+ (µ − T )2 . (4)
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 are used to establish the formula of those capability in-dices. A rough categorization of those indices is by consideration of the target value T . The first category includes Cp and Cpk,
which are independent of T . Process loss incurred by the depar-ture from the target is, however, neglected. The second category includes Cpm and Cpmk, which rectify the disadvantage by
tak-ing the target value into account. The limitation on ustak-ing those indices defined above is that they require the assumption that the quality characteristic measurements must be coming from nor-mal distributions. Process quality yield index Yq is proposed to
Traditionally, process yield Y , is defined as the percentage of the processed product units passing the inspections, which has for a long time been the most common and standard criteria used in the manufacturing industries for judging process performance. According to the manufacturing specifications placed on vari-ous key product characteristics, units are inspected and sorted into two categories: accepted (conforming items) and rejected (defectives). For product units rejected during the inspection, additional costs would be incurred to the factory for scrapping or reworking. All passed product units are treated equally and accepted by the producer. No additional cost to the factory is required. The definition of Y index is
Y=
USL
L SL
dF(x), (5)
where USL and L SL are the upper and the lower specification limits, respectively, and F(x) is the cumulative distribution func-tion of the measured characteristic X. The disadvantage of yield measure is that it does not distinguish the products that fall in-side of the specification limits. Customers do notice unit-to-unit differences in these characteristics, especially if the variance is large and/or the mean is offset from the target. To rectify this, a more accurate, complete and customer-oriented measure of yield, which is referred to as quality yield Yq, was proposed [4].
The index distinguishes the products within the specifications by increasing the penalty as the departure from the target increases. The quadratic loss function is incorporated with the yield meas-ure. Johnson [5] developed the relative expected loss Leto
pro-vide comparisons between processes, defined as:
Le= ∞ −∞ (x − T )2 d2 dF(x), (6)
whereσ2 is the process variance,µ is the process mean, T is the target value and d= (USL − LSL)/2 is the half specification width. The disadvantage of the Leindex is the difficulty in
set-ting a standard for the index since it increases from zero to infinity. The quality yield index Yqdiffers from the expected relative worth
index defined by Johnson [5] by truncating the deviation outside the specifications. With this truncation, the quality yield index will be between zero and one and thus has better interpretation. To illustrate basic differences among yield Y , quality yield Yq, and
process capability indices Cp, Cpk, Cpmand Cpmk, we calculated
their index values for some cases, as presented in Table 1. Quality yield can be treated as traditional yield minus trun-cated expected relative process loss within the specifications to offer an excellent opportunity to quantify how well a process can meet customer requirements. While yield is the proportion of conforming products, Q-yield can be interpreted as the pro-portion of “perfect” products. By relating to the yield measure, which is familiar to engineers, it is much easier for the engineers to understand and accept this capability measure. The advantage of the Yq index over the Le index is that the value of the
for-mer goes from zero to one. Similarly to the yield index, the Y
Table 1. Comparisons of yield, Q-yield and PCIs
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
measure, the ideal value of Yq is one, which provides the user
a clear concept about the standard. Similar to yield Y , the Yq
index does not rely on the normality assumption. Current prac-tices of measuring manufacturing capability by only evaluating the point estimates of capability indices have been severely criti-cized since it ignores sampling error. The sampling distribution and sampling errors of the estimated Q-yield have never been investigated due to their mathematical intractability. A decision maker, however, may be interested in the lower confidence bound on the quality yield rather than just the point estimate, which does not convey reliable information.
In this paper, we apply the bootstrap resampling technique to obtain the lower confidence bound on ˆYq for practical
pur-pose. Four types of bootstrap confidence intervals, including the standard bootstrap confidence interval (SB), the percentile boot-strap confidence interval (PB), the biased corrected percentile bootstrap confidence interval (BCPB), and the bootstrap-t (BT) methods will be conducted. The practitioners can use the results to perform quality testing and determine the process can repro-duce product items to meet the specified quality requirement. The lower confidence bound not only provides us information re-garding actual process performance, which is tightly related to both the fractions of defective units and customer quality loss, but is also useful in making reliable decisions for capability test-ing and monitortest-ing the performance of process departure for targets as well.
This paper is organized as follows. We first give a brief intro-duction on the quality yield index Yqand the sample estimator of
Yq. We then introduce the bootstrap estimation technique and the
definitions of the four bootstrap confidence intervals in Sect. 3. Subsequently, in Sect. 4, some simulations on four distributions (normal, student’s t, chi-square and lognormal) are conducted to examine the distribution behavior of the estimated Yq. For
il-lustrative purpose, a real-world application to the light emitting diode (LED) manufacturing process is presented in Sect. 5. An integrated computer program for calculating the bootstrap lower confidence bounds is given in the Appendix. Some concluding remarks are made in Sect. 6.
2 Estimation of yield and quality yield
The main idea of the quality yield index Yq is that it
target. It was suggested by Ng and Tsui [4] by connecting the proportion-conforming-based index Y and loss-function-based index Le. Unlike the yield index Y , the quality yield Yq focuses
on the ability of the process to cluster around the target by tak-ing the relative loss within the specifications into consideration. If the USL and L SL are the upper and lower specification lim-its, respectively, T is the target value, d is the half specification width, and F(x) is the cumulative distribution function of the measured characteristic, then the index Yqis defined as
Yq= USL L SL 1−(x − T ) 2 d2 dF(x). (7)
In practical applications, sample data must be collected to es-timate the index. A sample estimator based on a finite popu-lation of products was proposed by Ng and Tsui [4]. Suppose
X1, X2, . . ., Xndenote the sample measurements of product
char-acteristics. A natural estimator of Y and Yqmay be expressed as
ˆY = L SL≤Xi≤USL 1 n, (8) ˆYq= L SL≤Xi≤USL 1− (Xi− T )2/d2 n . (9)
In addition to point estimation, however, a decision maker may be interested in a lower limit on the quality yield from the pro-cess as well. The sampling distribution of ˆYq is then required
but, unfortunately, the derivation of the exact distribution of ˆYq
is mathematically intractable. Pearn et al. [6] constructed an ap-proximate lower confidence bound of the estimator ˆYq for very
low fraction of defectives under the assumption of normality. However, the calculation of the approximation is rather messy and cumbersome to undertake. Further, the accuracy of the ap-proximation has not been investigated.
Normal-based process capability indices such as Cp, Cpk,
Cpm and Cpmk do not measure process fallout for non-normal
process data accurately. In the literature, Somerville and Mont-gomery [7] presented an extensive study to illustrate how poorly the normally based capability indices perform as a predictor of process fallout when the process is non-normally distributed. If the normally based capability indices are still used to deal with non-normal process data, the values of the capability indices are incorrect and might misrepresent the actual product qual-ity. Although new capability indices have been developed for non-normal distributions, those indices are harder to compute and interpret, and are sensitive to data peculiarities such as bi-modality or truncation. Moreover, those indices do not explicitly account for the manufacturing cost or customer’s loss. If a pro-cess is clearly non-normal, there is some question as to whether any process index is valid or should even be calculated. To illus-trate the relationship between the squared loss function and some probability distributions, we plot four process distributions: nor-mal distribution, lognornor-mal distribution, student’s t distribution and chi-square distribution, respectively, with the loss function and under the true value of Yq= 0.6 (see Figs. 1–4).
Fig. 1. Distribution plots of normal distribution with the loss function under true Yq= 0.6
Fig. 2. Distribution plots of lognormal distribution with the loss function under true Yq= 0.6
Fig. 3. Distribution plots of t distribution with the loss function under true
Yq= 0.6
Note that most existing capability indices require the nor-mality assumption and they are generally defined based on the specification limits rather than the customer’s satisfactions. The advantage of using the Q-yield as process performance
meas-Fig. 4. Distribution plots of chi-square distribution with the loss function under true Yq= 0.6
ure is that it does not rely on the normal distribution assumption. High values of Q-yield are desirable, which can be viewed as im-proving product quality from the customer’s viewpoint. Further-more, Q-yield is more flexible because it compares 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.
3 The bootstrap methodology
Traditionally, statistical research work has relied on the central limit theorem and normal approximations to obtain standard er-rors and confidence intervals. These techniques are valid only when the statistic, or some known transformation of the statis-tic, is asymptotically normally distributed. Unfortunately, many real world processes are not normally distributed and this de-parture from normality could potentially affect these estimates. A major motivation for the traditional reliance on normal-theory methods has been computational tractability. Access to power-ful computation enables the use of statistics in new and varied ways. Idealized models and assumptions can now be replaced with more realistic modeling or by virtually model-free analyses. Much statistical work and data analysis is undertaken today by computers in ways that are too complicated for practical analyti-cal treatment. The new effects of these computational advances are probably best reflected in the recent enormous success of bootstrap methodology, which shows that many problems, previ-ously difficult to solve, can be conquered. For either normal or non-normal distributions, the bootstrap method could be applied to return valid inferential results required.
The essence of bootstrapping is the idea that in the absence of any other knowledge about a population, the distribution of values found in a random sample of size n from the popula-tion is the best guide to the distribupopula-tion in the populapopula-tion. By resampling observations from the observed data, the process of sampling observations from the population is mimicked. Instead of using a sample statistic to estimate a population parameter, as
is done within the framework of conventional parametric statis-tical tests, the bootstrap uses multiple samples derived from the original data to provide what in some instances may be a more accurate measure of the population parameter. Therefore, to ap-proximate what would happen if the population was resampled, it is sensible to resample the sample. In other words, the infi-nite population that consist of the n observed sample values, each with probability 1/n, is used to model the unknown real pop-ulation. The sampling is with replacement, which is the only difference in practice between bootstrapping and randomization in many applications.
The bootstrap, a data-based simulation technique for statisti-cal inference which introduced by Efron [8, 9] is a nonparamet-ric, computationally intensive but effective estimation method. The most common application of the bootstrap involves esti-mating a population standard error and/or confidence interval. In particular, one can use the sampling distribution of a statis-tic, while assuming that the sample is only representative of the population from which it is drawn, and that the observations are independent and identically distributed. The main merit of the nonparametric bootstrap is that it does not rely on any distribu-tional assumptions about the underlying population. The more ambiguous the information is to the researcher regarding the un-derlying population distribution, the more likely it is that the bootstrap may prove useful. Rather than using distribution fre-quency tables to compute approximate p probability values, the bootstrap method generates a unique sampling distribution based on the actual sample rather than the analytic methods. The for-mulation detail follows.
In this method, B new samples, each of the same size as the observed data, are drawn with replacement from the available sample. The statistic of interest is then calculated for each new set of resampled data, in our case say ˆYq1∗, ˆYq2∗, . . . , ˆYqB∗ , yield-ing a bootstrap distribution for the statistic, say ˆYq. Four types
of bootstrap confidence intervals, including the standard boot-strap confidence interval (SB), the percentile bootboot-strap confi-dence interval (PB), the biased corrected percentile bootstrap confidence interval (BCPB), and the bootstrap-t (BT) method in-troduced by Efron [10] and Efron and Tibshiraniwill [11] will be conducted in this paper. Assume the observations x1, x2, . . . , xn
to be a random sample of size n taken from a process. A boot-strap sample, denoted by x1∗, x2∗, . . . , x∗n, is a sample of size n drawn with replacement from the original sample. There are pos-sibly a total of nn such resamples. Each such sample is called a “bootstrap sample.” In our case, these resamples would then be used to calculate nn values of ˆYq∗. Each of these would be an estimate of Yq and the entire collection would constitute the
(complete) bootstrap distribution for ˆYq. Bootstrap sampling is
equivalent to sampling (with replacement) from the empirical probability distribution function. Thus, the bootstrap distribution of Yqis estimator of the distribution of Yq.
Due to the overwhelming computation time, it is not of prac-tical interest to choose nn such samples. Usually, in practice, only a random sample of nn possible resamples is drawn, the statistic is calculated for each of these, and the resulting empir-ical distribution is referred to as the bootstrap distribution of the
statistic. Empirical work [11] indicated that only rough minimum of 1000 bootstrap resamples are required for the procedure to be useful to calculate valid confidence limits for population param-eters. Throughout our discussion, it is assumed that B= 10 000 bootstrap resamples (each of the same size as the available data) are taken and B= 10 000 bootstrap estimate of Yqare calculated
and ordered from smallest to largest. The generic notations ˆYq
and ˆYq∗(i) will be used to denote the estimator of a Q-yield index
and the associated ordered bootstrap estimate. Construction of a two-sided(1−2α)100% confidence limit will be described. We note that a lower(1 − α)100% confidence limit can be obtained by using only the lower limit. If the calculated bootstrap lower confidence limit is found to be smaller than the predetermined in-dex value, we would judge that the process is incapable. Quality improvement activities will be initiated. Otherwise, the process is considered to be capable. Four kinds of confidence intervals can be derived.
3.1 Standard bootstrap (SB)
From the B bootstrap estimates ˆYq∗(i), the sample average and
the sample standard deviation can be obtained as
ˆY∗ q = 1 B B i=1 ˆY∗ q(i), (10) S∗Y q= 1 B− 1 B i=1 ˆY∗ q(i) − ˆYq∗ 2 . (11)
where ˆYq∗(i) is the ith bootstrap estimate. Actually the quantity S∗Y
qis an estimator of the standard deviation of ˆYqif the
distribu-tion of ˆYq is approximately normal. Thus, the(1 − 2α)100% SB
confidence interval for Yqcan be constructed as
ˆYq− zαS∗Yq, ˆYq+ zαS ∗ Yq , (12)
where ˆYqis the estimated Yqfor the original sample, and zαis the upperα quantile of the standard normal distribution.
3.2 The percentile bootstrap (PB)
From the ordered collection of ˆYq∗(i), the α percentage and 1 − α
percentage points are used to obtain the(1 − 2α)100% PB confi-dence interval for Yq,
ˆY∗
q(αB), ˆYq∗((1 − α)B)
. (13)
3.3 Biased-corrected percentile bootstrap (BCPB)
While the percentile confidence interval is intuitively appealing it is possible that due to sampling errors, the bootstrap distribu-tion may be biased. In other words, it is possible that bootstrap
distributions obtained only using a sample of the complete boot-strap distribution may be shifted higher or lower than would be expected. A three steps procedure is suggested to correct for the possible bias [9]. First, using the ordered distribution of ˆYq∗, calculate the probability p0= P[ ˆYq∗≤ ˆyq]. Second, we compute
the inverse of the cumulative distribution function of a stan-dard normal based upon p0 as z0= Φ−1(p0), pL= Φ(2z0− zα)pU= Φ(2z0+ zα), where Φ(·) is the standard normal
cu-mulative distribution function. Finally, executing these steps to obtain the BCPB confidence interval,
ˆY∗ q(pLB), ˆYq∗(pUB) . (14) 3.4 Bootstrap-t (BT)
By using bootstrapping to approximate the distribution of a statistic of the form T= ( ˆYq− Yq)/SYq, where ˆYq is an
es-timate of Yq, with estimated standard error SYq. The bootstrap
approximation in this case is obtained by taking bootstrap sam-ples from the original data values, calculating the corresponding estimates ˆYq∗and their estimated standard error, and hence find-ing the bootstrapped T -values T= ( ˆYq∗− ˆYq)/S∗Yq. The hope is
then that the generated distribution will mimic the distribution of
T . The(1 − 2α)100% BT confidence interval for Yqmay
consti-tute as ˆYq− tα∗S∗Yq, ˆYq− t ∗ 1−αS∗Yq , (15)
where tα∗ and t∗1−α are the upper α and 1 − α quantile of the bootstrap t-distribution respectively, i.e., by finding the values that satisfy the two equations P[( ˆYq∗− ˆYq)/S∗Yq> t
∗
α] = α and
P[( ˆYq∗− ˆYq)/S∗Yq > t1−α∗ ] = 1 − α, for the generated bootstrap
estimates.
In the literature, Franklin and Wasserman [12] investigated the lower confidence bounds for the capability indices, Cp, Cpk
and Cpm using the first three bootstrap methods. Some
simula-tions were conducted and a comparison was made among the three bootstrap methods based on the parametric estimates. The simulation results indicate that for normal processes the boot-strap confidence limits perform equally well as results obtained by Chou, Owen and Borego [13], Bissell [14], and Boyles [15]. And for non-normal processes the bootstrap estimates performed significantly better than other methods.
4 Distribution plot of the Q-yield estimator
In this section, some Monte Carlo simulations are conducted to study the behavior of the sampling distribution of the estimated
Yq, for several cases where the underlying process distributions
are normal, skewed, or heavy tailed. We consider two levels of
Yq, say, Yq= 0.9, Yq= 0.6, with underlying process distributions
1. Normal distribution with probability density function f(x) = (√2πσ)−1exp −(x − µ)2σ2 2 , (16)
with meanµ and variance σ2, for−∞ < x < ∞.
2. Lognormal distribution with probability density function of f(x) = (x√2πσ)−1exp −(ln x − µ)2 2σ2 , (17)
with meanµ = eα+β2/2 and varianceσ2= e2α+β2(eβ2− 1), for x> 0.
3. Student’s t distribution with degree of freedom k, where the probability density function tkis,
f(x) = Γ ((k + 1)/2) Γ(k/2) 1 √ kπ 1+x 2 k −(k+1)/2 , (18)
with meanµ = 0, for k > 1 and variance σ2= k/(k − 2), for k> 2, −∞ < x < ∞.
Fig. 5. Distribution plots of ˆYqfor normal distribution with n= 25, 50, 100,
300, 500 (bottom to top) under true Yq= 0.9
Fig. 6. Distribution plots of ˆYqfor normal distribution with n= 25, 50, 100,
300, 500 (bottom to top) under true Yq= 0.6
4. Chi-square distribution with degree of freedom k, where the probability density function ofχ2
k is f(x) = 1 Γ(k/2) 1 2 k/2 χk/2−1e−x/2, (19) with meanµ = k and variance σ2= 2k, k = 1, 2, . . .. For each distribution, we randomly generate N= 20 000 samples of sizes n= 25, 50, 100, 300, 500, then calculate the es-timated capability index Yq. Figures 5–12 plot the distribution of
ˆYqfor the two levels of Yq, Yq= 0.9, and Yq= 0.6, with four
pro-cess distributions, normal distribution, lognormal distribution, student’s t distribution and chi-square distribution, respectively. For moderate and large sample size n, the distributions of the estimated Q-yield index all appear to be normal. Therefore, for processes where large sample data may be collected (product items may be inspected by automatic inspection machines) and normal approximations may be used for capability testing. Oth-erwise, the proposed bootstrap methodology seems to be more
Fig. 7. Distribution plots of ˆYqfor lognormal distribution with n= 25, 50,
100, 300, 500 (bottom to top) under true Yq= 0.9
Fig. 8. Distribution plots of ˆYqfor lognormal distribution with n= 25, 50,
Fig. 9. Distribution plots of ˆYqfor t distribution and n= 25, 50, 100, 300, 500 (bottom to top) under true Yq= 0.9
Fig. 10. Distribution plots of ˆYqfor t distribution and n= 25, 50, 100, 300, 500 (bottom to top) under true Yq= 0.6
reliable to make statistical inference on the estimated Yq, when
one has no idea what the underlying distribution really is. The bootstrap method especially is superior to other methods when the process distribution significantly deviates from normality and the size of sample data is small.
5 An application for LEDs
We present a case study on the light emitting diode (LED) manu-facturing process to illustrate the usage of the bootstrap lower confidence bound on Yq. The case we investigated was taken
from a manufacturing factory located in the Science-Based In-dustrial Park, Taiwan, making LEDs. The application of LEDs is expanding rapidly since high intensity LEDs with a wide range of colors have been recently developed and become available, which enabled application of LEDs in a wide variety of areas in-cluding color displays, traffic signals, roadway signs (barricade lights), airport signaling and lighting. Two typical LED applica-tions including font display and white LED lamps are shown in
Fig. 11. Distribution plots of ˆYqfor chi-square distribution and n= 25, 50, 100, 300, 500 (bottom to top) under true Yq= 0.9
Fig. 12. Distribution plots of ˆYqfor chi-square distribution and n= 25, 50, 100, 300, 500 (bottom to top) under true Yq= 0.6
Fig. 13 and Fig. 14. As various LED applications are developed, accurate specifications of LED characteristics become increas-ingly important. However, serious discrepancy in measurement is gathered from different LED manufacturers and users. LEDs are unique light sources that are very different from lamps in
Fig. 13. LED application on font display
Fig. 14. LED application on white lamps
terms of physical size, flux level, spectrum, and spatial intensity distribution. A transfer of photometric scales from traditional lu-minous intensity standard lamps to LEDs is not a trivial task, and large uncertainties are involved. The temperature-dependent characteristics and a large variety of optical designs of LEDs make it even more difficult to reproduce measurements.
In order to solve this problem, the factory was requested to provide calibrated standard LEDs for luminous intensity and lu-minous flux, which should dramatically improve the accuracy of measurement at industry level. Thus, the factory develops the measurement technology and standards for LED luminous inten-sity and luminous flux measurements, and to establish calibration services for LEDs, thereby improving the accuracy and unifor-mity of LED measurements among optoelectronics and other industry. A photometric technique has been developed to de-termine the effective reference plane of a photometer with an uncertainty of 0.2 mm, using a photometric bench and a stable integrating sphere source instead of a tungsten filament lamp. With this method, any photometer head with unknown reference plane position can be calibrated for LED measurements at any distances longer than 10 cm within an uncertainty of less than 1%. The alignment of LEDs is still a major uncertainty compon-ent for luminous intensity. As described above, LEDs generally do not follow the inverse-square law, so setting the distances ac-curately is critical to achieve reproducible results. One method of setting the alignment is permanently mounting an LED in a mount that has a reference surface. The distance from the tip of the LED to the reference surface can be measured accurately. The angular alignment will not change because the reference sur-face will align the LED with the apparatus.
Typically, LEDs are not mounted in a permanent fixture, they are just bare LEDs. The widely accepted method of aligning the bare LEDs is along their mechanical axis, mainly because it can be done quickly. The factory tried two different methods of align-ing bare LEDs, one usalign-ing a mount that physically holds the LED by the sides and another using an optical aligning procedure. A mount that physically holds the sides has the advantages of
the permanent mount once the LED is in the fixture. The fix-ture can be reproducibly placed in and out of a holder such that the distances are well known. The LED is easily centered along the detector axis and switching from the test LED to a standard LED can be done very quickly. However, we found reproducibly mounting the bare LED in the fixture was difficult. The fixture re-lied on placing pressure on the sides of the LED, which caused the sides of the LEDs to become scratched and damaged. In add-ition, a new fixture had to be fabricated for each different style or size of LED.
A better method is aligning the bare LEDs optically. Using a fixed telescope, a point in space is defined along the detector axis. The detector is on a translational stage with an optical en-coder. The reference plane of the detector is moved to the point in space and then translated 100 mm or 316 mm away depend-ing on the condition. The bare LED is mounted by its contacts on a stage that has five degrees of freedom. The stage can rotate, translate in the X, Y, and Z directions and tip and tilt about the point in space defined by the fixed telescope. By examining the LED from the side, the tip of the LED is translated to the point in space, set parallel to the detector axis and adjusted vertically. An LED application on LCD backlighting package dimension are depicted in Figs. 15 and 16.
We have established a capability for calibrating the lumi-nous intensity of LEDs using the detector-based method. We have built a tentative measurement set up for LED measure-ments in the photometric bench and made the calibration service available for submitted LEDs. The measurement of LED lu-minous intensity currently has an overall uncertainty of 1.5% for LEDs with a special fixture, and 3% for normal bare LEDs with no alignment aids. A dedicated small photometric bench for LED measurements is to be built. Long-term stability and temperature dependence of these LEDs will be studied and stan-dard LEDs for luminous intensity are to be developed. LEDs are unique light sources and are very different from traditional lamps in terms of physical size, flux level, spectrum and spa-tial distribution. The transfer of photometric scales from lumi-nous intensity standard lamps to LEDs has not been trivial and
Fig. 15. The package dimensions drawing (top and side) of an LCD back-lighting application
Fig. 16. The package dimensions drawing (bottom and polarity) of an LCD backlighting application
large discrepancies among companies have been measured. The factory has established two measurement conditions for single element LEDs with diameters less than 10 mm. These two meas-urement techniques compare LED luminous intensities without strictly using point source conditions. The factory has started re-search programs to establish appropriate measurement methods and calibration standards for all photometric quantities of LEDs. In particular, the measurement of luminous intensity of LED sources will be focused in our study. We investigated a particular model of the LED product with the upper and the lower speci-fication limits of luminous intensity are set to USL= 90 mcd,
L SL= 40 mcd, and the target value is set to T = 65 mcd. If the
characteristic data does not fall within the tolerance (L SL, USL), the LED is said to be defective.
For the purpose of making use of the methodology more con-venient and accelerate the computation, an integrated S-PLUS computer program is developed (see Appendix) to calculate the bootstrap lower confidence bounds. The practitioners only need to input the manufacturing specification limits, USL, L SL, tar-get value T , and the collected sample data of size n. Then the estimated values ˆY , ˆYq and the four bootstrap lower confidence
bounds (SB, PB, BCPB, BT) of ˆYq may be obtained. Thus,
whether or not the process is capable may be determined. A total of 100 observations were collected from a stable process in the factory and are displayed in Table 2. Figure 17 dis-plays the histogram, and Fig. 18 disdis-plays the normal probability
Table 2. A total of 100 observations
62 58 52 55 58 48 76 69 86 55 55 44 49 57 55 45 51 57 89 45 66 67 58 49 68 69 69 59 71 45 68 65 57 75 56 68 47 55 56 68 62 68 61 68 88 41 70 68 57 45 59 63 85 56 45 66 67 64 53 41 78 78 56 43 64 55 46 59 51 79 67 88 68 48 69 55 88 48 67 88 85 57 57 57 43 65 49 59 86 68 57 46 57 64 60 55 75 72 49 67
Fig. 17. Histogram plot of the sample data of size n= 100
Fig. 18. Normal probability plot of the sample data of size n= 100
plot of these sample data. From Figs. 17 and 18, it is evident to conclude the data collected from the factory are not normal distributed. The data analysis results justify that the process is significantly away from the normal distribution. Proceeding with the calculations by running the integrated S-PLUS program with 95% confidence, we obtain the values of the sample estimators ˆYq= 0.7477 and the corresponding bootstrap lower confidence
bound (LCB) as Table 3.
We note that the estimated index values for all the four exten-sions are greater than 0.7. In fact, all 100 observations fall within the specification interval (L SL, USL) resulting that sample es-timators of yield ˆY= 1. From the producer’s point of view, the
Table 3. Summary of the four bootstrap lower confidence bounds
Type SB PB BCPB BT
LCB 0.7010 0.7005 0.7027 0.7015
proportion of conforming products is 100%. However, to quan-tify how well a process can meet customer requirements, the lower confidence bound of ˆYqis approximately 0.7 and can be
in-terpreted as the proportion of “perfect” products being approxi-mately 70%. From the corresponding lower confidence bounds on Yqbased on four bootstrap methods, 0.7010, 0.7005, 0.7027,
and 0.7015, an example of capability testing is that if the Q-yield requirement preprint on the contract Yq is set to 0.7, we may
only conclude that the process is marginally capable, with 95% of confidence.
6 Conclusions
Quality yield is a flexible index because it compares the qual-ity of different characteristics of a product on a single percentage scale, and indicates how close a product comes to meeting 100% customer satisfaction. Furthermore, comparing with the existing capability indices, these capability indices rely on the underly-ing assumption of normal distribution. Although new capability indices have been developed for non-normal distributions, those indices are harder to compute and interpret, and are sensitive to data peculiarities such as bimodality or truncation. Second, these indices do not explicitly account for the manufacturing cost or customer’s loss. Capability indices are generally defined with respect to the specification limits rather than the customer’s func-tional limits. If a process is clearly non-normal, there is some question as to whether any process index is valid or should even be calculated. In this paper, the nonparametric is computationally intensive but an effective estimation bootstrap method is applied to the Q-yield measure ˆYqto obtain the lower confidence bounds.
The lower confidence bound provides information regarding ac-tual process performance for both the fractions of defectives units and customer quality loss. The proposed approach makes it feasible for the engineers to perform approximate process quality testing using the calculated Yq.
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Appendix
S-PLUS program for four bootstrap lower confidence bounds #---# Input manufacturing specification limits # USL, LSL and the target value T
#---USL_90
LSL_40 Target_65
#---# Input the original sample data of size
# n = 100 collected from the factory
#---data0_c( 62, 58, 52, 55, 58, 48, 76, 69, 86, 55, 55, 44, 49, 57, 55, 45, 51, 57, 89, 45, 66, 67, 58, 49, 68, 69, 69, 59, 71, 45, 68, 65, 57, 75, 56, 68, 47, 55, 56, 68, 62, 68, 61, 68, 88, 41, 70, 68, 57, 45, 59, 63, 85, 56, 45, 66, 67, 64, 53, 41, 78, 78, 56, 43, 64, 55, 46, 59, 51, 79, 67, 88, 68, 48, 69, 55, 88, 48, 67, 88, 85, 57, 57, 57, 43, 65, 49, 59, 86, 68, 57, 46, 57, 64, 60, 55, 75, 72, 49, 67) #---# Function to calculate the estimated Y
# and Yq based on the given data
#---delta_(USL-LSL)/2 Q.yield_function(data){ N_length(data) indata_data[data<USL&data>LSL] m_length(indata) Y_m/N
Yq_Y-(sum(((indata-Target)/delta)^2)/N) return(Y,Yq)
}
#---# Calculate the estimate of Y and Y_q based on # the original sample data
#---Y.Estimate_Q.yield(data0)$Y Yq.Estimate_Q.yield(data0)$Yq Y.Estimate Yq.Estimate #---# Generate B = 10000 bootstrap resamples from # the original sample data
#---B_10000 Y.B_rep(0,B) Yq.B_rep(0,B) for (i in 1:B){ dataS_sample(data0,length(data0),replace=T) Y.B[i]_Q.qield(dataS){\$}Y Yq.B[i]_Q.qield(dataS){\$}Yq } #---# Calculate the four bootstrap lower confidence # bounds based on resampled data
#---Yq.SB.bootstrap.95LCB_Yq.Estimate -qnorm(0.95)*var(Yq.B)^0.5 Yq.PB.bootstrap.95LCB_quantile (Yq.B, probs = 0.05) p0_mean(Yq.B<=q.Estimate) z0_qnorm(1-p0) pL_pnorm(2*z0-qnorm(0.95)) Yq.BCPB.bootstrap.95LCB_quantile (Yq.B, probs = floor(pL*B)/B) qtB095_quantile((Yq.B-Yq.Estimate) /(var(Yq.B)^0.5), probs = 0.95) Yq.BT.bootstrap.95LCB_Yq.Estimate -qtB095*var(Yq.B)^0.5 Yq.SB.bootstrap.95LCB Yq.PB.bootstrap.95LCB Yq.BCPB.bootstrap.95LCB Yq.BT.bootstrap.95LCB The output of the S-PLUS program is:
The estimated Y and Yqbased on the original sample data:
> Y.Estimate = 1
> Yq.Estimate = 0.747744
Four bootstrap lower confidence bounds of Yq based on
re-sampled data:
> Yq.SB.bootstrap.95LCB = 0.7010094 > Yq.PB.bootstrap.95LCB = 0.700512 > Yq.BCPB.bootstrap.95LCB = 0.70272 > Yq.BT.bootstrap.95LCB = 0.7015304
