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Critical acceptance values and sample sizes of a variables sampling
plan for very low fraction of defectives
W.L. Pearn
a,∗, Chien-Wei Wu
baDepartment of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan bDepartment of Business Administration, Feng Chia University, 100 Wenhwa Road, Taichung 40724, Taiwan
Received 24 September 2003; accepted 10 August 2004 Available online 28 October 2004
Abstract
Acceptance sampling plans are practical tools for quality control applications, which involve quality contracting on product orders between the vendor and the buyer. Those sampling plans provide the vendor and the buyer rules for lot sentencing while meeting their preset requirements on product quality. In this paper, we introduce a variables sampling plan for unilateral processes based on the one-sided process capability indicesCPU (orCPL), to deal with lot sentencing problem with very low fraction of defectives. The proposed new sampling plan is developed based on the exact sampling distribution rather than approximation. Practitioners can use the proposed sampling plan to determine accurate number of product items to be inspected and the corresponding critical acceptance value, to make reliable decisions. We also tabulate the required sample sizen and the corresponding critical acceptance value C0for various-risks,-risks, and the levels of lot or process fraction of defectives that correspond to acceptable and rejecting quality levels.
䉷2004 Elsevier Ltd. All rights reserved.
Keywords: Acceptance sampling plan; Critical acceptance value; Fraction of defectives; Process capability indices
1. Introduction
Acceptance sampling plan has been one of the most prac-tical tools in statisprac-tical quality control applications, which involves quality contracting on product orders between the factories and customers. It provides the vendor and the buyer a general criterion for lot sentencing while meeting their pre-set requirements on product quality. A well-designed sam-pling plan can significantly reduce the difference between the required (expected) and the actual supplied product qual-ity. Acceptance sampling plan, however, cannot avoid the risk of accepting bad product lots or rejecting good product
∗Corresponding author. Tel.: 571-4261; fax.: +88-63-572-2392.
E-mail address:wlpearn@mail.nctu.edu.tw(W.L. Pearn).
0305-0483/$ - see front matter䉷2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.omega.2004.08.001
lots even when 100% inspection is implemented, because of human error and fatigue, we are never ensured that the decision will be the right one. Acceptance sampling plan is a statement regarding the required sample size for product inspection and the associated acceptance or rejection crite-ria for sentencing each individual lot. The critecrite-ria used for measuring the performance in an acceptance sampling plan, is usually based on the operating characteristic (OC) curve which quantifies the risk for vendors and buyers.
The OC curve plots the probability of accepting the lot against actual lot fraction defective, which displays the dis-criminatory power of the sampling plan. For product quality protection and company’s profit, both the vendor and the buyer would focus on certain points on the OC curve to reflect their benchmarking risk. The vendor (supplier) usu-ally would focus on a specific level of product quality, traditionally called acceptable quality level (AQL), which
W.L. Pearn, C.-W. Wu / Omega 34 (2006) 90 – 101 91 would yield a high probability for accepting a lot. The AQL
also represents the poorest level of quality for the vendor’s process that the consumer would consider acceptable as a process average. Therefore, a preferred sampling procedure would be one, which gives a high probability of acceptance at the AQL that is normally specified in the contract. The consumer would also focus on another point at the other end of the OC curve, traditionally called lot tolerance per-cent defective (LTPD). Alternate names for the LTPD are the rejecting quality level (RQL) and limiting quality level (LQL). The LTPD is the poorest level of quality that the consumer is willing to accept for an individual lot. Note that the LTPD is a level of quality specified by the buyer, repre-senting the specified low probability of accepting a lot with defect level as high as the LTPD.
A well-designed sampling plan must provide a probabil-ity of at least 1− of accepting a lot if the lot fraction of defectives is at the contracted AQL. The sampling plan must also provide a probability of acceptance no more than if the lot fraction of defectives is at the LTPD level, the desig-nated undesired level preset by the buyer. Thus, the accep-tance sampling plan must have its OC curve passing through those two designated points (AQL, 1− ) and (LTPD, ). There are a number of different ways to classify acceptance-sampling plans. One major classification is by attributes and variables. When a quality characteristic is measurable on a continuous scale and is known to have a distribution of a specified type, it may be possible to use as a substitute for an attributes sampling plan based on sample measurements such as the mean and the standard deviation of the sample. These variables sampling plans have the primary advantage that the same OC curve can be obtained with a smaller sam-ple then is required by attributes plan. The precise mea-surements required by a variables plan would probably cost more than the simple classification of items required by an attributes plan, but the reduction in sample size may more than offset this exact expense. Such saving may be espe-cially marked if inspection is destructive and the item is expensive (see Refs.[1,2]).
Guenther [3]developed a systematic search procedure, which can be used with published tables of binomial, hyper-geometric, and Poisson distributions to obtain the desired acceptance sampling plans. Stephens[4]provided a closed form solution for single sample acceptance sampling plans using a normal approximation to the binomial distribution. Hailey[5]presented a computer program to obtain mini-mum sample size single sampling plans based on either the Poisson or binomial distribution. Hald [6]gave a system-atic exposition of the existing statistical theory of lot-by-lot sampling inspection by attributes and provided some tables for the sampling plans. Other researches related to the clas-sical acceptance sampling plans include Jennett and Welch
[7], Wallis[8,9], Jacobson[10], Lieberman and Resnikoff
[11], Das and Mitra[12], Owen[13], Kao[14], Hamaker
[15], Bender[16], Govindaraju and Soundararajan[17], and Suresh and Ramanathan[18]. In addition to the graphical
procedure for designing sampling plans with specified OC curves, tabular procedures are also available for the same purpose. Duncan[19]gave a good description of these tech-niques. In this paper, we consider a variable sampling plan for product lots (or processes) with very low fraction of de-fectives. The proposed sampling plan is based on analytical exact formulas hence the decisions made are reliable.
2. Process capability indices approach
Process capability indices, includingCp,CPU,CPL, and
Cpk, have been popularly used in the manufacturing industry to measure whether a process is capable of reproducing product items within the specified manufacturing tolerance. Those indices provide common quantitative measures on process potential and performance[20–23], are defined in the following, where USL and LSL are the upper and lower specification limits, respectively, is the process mean, and
is the process standard deviation.
Cp=USL6− LSL , CPU= USL− 3 , CPL= − LSL 3 , Cpk= min USL− 3 , − LSL 3 . WhileCp andCpkare appropriate measures for processes with two-sided specifications (which require both USL and LSL),CPU and CPL have been designed specifically for processes with one-sided specifications (which require only USL or LSL). The indexCPU measures the capability of a smaller-the-better process with an upper specification limit USL, whereas the indexCPL measures the capability of a larger-the-better process with a lower specification limit LSL.
For normally distributed processes with one-sided speci-fication limit USL, the process yield is:
P (X < USL) = P X − 3 < USL− 3 = P 1 3Z < CPU = P (Z < 3CPU) = (3CPU),
where Z follows the standard normal distribution N(0, 1) with the cumulative distribution function (x) = (2)−1/2−∞x exp(−t2/2)dt. Similarly, for normally dis-tributed processes with one-sided specification limit LSL, the process yield can be calculated as P (X > LSL) = P (−Z/3 < CPL) = 1 − (−3CPL) = (3CPL). For con-venience of presentation, we letCI denote eitherCPU or CPL. Therefore, the corresponding non-conforming units in parts per million (NCPPM) for a well controlled nor-mally distributed process can be expressed as NCPPM
= p = 106× [1 − (3C
I)]. Thus, process capability in-dicesCIprovide an exact measure of process yield.Table 1
Table 1
VariousCIvalues and the corresponding NCPPM
CPUorCPL NCPPM CPUorCPL NCPPM 1.00 1349.90 1.50 3.40 1.15 280.29 1.60 0.7933 1.25 88.42 1.67 0.2722 1.30 48.10 1.70 0.1698 1.33 33.04 1.90 0.0060 1.45 6.81 2.00 0.0010
In practice, sample data must be collected in order to calculate those indices since the process mean and standard deviation are usually unknown. To estimate the indices CPU and CPL, Chou and Owen [24] considered ˆCPU and ˆCPL, the natural estimators of CPU and CPL,
which are defined below, where ¯X =ni=1Xi/n is the sample mean, andS2= (n − 1)−1ni=1(Xi− ¯X)2is the sample variance, which may be obtained from a process that is demonstrably stable (under statistical control). Under the normality assumption, the estimator ˆCPUis distributed
as(3√n)−1tn−1(), where tn−1() is a non-central t dis-tribution withn − 1 degrees of freedom and non-centrality parameter = 3√nCPU. The estimator ˆCPL has the same
sampling distribution as ˆCPUbut with = 3 √
nCPL.
How-ever, both estimators ˆCPUand ˆCPL are biased. Pearn and
Chen [25] showed that by adding the correction factor
bn−1=[2/(n−1)]1/2[(n−1)/2]/[(n−2)/2] to ˆC PUand ˆCPL, we could obtain unbiased estimators bn−1ˆCPU and
bn−1ˆCPL which have been denoted as ˜CPUand ˜CPL. That
is, E( ˜CPU) = CPU, and E( ˜CPL) = CPL. Sincebn−1< 1,
then Var( ˜CPU) < Var( ˆCPU) and Var( ˜CPL) < Var( ˆCPL).
And since the estimators ˜CPUand ˜CPLare only based on the
complete and sufficient statistics (X, S2), it can conclude that ˜CPUand ˜CPLare the uniformly minimum variance
un-biased estimators (UMVUEs) ofCPUandCPL, respectively.
Therefore, in practice using the UMVUEs ˜CPUand ˜CPLto
calculate the capability measures would be desirable. ˆCPU=
USL− ¯X
3S , ˆCPL=
¯X − LSL 3S .
To test whether the process meets the capability require-ment, we consider the following testing hypothesis with H0: CIC (the process is incapable), versus the alternative H1: CI> C (the process is capable). Thus, we may consider the test∗(x) = 1 if CI> C0, and∗(x) = 0, otherwise. The test∗rejects the null hypothesis if ˜CI> C0, with type I error(C0) = , the chance of incorrectly judging an inca-pable process as a cainca-pable one. Thus, the power of the test can be calculated as (CI) = P ( ˜CIC0| CI) = P tn−1()3 √ nC0 bn−1 , which under H0 forECI(∗(x)) = , we can obtain that
the critical value C0 = bn−1(3√n)−1tn−1,(), where = 3√nCI and tn−1,() is the upper quantile of
non-central t distribution with n − 1 degrees of freedom satisfiesP (tn−1()tn−1,()) = . And from the proba-bility density function of ˜CI, we let
(x) =f (x, CI) f (x, CI) = ∞ 0 y(n−2)/2exp −1 2 y + 3x√ny bn−1√n−1− 3 √ nCI 2 dy ∞ 0 y(n−2)/2exp −1 2 y + 3x√ny bn−1√n−1− 3 √nCI2 dy .
Since forCI> CI> 0 , (x) is a non-decreasing function of x, then {(f˜C
I(x, CI) | CI> 0} has monotone likelihood
ratio (MLR) property in ˜CI. Therefore, we can conclude that the test∗ is uniformly most powerful (UMP) test of its sizeECI(∗(x) | H0) = . In fact, the decision rule of the
UMP test can be constructed as The lot is accepted, if ˜CIbn−1
3√ntn−1,( = 3 √nC
I). The lot is rejected, otherwise.
3. Designing variables sampling plan based onCPU
andCPL
Consider a variables sampling plan to control the lot or process fraction defective (or nonconforming). Since the quality characteristic is a variables, there will exist either an USL or a LSL, or both, that define the acceptable values of this parameter. As indicated earlier, selection of a meaning-ful critical value for a capability test requires specification of an AQL and a LTPD for theCIvalue. The AQL is simply a standard against which to judge the lots. It is hoped that the vendor’s process will operate at a fallout level that is considerably better than the AQL. Both the vendor and the buyer will lay down their requirements in the contract: the former demands that not too many “good” lots shall be re-jected by the sampling inspection, while the latter demands that not too many “bad” lots shall be accepted. A sampling plan attempts will be chosen to meet these somewhat op-posing requirements. Let(AQL, 1 − ) and (LTPD, ) be the two points on the OC curve of interest. To determine whether a given process is capable, we can first consider the following testing hypothesis:
H0: p = AQL (process is capable), H1: p = LTPD (process is not capable).
Process capability index is a function of process parame-ters and manufacturing specifications. It measures the abil-ity of the process to reproduce product units that meet the specifications. For processes with one-sided specification limits, thus,CIcan be used as a quality benchmark for ac-ceptance of a product lot. That is, the null hypothesis with process fraction of defectives, H0: p = AQL, is equivalent to test process capability index with H0: CICAQL, where
W.L. Pearn, C.-W. Wu / Omega 34 (2006) 90 – 101 93 CAQLis the level of acceptable quality forCIindex
corre-spond to the lot or process fraction of nonconformities AQL as−1(1 − AQL × 10−6)/3. For instance, if the propor-tion defectivep = AQL of vendor’s product is less than 88 NCPPM, then the probability of consumer accept the lots will greater than 100(1−)%. On the other hand, if the pro-portion defective of vendor’s product,p = LTPD, is more than 1350 NCPPM, then the probability of consumer would accept no more than 100%. Thus, from the relationship be-tween the index value and fraction of defectives, we could obtain the equivalentCAQL= 1.25 and the CLPTD= 1.00. Then, the required inspection sample sizen and critical ac-ceptance valueC0 for the sampling plans are the solution to the following two nonlinear simultaneous equations.
P {Accepting the lot|proportion defective p = AQL}1 − ,
P {Accepting the lot|proportion defective p = LTPD}.
As described before, the sampling distribution of ˜CIis dis-tributed asbn−1(3√n)−1tn−1(), where tn−1() is a non-centralt distribution with n − 1 degrees of freedom and non-centrality parameter = 3√nCI. Thus, the probability of accepting the lot can be expressed as
A(CI) = P ( ˜CIC0| CI= C) = P tn−1()3 √ nC0 bn−1 .
Therefore, the required inspection sample sizen and crit-ical acceptance valueC0of ˜CIfor the sampling plan are the solution to the following two nonlinear simultaneous equa-tions (1) and (2). P tn−1(1) 3√nC0 bn−1 1 − , (1) P tn−1(2) 3√nC0 bn−1 , (2)
where1=3√nCAQLand2=3√nCLTPD,CAQL> CLPTD. We note that the required sample sizen is the smallest pos-sible value ofn satisfying Eqs. (1) and (2), and determining then as sample size, where n means the least integer greater than or equal ton. Moreover, to illustrate how we solve the above two nonlinear simultaneous equations (1) and (2), let S1(n, C0) = 1 2n−32 n−1 2 ∞ 0 x n−2e−x2/2 1 √ 2 × 3√nC0x/(bn−1 √ n−1) 0 × exp −1 2(u − 1) 2 du dx − (3)
Fig. 1. Surface plot ofS1andS2.
Fig. 2. Contour plot ofS1andS2.
S2(n, C0) = 1 2n−32 n−1 2 ∞ 0 x n−2e−x2/2 1 √ 2 × 3√nC0x/(bn−1 √ n−1) 0 × exp −1 2(u − 2) 2 du dx − (1 − ). (4) Figs. 1 and 2 display the surface and contour plots of Eqs. (3) and (4) simultaneously with -risk = 0.10 and -risk = 0.10 for CAQL= 1.50 and CLPTD= 1.00,
respectively. From Fig. 2, we can see that the interac-tion of S1(n, C0) and S2(n, C0) contour curves at level 0 is (n, C0) = (24.49, 1.2200), which is the solution to
nonlinear simultaneous equations (1) and (2). That is, in this case, the minimum required sample size isn = 25, and the critical acceptance value is C0= 1.2200 for the sampling plan based on the one-sided capability indexCI.
To investigate the behaviour of the critical acceptance values and required sample sizes with various parameters, we perform extensive calculations to obtain the solution of (1) and (2). The results indicated that the larger of the risks which producer or customer would suffer, the smaller sample sizen is required for inspection. This phenomenon can be interpreted intuitively, as if we would want the chance of wrongly concluding a bad lot (process) as a good one, or a good lot (process) as a bad one, to be smaller, we would need more sample information to make the judgement. Further, for fixed , risks and CLTPD, the required sample sizes become smaller when theCAQL becomes larger. This can also be explained by the same reasoning, since the judgement will be more correct with a larger value of the difference between the CAQL and CLTPD. For practical applications purpose, we calculate and tabulate the critical acceptance values and sample sizes required for the sampling plans, with commonly used , , CAQLandCLTPD. The values of(n, C0) for producer’s -risk = 0.01(0.01)0.10, buyer’s -risk = 0.01(0.01)0.10, with various benchmarking quality levels,(CAQL, CLPTD)= (1.25, 1.00), (1.45, 1.00), (1.60, 1.00), (1.45, 1.25), (1.60, 1.25) and (1.60, 1.45) are displayed in the Appendix. As an example, if the benchmarking quality level(CAQL, CLPTD) is set to (1.45, 1.00) with producer’s -risk = 0.01 and buyer’s -risk = 0.05, then the corresponding sample size and critical acceptance value can be calculated as (n, C0) = (66, 1.1749). The lot will be accepted if the 66 inspected product items yield measurements with ˜CI1.1749. Otherwise, we do not have sufficient informa-tion to conclude that the process meets the present capability requirement. In this case, the buyer will reject the lot.
4. Discussions and comparisons
An approximate approach based on measurements sim-ilar to process capability indices, was used to designed a variables sampling plan, which was proposed to deal with the lot sentencing problem[1,2,15]. The approximation used the statistic,
ZLSL= ¯X − LSLS .
The relationship between ˜CPL and ZLSL (or ˜CPU and ZUSL) is that ˜CPL is a constant multiplied by theZLSL. Clearly, ˜CI> C0 ⇔ ZLSL> k, with k = 3C0/bn−1. Taking approximation approach, the values of n and k can be obtained from the following formulas for given p1, p2, , and . Results should always be rounded up. Formula forn depends upon the given knowledge of the
standard deviation. k =zzp2+ zzp1 z+ z , n = zz + z p1− zp2 2 , if is known, n = 1+ k 2 2 z + z zp1− zp2 2 , if is unknown, where the z’s are the normal deviates the probability of exceeding which are thep1, p2, , and . Formulas of n and k are based on the assumption that ¯X ± ks is approximately normally distributed with a mean of ± k and a standard deviation equal approximately to
1 n+ k2 2n.
In addition to the formulas derived by Wallis[8,9] Jacob-son[10] developed the useful nomograph. Table ofn and
k for some given p1’s and p2’s is also provided (can be
found in Statistical Research Group, Columbia University, Techniques of Statistical Analysis,[26, pp. 22–25]).
We compared the sample sizes(n) and the corresponding critical acceptance value(C0) based on the existing
approx-imation and our proposed (exact) method. We summarized
(n, C0) values for various CAQL, CLPTD, and the risks of
producer and customer (i.e.p1, p2, , and ), as given in
the followingTables 2and3.Table 2displays some cases with < . It is noted that for such cases the existing ap-proximation requires larger sample size (hence more cost) and larger critical acceptance values than the proposed exact approach. For such cases, the existing approximation tends to in favor of rejecting the lots, thus provides more pro-tection to the customer (producer will suffer unfairly more risk).Table 3displays some cases with > . It is noted that for those cases, the existing approximation requires smaller sample size and smaller critical acceptance values than the proposed exact approach. For such cases, the existing ap-proximation tends to in favor of accepting the lots, thus pro-vides more protection to the producer (customer will suf-fer unfairly more risk in this case). The proposed sampling plan is based on the analytical exact approach, which pro-vides the vendor and the buyer a fair and accurate criterion for lot sentencing, which significantly reduces the differ-ence between the expected and the actual supplied product quality.
5. An application example
Electrically erasable programmable read-only memory (EEPROM) chip is a user-modifiable read-only memory chip
W.L. Pearn, C.-W. Wu / Omega 34 (2006) 90 – 101 95 Table 2
(n, C0) values for= 0.01,= 0.05 and 0.10 with various (CAQL,CLTPD)
(,) CAQL= 1.25 CAQL= 1.45 CAQL= 1.60 CAQL= 1.45 CAQL= 1.60
CLTPD= 1.00 CLTPD= 1.00 CLTPD= 1.00 CLTPD= 1.25 CLTPD= 1.25 n C0 n C0 n C0 n C0 n C0 (0.01,0.05) Exact 185 1.0997 66 1.1749 41 1.2280 398 1.3305 142 1.3880 Approx. 194 1.1420 71 1.2502 45 1.3288 413 1.3647 151 1.4478 (0.01,0.10) Exact 150 1.0843 53 1.1465 33 1.1907 325 1.3185 116 1.3673 Approx. 164 1.1559 61 1.2742 39 1.3600 346 1.3760 128 1.4670 Table 3
(n, C0) values for= 0.05(0.01)0.10,= 0.01 with CAQL= 1.25 and CLTPD= 1.00
Exact Approximation n C0 n C 0= bn−1× k/3 0.05 0.01 193 1.1423 182 1.0990 0.06 0.01 185 1.1456 173 1.0954 0.07 0.01 179 1.1488 165 1.0920 0.08 0.01 173 1.1516 159 1.0890 0.09 0.01 167 1.1541 153 1.0860 0.1 0.01 162 1.1566 147 1.0832
that can be erased and reprogrammed (written onto) repeat-edly through the application of higher electrical voltage. It is usually used in portable phones, PHS phones, com-pact portable terminals, consumer products (such as cord-less phones and audio systems); industrial equipment in-cluding measuring instruments and PLCs; OA products such as printers and scanners, in-house telephone switches, and other communication equipment. The output leakage current (OLC) is an essential product quality characteristic, which has significant impact to product quality. For the output leak-age current of a particular model of EEPROM, the upper specification limit, USL, is set to 5A.
In a purchasing contract, a minimum value of the PCI is usually specified. If the prescribed minimum value of the PCI fails to be met, the process is determined to be incapable. Otherwise, the process will be determined to be capable. For processes with one-sided specifications, some minimum ca-pability requirements have been recommended for specific process types must run under some designated quality con-ditions. In particular, 1.25 for existing processes, and 1.45 for new processes; 1.45 also for existing processes on safety, strength, or critical parameter, and 1.60 for new processes on safety, strength, or critical parameter. The recommended guidelines for minimum quality requirements and the cor-responding parts per million (PPM) of non-conformities (NC) for those processes can be found in Montgomery[2].
In recent years, many companies have adopted criteria for evaluating their processes that include process capability objectives that are more stringent than those recommended minimums above. For example, the “Six-Sigma” program pioneered by Motorola essentially requires that when the process mean is in control, it will not be closer than six stan-dard deviations from the nearest specification limit. Thus, in effect, requires that the process capability ration will be at least 2.0.
To illustrate how the sampling plan can be established and applied to the actual data collected from the factories, we consider the following example taken from a company man-ufacturing and designing standard Flash Memory EEPROM and Mixed-Signal products, such as, PLL, ADC DAC, and many others. The manufacturing specifications for a 128-bit EEPROM chip, has an upper specification limit USL=5 A for the output leakage current which are mentioned before. If the OLC is greater than 5A, then the EEPROM chip is considered to be nonconforming product, and will not be used to make the EEPROM chip of that particular model.
The capability requirement for this particular model of EEPROM chip was defined as “Capable” ifCPU> 1.60. In the contract, theCAQLandCLTPD are set to 1.60 and 1.25 with the-risk = 0.01, and -risk = 0.05 respectively. That is, the sampling plan must provide a probability of at least 99% of accepting the lot if the lot proportion defective is
T able 4 The sample data with 142 observ ations (unit: A) 4.1 4 3.914 3.993 3. 39 3. 2 4.201 4.066 4.049 4.210 4.247 4.106 4.650 3. 47 4.420 4.216 3.746 4.5 9 3.945 3. 39 3.342 4.175 4.1 3.644 3.946 4.09 3.696 3.729 4.024 3.975 3.7 2 4.211 3. 44 3.931 4.091 4.057 3.761 3.965 3.976 3.94 4.154 4.156 4.316 3. 7 3.917 3.953 4.145 3.91 4.00 4.0 4 4.170 4.042 3.906 4.26 4.241 4.153 3.620 4.139 3. 2 3 . 2 4 3.752 4.610 4.0 2 3.571 4.015 3. 3 3. 23 4.233 3.905 4.2 9 3.761 4.059 4.333 3.921 3. 30 3. 25 4.040 4.715 4.123 3. 64 4.103 3.957 4.4 0 3.717 3.921 4.515 3.666 3. 74 3.695 4.146 4.025 3.7 4 4.10 4.320 4.127 3. 74 4.191 4.12 4.045 4.2 2 3 . 7 3 4.245 4.279 4.301 3.713 4.046 3.619 4.356 3. 25 3.763 3. 61 4.130 4.075 3. 04 3. 70 3.96 3.943 4.637 3.745 4.199 4.139 3.7 3 4.3 9 3.442 3.965 4.025 4.166 4.123 3.955 3.773 4.0 6 4.191 3.9 5 3.994 4.005 4.541 4.147 3.767 3.970 3.770 4.324 3. 6 4.140
at theCAQL= 1.60 (which is equivalent to AQL = 0.79 NCPPM), and also provide a probability of no more than 5% of accepting the lot if the lot proportion defective is at the CLTPD= 1.25 (which is equivalent to LTPD = 88 NCPPM). Therefore, by checkingTable 5in this Appendix we find the required sample sizes and critical acceptance value(n, C0)=
(142, 1.3880). Hence, 142 inspected, EEPROM chips are
taken from the lot randomly and the observed measurements are displayed inTable 4. Based on these inspections, we obtain that
X = 4.0248, S = 0.2407, ˜CPU= bn−1×
USL− ¯X
3S = 1.3433.
Since the sample estimator, 1.3433, is smaller than the critical acceptance value 1.3880 of the sampling plan, the buyer will reject the lot.
6. Concluding remarks
Acceptance sampling plan basically consists of a sample size(n) and an acceptance criterion (C0). Since the sam-pling cannot guarantee that every defective item in a lot will be inspected, then the sampling involves risks of not ade-quately reflecting the quality conditions of the lot. Such risk is even more significant as the rapid advancement of the manufacturing technology and stringent customers demand is enforced. Particularly, when the fraction of defectives is very low, such as in PPM, the required number of inspec-tion items must be enormously large in order to adequately reflecting the actual lot quality. In this paper, we introduce a variables sampling plan for one-sided processes based on the uniformly most powerful test of the capability indices CPU (or CPL), to deal with lot sentencing problem with very low fraction of defectives. The proposed new sampling plan is developed based on the exact sampling distribution rather than approximation. Therefore, practitioners can de-termine the number of required inspection units and the crit-ical acceptance value, and to make more accurate decision. To illustrate how the sampling plan can be established and applied to the actual data collected from the factories, a real world application to the EEPROM manufacturing process is also provided.
Appendix
The values of (n, C0) for producers -risk buyers -risk with various benchmarking quality levels are shown in Tables 5and6.
W .L. P earn, C.-W . W u /Ome ga 34 (2006) 90 – 101 97 Table 5
(n, C0) values for= 0.01(0.01)0.050,= 0.01(0.01)0.10 with selected (CAQL, CLTPD)
CAQL= 1.25 CAQL= 1.45 CAQL= 1.60 CAQL= 1.45 CAQL= 1.60 CAQL= 1.60 CLTPD= 1.00 CLTPD= 1.00 CLTPD= 1.00 CLTPD= 1.25 CLTPD= 1.25 CLTPD= 1.45 n C0 n C0 n C0 n C0 n C0 n C0 0.01 0.01 259 1.1216 93 1.2145 59 1.2837 554 1.3480 200 1.4192 1229 1.5216 0.02 228 1.1137 82 1.2006 51 1.2624 489 1.3417 176 1.4081 1096 1.5170 0.03 209 1.1080 75 1.1904 47 1.2499 449 1.3372 161 1.4000 1015 1.5137 0.04 195 1.1034 70 1.1821 44 1.2395 421 1.3337 151 1.3940 957 1.5112 0.05 185 1.0997 66 1.1749 41 1.2280 398 1.3305 142 1.3880 910 1.5089 0.06 176 1.0962 63 1.1691 39 1.2197 379 1.3277 136 1.3838 872 1.5069 0.07 168 1.0928 60 1.1629 37 1.2108 363 1.3252 130 1.3792 839 1.5051 0.08 162 1.0901 57 1.1562 36 1.2060 349 1.3229 125 1.3752 810 1.5034 0.09 156 1.0873 55 1.1515 34 1.1960 337 1.3207 120 1.3709 784 1.5018 0.10 150 1.0843 53 1.1465 33 1.1907 325 1.3185 116 1.3673 760 1.5003 0.02 0.01 231 1.1290 84 1.2281 53 1.3001 494 1.3541 179 1.4297 1084 1.5263 0.02 202 1.1211 73 1.2132 46 1.2800 432 1.3477 156 1.4183 959 1.5217 0.03 184 1.1152 67 1.2037 42 1.2665 395 1.3432 143 1.4107 884 1.5184 0.04 172 1.1109 62 1.1947 39 1.2551 369 1.3397 133 1.4041 829 1.5157 0.05 162 1.1069 58 1.1868 37 1.2467 347 1.3364 125 1.3984 786 1.5134 0.06 154 1.1034 55 1.1803 35 1.2377 330 1.3336 119 1.3937 750 1.5114 0.07 146 1.0997 52 1.1733 33 1.2280 315 1.3310 113 1.3886 719 1.5095 0.08 140 1.0967 50 1.1683 32 1.2228 302 1.3285 108 1.3841 692 1.5077 0.09 135 1.0941 48 1.1630 30 1.2116 290 1.3262 104 1.3803 668 1.5061 0.10 130 1.0913 46 1.1574 29 1.2057 280 1.3241 100 1.3762 647 1.5045 0.03 0.01 215 1.1345 78 1.2370 50 1.3133 458 1.3584 167 1.4374 996 1.5296 0.02 187 1.1265 68 1.2228 43 1.2926 399 1.3521 145 1.4261 877 1.5250 0.03 170 1.1207 62 1.2128 39 1.2784 363 1.3475 131 1.4175 805 1.5217 0.04 158 1.1162 57 1.2034 36 1.2662 337 1.3438 122 1.4112 753 1.5190 0.05 148 1.1119 54 1.1971 34 1.2573 317 1.3406 115 1.4059 712 1.5167 0.06 140 1.1082 51 1.1903 32 1.2476 300 1.3376 108 1.4000 678 1.5146 0.07 133 1.1048 48 1.1829 30 1.2370 286 1.3350 103 1.3955 648 1.5127 0.08 128 1.1021 46 1.1775 29 1.2313 274 1.3326 99 1.3917 623 1.5109 0.09 122 1.0987 44 1.1719 28 1.2253 263 1.3303 94 1.3865 600 1.5092 0.10 118 1.0963 42 1.1658 27 1.2190 253 1.3281 91 1.3833 580 1.5076
W .L. P earn, C.-W . W u /Ome ga 34 (2006) 90 – 101
CAQL= 1.25 CAQL= 1.45 CAQL= 1.60 CAQL= 1.45 CAQL= 1.60 CAQL= 1.60 CLTPD= 1.00 CLTPD= 1.00 CLTPD= 1.00 CLTPD= 1.25 CLTPD= 1.25 CLTPD= 1.45 n C0 n C0 n C0 n C0 n C0 n C0 0.04 0.01 203 1.1387 74 1.2446 47 1.3218 432 1.3619 157 1.4429 933 1.5323 0.02 175 1.1305 64 1.2300 41 1.3034 374 1.3555 136 1.4318 817 1.5276 0.03 159 1.1249 58 1.2196 37 1.2887 340 1.3510 123 1.4235 748 1.5243 0.04 147 1.1201 54 1.2117 34 1.2762 315 1.3473 114 1.4170 698 1.5217 0.05 138 1.1161 50 1.2029 32 1.2668 295 1.3440 107 1.4114 658 1.5193 0.06 131 1.1128 47 1.1956 30 1.2566 279 1.3411 101 1.4061 626 1.5173 0.07 124 1.1091 45 1.1904 28 1.2454 265 1.3384 96 1.4014 598 1.5153 0.08 118 1.1057 43 1.1848 27 1.2393 253 1.3358 92 1.3974 573 1.5135 0.09 113 1.1027 41 1.1788 26 1.2329 243 1.3336 88 1.3930 551 1.5118 0.10 109 1.1002 39 1.1724 25 1.2261 233 1.3312 84 1.3884 531 1.5101 0.05 0.01 193 1.1423 71 1.2514 45 1.3302 411 1.3649 150 1.4482 883 1.5346 0.02 167 1.1345 61 1.2365 39 1.3112 355 1.3586 129 1.4368 771 1.5300 0.03 151 1.1288 55 1.2257 35 1.2960 321 1.3540 117 1.4290 703 1.5267 0.04 139 1.1238 51 1.2176 32 1.2828 297 1.3503 108 1.4223 655 1.5240 0.05 130 1.1197 48 1.2108 30 1.2730 278 1.3470 101 1.4165 616 1.5216 0.06 123 1.1162 45 1.2033 28 1.2621 262 1.3440 95 1.4110 585 1.5196 0.07 117 1.1129 42 1.1951 27 1.2562 249 1.3414 90 1.4060 558 1.5176 0.08 111 1.1094 40 1.1891 26 1.2500 238 1.3390 86 1.4018 534 1.5158 0.09 106 1.1063 39 1.1860 25 1.2434 227 1.3364 82 1.3972 513 1.5141 0.10 102 1.1036 37 1.1793 24 1.2364 218 1.3341 79 1.3936 494 1.5124
W .L. P earn, C.-W . W u /Ome ga 34 (2006) 90 – 101 99 Table 6
(n, C0) values for= 0.06(0.01)0.10,= 0.01(0.01)0.10 with selected (CAQL, CLTPD)
CAQL= 1.25 CAQL= 1.45 CAQL= 1.60 CAQL= 1.45 CAQL= 1.60 CAQL= 1.60 CLTPD= 1.00 CLTPD= 1.00 CLTPD= 1.00 CLTPD= 1.25 CLTPD= 1.25 CLTPD= 1.45 n C0 n C0 n C0 n C0 n C0 n C0 0.06 0.01 185 1.1456 68 1.2568 44 1.3396 393 1.3675 144 1.4528 841 1.5367 0.02 159 1.1376 59 1.2432 37 1.3170 338 1.3611 124 1.4418 732 1.5321 0.03 144 1.1321 53 1.2323 34 1.3053 306 1.3567 112 1.4338 666 1.5288 0.04 133 1.1275 49 1.2239 31 1.2920 282 1.3529 103 1.4270 619 1.5261 0.05 124 1.1232 45 1.2145 29 1.2820 264 1.3498 96 1.4210 582 1.5238 0.06 117 1.1196 43 1.2093 27 1.2709 249 1.3469 90 1.4154 551 1.5217 0.07 110 1.1157 40 1.2008 26 1.2649 236 1.3441 86 1.4113 525 1.5197 0.08 105 1.1126 38 1.1946 25 1.2585 224 1.3414 82 1.4069 502 1.5179 0.09 100 1.1094 37 1.1914 23 1.2445 214 1.3390 78 1.4022 481 1.5161 0.10 96 1.1066 35 1.1844 22 1.2368 205 1.3366 74 1.3971 463 1.5145 0.07 0.01 179 1.1488 66 1.2626 42 1.3449 378 1.3699 139 1.4572 806 1.5386 0.02 153 1.1408 56 1.2471 36 1.3251 325 1.3637 119 1.4459 699 1.5340 0.03 138 1.1352 51 1.2377 32 1.3090 293 1.3592 107 1.4378 635 1.5308 0.04 127 1.1304 47 1.2291 30 1.2997 270 1.3555 98 1.4307 589 1.5281 0.05 118 1.1261 43 1.2194 28 1.2895 252 1.3523 92 1.4255 552 1.5257 0.06 111 1.1223 41 1.2141 26 1.2781 237 1.3493 86 1.4197 522 1.5236 0.07 105 1.1188 39 1.2083 25 1.2719 224 1.3465 82 1.4154 497 1.5217 0.08 100 1.1157 37 1.2021 23 1.2583 213 1.3439 78 1.4109 475 1.5199 0.09 95 1.1123 35 1.1953 22 1.2508 203 1.3414 74 1.4060 455 1.5181 0.10 91 1.1094 33 1.1879 21 1.2428 195 1.3393 71 1.4021 437 1.5164 0.08 0.01 173 1.1516 64 1.2676 41 1.3521 365 1.3722 134 1.4609 775 1.5404 0.02 147 1.1435 54 1.2519 35 1.3322 313 1.3661 115 1.4501 670 1.5358 0.03 133 1.1381 49 1.2423 31 1.3158 281 1.3615 103 1.4418 607 1.5326 0.04 122 1.1333 45 1.2335 29 1.3063 259 1.3579 95 1.4354 562 1.5299 0.05 113 1.1289 42 1.2260 27 1.2959 241 1.3546 88 1.4292 527 1.5276 0.06 107 1.1256 39 1.2178 25 1.2841 227 1.3517 83 1.4242 497 1.5254 0.07 101 1.1220 37 1.2118 24 1.2777 214 1.3488 78 1.4188 472 1.5235 0.08 96 1.1188 35 1.2052 23 1.2709 203 1.3462 74 1.4141 451 1.5217 0.09 91 1.1154 34 1.2017 22 1.2636 194 1.3439 71 1.4103 431 1.5199 0.10 87 1.1124 32 1.1942 21 1.2558 185 1.3414 68 1.4063 414 1.5183
W .L. P earn, C.-W . W u /Ome ga 34 (2006) 90 – 101
CAQL= 1.25 CAQL= 1.45 CAQL= 1.60 CAQL= 1.45 CAQL= 1.60 CAQL= 1.60 CLTPD= 1.00 CLTPD= 1.00 CLTPD= 1.00 CLTPD= 1.25 CLTPD= 1.25 CLTPD= 1.45 n C0 n C0 n C0 n C0 n C0 n C0 0.09 0.01 167 1.1541 62 1.2720 40 1.3586 354 1.3744 130 1.4646 747 1.5420 0.02 143 1.1465 53 1.2578 34 1.3384 302 1.3682 111 1.4537 644 1.5375 0.03 128 1.1407 47 1.2461 30 1.3217 271 1.3637 99 1.4452 582 1.5343 0.04 117 1.1358 43 1.2370 28 1.3121 249 1.3601 91 1.4387 538 1.5316 0.05 109 1.1318 40 1.2293 26 1.3013 232 1.3569 85 1.4332 504 1.5293 0.06 102 1.1278 38 1.2237 24 1.2892 217 1.3538 80 1.4282 475 1.5272 0.07 97 1.1248 36 1.2176 23 1.2825 205 1.3510 75 1.4227 451 1.5253 0.08 92 1.1215 34 1.2109 22 1.2754 195 1.3486 71 1.4178 430 1.5235 0.09 87 1.1179 32 1.2037 21 1.2678 185 1.3459 68 1.4139 411 1.5217 0.10 83 1.1149 31 1.1998 20 1.2596 177 1.3436 65 1.4098 394 1.5200 0.10 0.01 162 1.1566 60 1.2760 33 1.1907 343 1.3764 126 1.4680 722 1.5436 0.02 138 1.1489 51 1.2615 29 1.2057 292 1.3703 107 1.4569 620 1.5391 0.03 124 1.1434 46 1.2516 27 1.2190 262 1.3659 96 1.4490 560 1.5360 0.04 113 1.1384 42 1.2425 25 1.2261 240 1.3622 88 1.4424 517 1.5333 0.05 105 1.1343 39 1.2347 24 1.2364 223 1.3589 82 1.4368 483 1.5310 0.06 99 1.1309 37 1.2290 22 1.2368 209 1.3560 77 1.4317 455 1.5289 0.07 93 1.1272 34 1.2195 21 1.2428 197 1.3532 72 1.4260 431 1.5269 0.08 88 1.1238 33 1.2161 21 1.2558 187 1.3507 68 1.4211 411 1.5252 0.09 84 1.1209 31 1.2087 20 1.2596 178 1.3482 65 1.4170 392 1.5234 0.10 80 1.1177 30 1.2047 19 1.2623 170 1.3459 62 1.4127 376 1.5217
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