Acceptance Sampling

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Acceptance Sampling

William A. Brenneman a; William R. Myers a

a Department of Biometrics and Statistical Sciences, Health Care Research Center, Procter & Gamble,

Mason, Ohio, U.S.A.

Online Publication Date: 23 April 2003

To cite this Section Brenneman, William A. and Myers, William R.(2003)'Acceptance Sampling',Encyclopedia of Biopharmaceutical Statistics,1:1,1 — 8

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Acceptance Sampling

William A. Brenneman William R. Myers

Procter & Gamble, Mason, Ohio, U.S.A.

INTRODUCTION

In general, acceptance sampling is a statistical tool used to help make decisions concerning whether or not a batch (or lot) of product should be released for consumer consumption or use. Acceptance sampling, which will be discussed in this paper, should not be confused with the sampling plans and acceptance criteria mentioned in the United States Pharmacopeia 25 and National Formulary 20.[1] While acceptance sampling has been used exten-sively in many industries over the past 60 years, in far too many cases, inspection sampling and quality have been thought of as being synonymous. This is simply not the case because quality stems from a well-designed product produced from a well-designed and capable process. Once the product has been produced and the quality of the product has been fully determined, acceptance sampling can only give us a level of assurance that the product quality is at a certain level and that an Acceptable Quality Level (AQL) will be exposed to consumers. For this reason, the focus in the quality arena has changed from the inspection of finished product quality to the con-tinuous improvement of the product design and pro-cess performance.

While most of any quality efforts should be put to-ward the proactive approach found in a continuous qua-lity improvement program, acceptance sampling still plays a practical role. Situations for which acceptance sampling may be useful include: incoming inspection with a new supplier for which little is known about its capability or quality history; new processes where sta-tistical control is trying to be achieved; processes where the quality is relatively poor; and processes that expe-rience flare-ups or special causes. Furthermore, accept-ance sampling is required in current good manufacturing practice (cGMP) in pharmaceutical research and devel-opment.[2] However, the most attractive feature of ac-ceptance sampling is that it allows one to balance two risks: the risk to the producer and the risk to the con-sumer. These risks need to be fully understood in order to appreciate acceptance sampling methodology.

An example will be used in order to provide the basic ideas behind acceptance sampling, giving special atten-tion to the producer and the consumer risks. It turns out

that these two types of risks uniquely determine a plan, and a thorough understanding of these basic principles will help in understanding more complex plans. Double sampling plans as well as more complex sampling plans are discussed briefly. Variables sampling plans are in-troduced, and the derivation of their acceptance regions is discussed. Finally, some common misconceptions asso-ciated with acceptance sampling plans are presented, and thoughts concerning the use of ANSI/ASQC[3] sampl-ing procedures are given. Theoretical justification for acceptance sampling is outside the scope of this entry but can be found in Wallis,[4] Schilling,[5] and Ste-phens.[6] For a historical look at inspection sampling, see Sherman.[7] Acceptance sampling is reviewed ex-tensively from a Bayesian viewpoint by Wetherill,[8] Hald,[9] Moskowitz and Berry,[10] and Thyregod.[11] Bayesian cost models are reviewed by Hald,[12] Guen-ther,[13] Lorenzen,[14] and Wadsworth and Olsen.[15] Situations where the fraction nonconforming is very low are discussed in Hahn,[16] Pesotchinsky,[17] and Moreno and Reeves.[18] Suich[19] talks about the effect of inspection errors on the performance of accept-ance sampling plans. Finally, the debate concerning all (100%) or nothing (0%) inspection is given by Vander Weil and Vardeman.[20]

ACCEPTANCE SAMPLING TERMINOLOGY There are two types of risks that go into creating or evaluating any acceptance sampling plan. The producer’s risk is the probability that a lot of product that is at an AQL is rejected by the plan. The AQL is the maximum percent nonconforming that can be considered satisfact-ory as a process average. On the other hand, the con-sumer’s risk is the probability that a lot with unaccept-able quality is accepted by the plan. The unacceptunaccept-able quality level is frequently referred to as the Lot Tolerance Percent Defective (LTPD). These four quantities uniquely determine an acceptance sampling plan and can be thought of as the properties (or characteristics) of the plan. In formal hypothesis testing, the terminology used for the producer and the consumer risks are the Type I Encyclopedia of Biopharmaceutical Statistics 1 DOI: 10.1081/E-EBS 120007571

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and Type II error rates. In the acceptance sampling literature, plans based on binary data (pass/fail) are re-ferred to as attribute acceptance sampling plans, while plans based on continuous data are referred to as variable acceptance sampling plans. This common terminology will be used throughout. The attribute data could be for the physical evaluation of drug product (e.g., capsule damage, empty capsules, cracked or broken tablets, and coating defects), packaging material (e.g., illegible print) or incoming raw material from a supplier. Examples of variable data are tablet weight, tablet hardness, and bot-tle measurements.

ACCEPTANCE SAMPLING PLANS FOR ATTRIBUTES

Acceptance sampling plans can be determined from the four quantities: producer’s risk, AQL, consumer’s risk, and LTPD. For example, consider the creation of a single sampling plan for which the response of interest is an attribute. A single sampling plan then requires determining a sample size n and an acceptance num-ber c. The plan is carried out by counting the numnum-ber of nonconforming (or defective) units d in a sample of size n and then accepting the lot if d c. The sample size and the acceptance number are chosen so that the plan will have the specified properties: pro-ducer’s risk (a), AQL ( p1 = AQL/100), consumer’s

risk (b), and LTPD ( p2= LTPD/100). Assuming that

the binomial distribution is an appropriate distribution for modeling the sampling procedure, the sample size n and the acceptance number c are the solutions to the two equations: 1 a ¼ X c d¼ 0 n! d!ðn dÞ!p d 1ð1 p1Þnd ð1Þ b ¼ X c d¼ 0 n! d!ðn dÞ!p d 2ð1 p2Þnd ð2Þ

To be specific, suppose a single sampling plan for cracked tablets calls for a producer’s risk of 0.05, an AQL of 1%, a consumer’s risk of 0.10, and an LTPD of 5%. Then a sample size of 132 tablets and an acceptance number of 3 are required. To carry out the plan, one would randomly sample 132 tablets from a lot and count the number of cracked tablets found. In practice, this is often done on line. If the number of cracked tablets found is three or less, then the lot is accepted. Otherwise, the lot is rejected.

ACCEPTANCE SAMPLING GRAPHS

An important graph to construct for any sampling plan is called the Operating Characteristic (OC) curve. The OC curve depicts the probability of accepting a lot for a given true, but unknown, defect level. The defect level p is placed on the horizontal axis and the probability of acceptance is placed on the vertical axis. For a single sampling plan for attributes, the probability of acceptance is the probability that d is less than or equal to c, and is given by the expression:

Pa ¼ Pfd cg ¼ Xc d¼ 0 n! d!ðn dÞ!p d_{ð1 pÞ}nd ð3Þ

Fig. 1 shows the OC curve for the sampling plan where the sample size is 132 and the acceptance number is 3. The OC curve shows how well a sampling plan does at discriminating between lots with various defect levels. For example, if the true defect level is 2%, then the proposed acceptance sampling plan will have approxi-mately a 73% probability of accepting the lot. That is, if 100 lots from a process that produces 2% defective products are evaluated by the sampling plan, then we can expect, on average, to accept 73 of the lots and reject 27 of them. For a discussion on the effect of a sample size n and an acceptance number c on the OC curve, see Montgomery[21]or Stephens.[6]

Another important graph is one that depicts the Average Outgoing Quality (AOQ). The AOQ is the average lot quality that one would expect over time from a process producing fraction defective p. The AOQ only makes sense when the rejected lots are subject to 100% inspection and all nonconforming items are either removed or replaced with conforming items. Such sampling programs are called rectification inspection programs because the action of 100% inspection affects the final product quality. Because the only time defective items are released is when a lot is accepted, it is easy to see that the formula for AOQ is given by:

AOQ ¼ PapðN nÞ

N ð4Þ

The AOQ curve for the c = 3, n = 132 plan where the lot size is N = 100,000 units is given in Fig. 2.

Note that the AOQ curve obtains a maximum of 0.015 when the fraction defective is 0.023. This maximum is called the Average Outgoing Quality Limit (AOQL) and it represents the poorest average quality level achievable under a rectification inspection program. This means that no matter what quality level the process is producing, the average level of quality over a large number of lots will be no greater than 1.5%.

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The last graph that needs mentioning is the Average Total Inspection (ATI) graph. This graph shows the average number of samples that need to be taken under a rectification inspection program. If lots contain no de-fective items, then the number of samples per lot will be n, while if the lots contain all defective items then the number of samples will be N. If the proportion of de-fective items is between these two extremes, then the number of samples will be between n and N. In this case, the ATI per lot is given by the equation:

ATI ¼ n þ ð1 PaÞðN nÞ ð5Þ

The ATI graph for the c = 3, n = 132 plan where the lot size is N = 100,000 units is given in Fig. 3.

DOUBLE SAMPLING: NOT AN AD HOC EXTENSION OF SINGLE SAMPLING

Up to this point, we have discussed single sampling plans, where a decision on the lot is based on one sample. However, there are formal double sampling plans where-by, based on the results observed in the initial sample, a second sample may be required. A double sampling plan consists of the following components:

. Sample size of the first sample (n1);

. _{Acceptance number for first sample (c}₁_); . _{Rejection number for first sample (r}₁_); . _{Sample size of the second sample (n}₂_);

. _{Acceptance number for the first and second samples}

combined (c2); and

Fig. 2 The AOQ curve for a single acceptance sampling plan with producer’s risk = 5%, AQL = 1%, consumer’s risk = 10%, and LTPD = 5%. Sample size, n = 132; acceptance number, c = 3; and lot size, N = 100,000.

Fig. 1 The OC curve for a single acceptance sampling plan with producer’s risk = 5%, AQL = 1%, consumer’s risk = 10%, and LTPD = 5%. Sample size, n = 132; and acceptance number, c = 3.

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. Rejection number for the first and second samples combined (r2; r2 = c2+ 1).

Fig. 4 provides a schematic of the double sampling acceptance plan. The producer’s risk, AQL, consumer’s risk, and LTPD are the necessary input, as in the case of single sampling, in order to develop a double sampl-ing plan.

The main advantage of double sampling over single sampling is that in most cases, double sampling will reduce the average number of samples (Average Sample Number, ASN) taken. The greatest potential for reducing the average number of samples taken in a double sam-pling plan is when samsam-pling is stopped (called curtail-ment) when the total number of defects is greater than the second rejection number. To see the relationship between ASN for single and double sampling plans, consider the ASN curve graphed in Fig. 5. When curtailment is fol-lowed, gains in sample size reduction can be made over

single sampling plans, but when complete inspection is performed, the gains may not be realized.

Another advantage of double sampling plans is that they may give the user a psychological advantage knowing that if the first sample does not allow for acceptance, then the second sample may. The disadvantage of double sampling plans is that they are more compli-cated to administer.

Double sampling is not an ad hoc extension of single sampling. For example, let us assume that a quality assu-rance organization instituted a single sampling plan, but decided to take a second sample after the number of non-conforming units observed was greater than the predefined acceptance number. The result of this type of ad hoc double sampling plan is an inflation of the consumer’s risk. Double sampling plans can be extended to multiple sampling plans, whereby more than two samples could possibly be taken. An even greater extension to double and multiple sampling plans is the sequential sampling plan.

Fig. 4 Operating flow chart for double acceptance sampling program.

Fig. 3 The ATI curve for a single acceptance sampling plan with producer’s risk = 5%, AQL = 1%, consumer’s risk = 10%, and LTPD = 5%. Sample size, n = 132; acceptance number, c = 3; and lot size, N = 100,000.

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Schilling[5]and Stephens[6]provide an in-depth discussion on these types of sampling plans.

ACCEPTANCE SAMPLING PLAN FOR VARIABLES

Variable acceptance sampling plans are used when the quality characteristic can be measured on a continuous scale. Examples of variables are tablet weight, tablet hardness, and bottle measurements. Variables contain much more information about the quality level than do attributes. For example, if the weight of a tablet is to be between 2 and 6 mg, then reporting that a tablet weighs 2.1 mg is more informative than reporting that the tablets weight is within specification. The fact that much more information can be gained from a variable over an at-tribute is reason enough to argue that one should never artificially change a variable into an attribute.

The increase in information of a variable over an attribute is the main advantage of variables sampling over attributes sampling and leads to much smaller sample sizes for variables plans over attributes plans. For exam-ple, consider the example again where it is required to have a plan with a producer’s risk of 0.05, an AQL of 1%, a consumer’s risk of 0.10, and an LTPD of 5%. The variable sampling plan requires only 70 samples while the corresponding attributes sampling plan requires 132 samples.

The main advantage of attributes sampling over va-riables sampling is that in general, attributes sampling is less time-consuming and takes less expensive testing equipment. Also, the amount of record keeping and the calculations for variables are often thought of as being more difficult to manage than for attributes. Occasionally,

the advantage of simplicity of attribute plans can offset the reduction in sample size due to variables. Another advantage of attribute sampling is that the mathematical assumptions about how quality varies from one item to another are fulfilled as long as a random sample is taken from the lot. On the other hand, the mathematical as-sumption of variables plans is that quality varies from one item to another in the form of a normal distribution. Whether or not the normal distribution adequately fits the underlying distribution of a variable quality charac-teristic should be determined through appropriate sta-tistical methods.

Designing a variables sampling plan is similar to designing an attribute sampling plan in that the following four quantities need to be specified: producer’s risk (a), AQL ( p1 = AQL/100), consumer’s risk (b), and LTPD

( p2 = LTPD/100). From these values, the sample size n

and the constant k, which are then used to carry out the plan, can be determined. To carry out the plan, a sample of size n is taken and then the sample mean x¯ and sample standard deviation s are calculated. Then for a quality characteristic with a single specification limit, say an USL, the criteria for acceptability is that x¯ + ks USL. The formula for k is given by:

k ¼ zp2zaþ zp1zb

zaþ zb

ð6Þ

where zp is the standard normal quantile of the order

(1 p). That is, if z N(0,1), then p = Pr(Z > zp). The

formula for n depends on whether or not the variance is assumed to be known or unknown. In general, the recommendation is to assume that the variance is unknown and needs be estimated from the sample because in practice, the variance can change from lot to lot. However, if one has a very large amount of historical data

Fig. 5 The ASN curve for complete double sampling, curtailed double sampling, and single sampling plans with properties: producer’s risk = 5%, AQL = 1%, consumer’s risk = 10%, and LTPD = 5%.

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and can show that the variance is constant from lot to lot, then one may wish to assume that the variance is known by using an estimate from the historical data. The result of assuming that the variance is known is a reduction in sample size. When the variance is assumed to be known, the formula for n is given by:

n ¼ zaþ zb zp1 zp2

2

ð7Þ

and when the variance is assumed to be unknown, the formula for n is given by:

n ¼ 1þk 2 2 zaþ zb zp1 zp2 2 ð8Þ

Single-Sided Specification Limits

For single-sided specifications, the acceptance regions can be characterized algebraically.

Sigma known testing criteria

USL: Accept if x + ks USL; reject otherwise. LSL: Accept if x ks LSL; reject otherwise. Sigma unknown testing criteria

USL: Accept if x + ks USL; reject otherwise. LSL: Accept if x ks LSL; reject otherwise.

Two-Sided Specification Limits

When there are two-sided specification limits, then more care needs to be taken in finding the acceptance region.

Simply taking the intersection of an LSL acceptance region and a USL acceptance region will not always give the correct acceptance region.

A two-sided acceptance region is most easily shown on a graph. Fig. 6 shows the acceptance region for the plan, assuming unknown variance, with a producer’s risk of 0.05, an AQL of 1%, a consumer’s risk of 0.10, and an LTPD of 5%. For each sample taken, the sample mean and the sample standard deviation are plotted and if the point lies below the curve and above the horizontal axis, then the lot is accepted.

Wallis[4] provides the mathematical algorithm for finding the acceptance region boundary. The first step is to compute k and n from Eqs. 6 and 7, or Eq. 8. In this case, the p1and p2values represent the fraction of items

outside of the interval LSL – USL. The second step is to find the number p0such that zp0 ¼ k. The third step is to

divide p0into two parts p0’ and p0@ such that p0 = p0’ + p0@.

The fourth step is to calculate:

x ¼ USL*zp 0 0þ LSL*zp 00 0 zp0 0þ zp 00 0 and s ¼ USL LSL zp0 0þ zp 00 0 ð9Þ

which are points that lie on the acceptance region boundary; another point on the acceptance region boundary is (x’, s) where x’ = USL + LSL x.

COMMON ACCEPTANCE SAMPLING MISCONCEPTIONS

One of the most common misconceptions about accept-ance sampling plans concerns the misinterpretation of the

Fig. 6 The acceptance region for a variables sampling plan with two-sided specifications (LSL = 10 and USL = 20) and producer’s risk = 5%, AQL = 1%, consumer’s risk = 10%, and LTPD = 5%.

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AQL. Many users of sampling plans interpret the AQL as if it were the LTPD. For example, a user of a sampling plan with an AQL equal to 1% and a producer’s risk equal to 5% will often make the incorrect statement that one is 95% confident that the accepted lots have a defect level less than 1%. The correct confidence statement about the defect level of accepted lots needs to be based on the LTPD and the consumer’s risk. For example, if the sampling plan just mentioned has an LTPD equal to 5% and a consumer risk equal to 10%, then one can say with 90% confidence that the lots that are accepted have a defect level less than 5%.

Another misconception about sampling plans is that sample size should be proportional to lot size. One in-correct rule of thumb for developing sampling plans is that the sample size should be equal to 10% of the lot size with an acceptance number of zero. The presump-tion is that following this type of rule provides plans with identical properties for any lot size. However, this is not the case as depicted in Fig. 7, where we have OC curves for two zero acceptance sampling plans that follow the 10% rule. The first plan is derived by as-suming a lot size of 10,000 with a sample size of 1000 and the second plan is derived by assuming a lot size of 1000 with a sample size of 100. In order for these two plans to have identical properties, their associated OC curves would need to be identical. It is clear from the graph that this is not the case and that the sampling plan with a sample size of 1000 is much more sensitive to rejecting ‘‘poor-quality’’ lots.

Another misconception encountered is the square root of N + 1 (or square root of N) rule for determining the sample size for a sampling plan, where N is the lot size. There is no apparent statistical justification for this rule to be used in general.

The final misconception is that in general, lot size should be used to determine a sampling plan. Theoret-ically, because all lots are of finite size, the hypergeo-metric distribution is the one that provides the correct model. The hypergeometric distribution does take into account the lot size and so any acceptance sampling plan developed under the hypergeometric distribution will need to take lot size into consideration. However, in most applications, the binomial distribution approximates the hypergeometric distribution so closely that lot size does not play a significant role in determining a sampling plan. A general guideline to determine if lot size should be taken into consideration is to calculate n/N, where n is the sample size and N is the lot size, and to see if this quantity is less than or equal to 0.1. That is, if the sample size is greater than 10% of the lot size, then the lot size should be considered when determining a sampling plan. However, a reasonable approximation is obtained even beyond this guideline.

THOUGHTS ON THE ANSI/ASQC SAMPLING PROCEDURES

Another popular method for developing an acceptance sampling plan is to use the ANSI/ASQC Z1.4-1993: Sampling Procedures and Tables for Inspection by Attributes,[3] formally the MIL STD 105E. This set of tables allows one to derive an acceptance sampling plan that is based on batch size, inspection level, and AQL. Experience has shown that many of those who implement these plans do not completely understand their properties. These sampling plans are AQL-oriented, in that they focus attention on the producer’s risk. This means that as long as the process average of nonconforming units is at

Fig. 7 The OC curve for two zero acceptance sampling plans for which sample size is equal to 10% of lot size.

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or below the prespecified AQL, the probability of ac-ceptance will be high (at or above 90%). However, these sampling plans do not directly address the consumer’s risk and, consequently, do not necessarily reject ‘‘poor-quality’’ lots with high probability. This should be clearly understood by those who use these sampling plans. In order to understand the consumer’s risk, one needs to observe the OC curve for the given sampling plan. There are OC curves available in the ANSI/ASQC Z1.4-1993. Also, a common misuse of ANSI/ASQC Z1.4 is failure to use the switching rule. The general concept of the switching rule is that if lots are continuously accepted, then a ‘‘reduced’’ sampling plan is instituted. However, if lots are continuously rejected, then a ‘‘tightened’’ sampling plan is established. By invoking the switching rules, the consumer risk will be reduced. Consequently, switching rules should be implemented when using these procedures. Finally, we want to mention that tables for deriving variables sampling plans can be found in the ANSI/ASQC Z1.9-1993: Sampling Procedures and Tables for Inspection by Variables for Percent Non-conforming.[22]

CONCLUSION

Acceptance sampling still has a role in the pharmaceutical industry. In reality, inspection is seen to have value when compared to the catastrophic results of performing no inspection when an unsatisfactory level of quality exists. In addition, acceptance sampling is an audit tool to ensure that the output of a process conforms to requirements. Therefore, it is essential for those implementing accept-ance sampling plans to fully understand their properties and to be aware of their common misconceptions. Experience shows that the most common misconception is that the AQL has the interpretation of the LTPD. This may lead to overconfidence in the acceptance sampling plans’ ability to reject lots of poor quality. This certainly applies when one selects an acceptance sampling plan from the ANSI/ASQC Sampling Procedure.

REFERENCES

1. United States Pharmacopeia 25 and National Formulary 20; United States Pharmacopeial Convention: Rockville, MD, 2001.

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3. ANSI/ASQC Z1.4-1993: American National Standard: Sampling Procedures and Tables for Inspection by Attributes; ASQ Quality Press: Milwaukee, WI, 1993. 4. Wallis, W.A. Use of Variables in Acceptance Inspection for

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6. Stephens, K.S. The Handbook of Applied Acceptance Sampling: Plans, Procedures and Principles; ASQ Quality Press: Milwaukee, WI, 2001.

7. Sherman, W.H. Inspection Through the Years. ASQC 50th Annual Quality Congress Proceedings; 1996; 571 – 574, (May).

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16. Hahn, G.J. Estimating the percent nonconforming in the accepted product after zero defect sampling. J. Qual. Technol. 1986, 18 (3), 182 – 188.

17. Pesotchinsky, L. Plans for very low fraction nonconform-ing. J. Qual. Technol. 1987, 19 (4), 191 – 196.

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19. Suich, R. The effects of inspection errors on acceptance sampling for nonconformities. J. Qual. Technol. 1990, 22 (4), 314 – 318.

20. Vander Wiel, S.A.; Vardeman, S.B. A discussion of all-or-none inspection policies. Technometrics 1994, 36 (1), 102 – 109.

21. Montgomery, D.C. Introduction to Statistical Quality Control; John Wiley: New York, 1997.

22. ANSI/ASQC Z1.9-1993: American National Standard: Sampling Procedures and Tables for Inspection by Variables for Percent Nonconforming; ASQ Quality Press: Milwaukee, WI, 1993.