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Acceptance sampling
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Acceptance sampling uses statistical sampling to determine whether to accept or reject a production lot of material. It has been a common quality control technique used in industry.
It is usually done as products leave the factory, or in some cases even within the factory. Most often a producer supplies a consumer with several items and a decision to accept or reject the items is made by determining the number of defective items in a sample from the lot. The lot is accepted if the number of defects falls below where the acceptance number or otherwise the lot is rejected.[1]
In general, acceptance sampling is employed when one or several of the following hold:[2]
- testing is destructive;
- the cost of 100% inspection is very high; and
- 100% inspection takes too long.
A wide variety of acceptance sampling plans is available. For example, multiple sampling plans use more than two samples to reach a conclusion. A shorter examination period and smaller sample sizes are features of this type of plan. Although the samples are taken at random, the sampling procedure is still reliable.[3]
History
[edit]Acceptance sampling procedures became common during World War II. Sampling plans, such as MIL-STD-105, were developed by Harold F. Dodge and others and became frequently used as standards.
More recently, quality assurance broadened the scope beyond final inspection to include all aspects of manufacturing. Broader quality management systems include methodologies such as statistical process control, HACCP, six sigma, and ISO 9000. Some use of acceptance sampling still remains.
Rationale
[edit]Sampling provides one rational means of verification that a production lot conforms to the requirements of technical specifications. 100% inspection does not guarantee 100% compliance and is too time-consuming and costly. Rather than evaluating all items, a specified sample is taken, inspected or tested, and a decision is made about accepting or rejecting the entire production lot.
Sampling plans have known risks: an acceptable quality limit (AQL) and a rejectable quality level, such as lot tolerance percent defective (LTDP), are part of the operating characteristic curve of the sampling plan. These are primarily statistical risks and do not necessarily imply that a defective product is intentionally being made or accepted. Plans can have a known average outgoing quality limit (AOQL).
Acceptance sampling for attributes
[edit]A single sampling plan for attributes is a statistical method by which the lot is accepted or rejected on the basis of one sample.[4] Suppose that we have a lot of sizes ; a random sample of size is selected from the lot; and an acceptance number is determined. If it is found the number of nonconforming is less than or equal to , the lot is accepted; and if the number of nonconforming is greater than , the lot is not accepted. The design of a single sampling plan requires the selection of the sample size and the acceptance number .
MIL-STD-105 was a United States defense standard that provided procedures and tables for sampling by attributes (pass or fail characteristic). MIL-STD-105E was cancelled in 1995 but is available in related documents such as ANSI/ASQ Z1.4, "Sampling Procedures and Tables for Inspection by Attributes." Several levels of inspection are provided and can be indexed to several AQLs. The sample size is specified and the basis for acceptance or rejection (number of defects) is provided. MIL-STD-1916 is currently the preferred method of sampling for all Department of Defense (DoD) contracts.
Variables sampling plan
[edit]When a measured characteristic produces a number, other sampling plans, such as those based on MIL-STD-414, are often used. Compared with attribute sampling plans, these often use a smaller sample size for the same indexed AQL.
See also
[edit]References
[edit]- ^ Kreyszig, Erwin (2006). Advanced Engineering Mathematics, 9th Edition. Wiley. p. 1248. ISBN 978-0-471-48885-9.
- ^ Eraldo Banovac, Dražan Kozak. "An analytic review of the characteristics of the Lot Acceptance Sampling Plans used for acceptance of large lots". International Review of Electrical Engineering (I.R.E.E.), Vol. 3, No. 6, November–December 2008, pp. 1070-1076.
- ^ Eraldo Banovac, Igor Kuzle. Applicability of the LASPs in the electric-power industry. Proceedings of the International IEEE Conference EUROCON 2009, Saint Petersburg, Russia, May 18–23, 2009, pp. 1152-1157.
- ^ Montgomery, D. C. (2009). Statistical Quality Control: A Modern Introduction, Wiley, ISBN 978-0-470-23397-9
Books
[edit]- Pyzdek, T, "Quality Engineering Handbook", 2003, ISBN 0-8247-4614-7
- De Feo, J. A., "Juran's Quality Handbook", 2016, ISBN 978-1-25964-361-3
- ASTM E105 Standard Practice for Probability Sampling of Materials
- ASTM E122 Standard Practice for Calculating Sample Size to Estimate, With a Specified Tolerable Error, the Average for Characteristic of a Lot or Process
- ASTM E141 Standard Practice for Acceptance of Evidence Based on the Results of Probability Sampling
- ASTM E1402 Standard Terminology Relating to Sampling
- ASTM E1994 Standard Practice for Use of Process Oriented AOQL and LTPD Sampling Plans
- ASTM E2234 Standard Practice for Sampling a Stream of Product by Attributes Indexedby AQL
- Sampling procedures for inspection by attributes, ISO 2859-1:1999
- Sampling procedures for inspection by attributes, JIS Z 9015-1:2006
External links
[edit]- Acceptance Sampling Calculators (SQC Online) (A subscription fee is required to use the calculators. The "free" calculations have locked features.)
Acceptance sampling
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Fundamentals
Definition and Purpose
Acceptance sampling is a statistical quality control procedure in which a random sample is selected from a lot or batch of products to evaluate whether the entire lot meets predefined quality criteria, resulting in either acceptance or rejection of the lot.[7][8] This method applies to both attribute sampling, which assesses discrete characteristics like the presence of defects, and variables sampling, which measures continuous traits such as dimensions or weights.[4] Key terminology includes the lot, defined as the aggregate batch submitted for inspection; the sample, a randomly drawn subset from the lot for examination; the acceptance number (often denoted as $ c $ or Ac), the maximum allowable number of defects or nonconformities in the sample for the lot to be accepted; and the rejection number (denoted as $ r $ or Re), the threshold of defects that triggers lot rejection.[7][8] These elements form the basis of sampling plans, which specify sample size and decision rules to ensure representative evaluation.[9] The primary purpose of acceptance sampling is to balance the costs of inspection against the risks associated with quality decisions, thereby reducing the need for resource-intensive 100% inspection while upholding acceptable quality standards.[10] It mitigates the producer's risk ($ \alpha \beta $), the probability of accepting a lot exceeding the lot tolerance percent defective (LTPD).[11][8] By serving as an intermediate approach between no inspection and full inspection, it efficiently determines lot acceptability without estimating overall quality.[9] This technique emerged as an alternative to complete inspection amid wartime production constraints during World War II, when rapid output was prioritized.[12] Operating characteristic (OC) curves illustrate the performance of these plans by plotting the probability of lot acceptance against varying defect levels, aiding in risk assessment.[13]Key Concepts
Acceptance sampling relies on several core parameters to define quality thresholds and associated risks, ensuring that sampling plans balance efficiency with reliability in quality assurance. The Acceptable Quality Limit (AQL) represents the maximum percentage of defects that is considered tolerable for a process, serving as the baseline for designing sampling plans where lots at or below this level have a high probability of acceptance.[7] For instance, an AQL of 1% indicates that lots with 1% or fewer defects are likely to be accepted, reflecting a satisfactory process average over a series of lots.[8] In contrast, the Lot Tolerance Percent Defective (LTPD) specifies the poorest quality level in an individual lot that should be rejected with high probability, typically 90% (corresponding to a consumer's risk β of 0.10), and is expressed as a percentage defective associated with a low consumer risk.[7][14] An example is an LTPD of 4%, where lots exceeding this defect rate are expected to be rejected to protect the consumer from poor quality.[8] These quality levels are evaluated through the lens of producer's and consumer's risks, which quantify the errors inherent in sampling decisions. The producer's risk (α), or Type I error, is the probability of incorrectly rejecting a good lot at the AQL, typically set at 0.05 to ensure at least 95% acceptance for satisfactory quality.[15] The consumer's risk (β), or Type II error, is the probability of accepting a bad lot at the LTPD, commonly valued at 0.10, meaning a 10% chance of passing unacceptable quality.[14] The discrimination ratio, defined as the ratio of LTPD to AQL, measures a plan's ability to distinguish between acceptable and unacceptable quality levels; a higher ratio, such as 4:1, indicates stronger differentiation between the two thresholds. Another important metric is the Average Outgoing Quality (AOQ), which estimates the expected proportion of defects in the product shipped after sampling and any rectification of rejected lots.[7] This value helps assess the overall quality protection provided by the sampling plan, particularly when rejected lots are fully inspected and defects are replaced, resulting in an AOQ that peaks at an intermediate defect level before declining.[8] These concepts underpin the operating characteristic (OC) curves used to evaluate plan performance.[8]Historical Development
Origins in Quality Control
The roots of acceptance sampling trace back to pre-20th century manufacturing practices, where product inspection emerged as a fundamental aspect of the emerging factory system in Great Britain during the mid-1750s, emphasizing manual checks to ensure basic conformity amid the onset of the Industrial Revolution.[16] These early efforts relied on 100% inspection by skilled craftsmen or overseers, but lacked statistical rigor, often leading to inefficiencies in large-scale production.[17] Formalization of quality control practices began in the early 1900s, with significant advancements at Bell Laboratories, where physicist Walter Shewhart developed control charts in the 1920s to monitor process variation statistically, laying the groundwork for sampling-based inspection over exhaustive checking.[2] Shewhart's work at Western Electric, a Bell Labs affiliate, shifted focus from reactive inspection to proactive statistical methods, influencing the transition toward acceptance sampling as a tool for efficient quality assurance. The catalyst for widespread adoption of acceptance sampling occurred during World War II, when statisticians Harold F. Dodge and Harry G. Romig at Bell Laboratories developed sampling plans in the 1930s and 1940s to alleviate inspection bottlenecks in high-volume munitions production, enabling faster throughput without compromising reliability.[18] These plans, initially focused on attribute inspection—classifying items as conforming or nonconforming—were designed to balance producer and consumer risks under wartime pressures.[19] In response, the U.S. military adopted Army Ordnance sampling tables in the early 1940s for ordnance inspection, standardizing procedures that supported massive wartime output.[20] Early acceptance sampling methods were primarily limited to attribute-based approaches, which provided binary outcomes but offered less precision for measuring process variability compared to later variables sampling plans that incorporated quantitative measurements.[4] This attribute focus suited the immediate needs of wartime inspection but highlighted the need for more sophisticated techniques in post-war industrial applications.[21]Evolution and Key Contributors
Following World War II, the U.S. military formalized acceptance sampling procedures to ensure consistent quality in procurement, issuing MIL-STD-105A in 1950 as the first standardized tables for attribute sampling plans based on acceptable quality limits (AQL).[22] This standard, later revised through MIL-STD-105E in 1989 and superseded by the civilian ANSI/ASQ Z1.4 in 1991, provided single, double, and multiple sampling schemes for lot inspection.[22] Complementing this, MIL-STD-414 was released in 1957, offering variables sampling plans that estimate quality characteristics like mean and variance from sample data, with its civilian counterpart ANSI/ASQ Z1.9 following in 2003.[2] Key figures advanced these foundations significantly. Harold F. Dodge and Harry G. Romig, statisticians at Bell Laboratories, developed comprehensive single and double sampling inspection tables in their 1959 book Sampling Inspection Tables: Single and Double Sampling, which influenced military standards and emphasized practical tables for rectifying inspection. Walter A. Shewhart, also from Bell Labs, integrated acceptance sampling with his pioneering control charts from the 1920s, promoting a shift from inspection-only approaches to process control that complemented sampling for ongoing quality monitoring.[2] Eugene L. Grant and Richard S. Leavenworth expanded on these in their influential 1979 textbook Statistical Quality Control (4th edition), detailing economic considerations and broader applications of sampling plans within quality systems.[23] During the 1960s and 1980s, acceptance sampling evolved toward economic optimization and computational efficiency. Researchers introduced models to design plans minimizing total inspection costs, balancing sampling risks and quality protection, as seen in works on sequential and adaptive schemes. Computer-aided tools emerged in the late 1970s and 1980s, enabling simulation of operating characteristics and custom plan generation, reducing reliance on pre-tabulated standards. Internationally, ISO 2859 was established in 1974 (with key revisions through the 1980s, including Part 1 in 1989), harmonizing attribute sampling globally and aligning with AQL-based systems like MIL-STD-105. By the 2020s, acceptance sampling has integrated with methodologies like Six Sigma, where it supports the Measure and Analyze phases of DMAIC for lot acceptance decisions.[24] Emerging adaptive sampling approaches dynamically adjust plans based on process yield and quality loss to optimize inspection, though core statistical plans like those in ANSI/ASQ Z1.4 remain foundational.[25]Theoretical Foundations
Statistical Rationale
Acceptance sampling provides a probabilistic foundation for inferring the quality of an entire lot based on a representative sample, allowing decisions on acceptance or rejection without inspecting every unit. This inference relies on statistical distributions that model the occurrence of defects in the sample: the hypergeometric distribution for exact calculations in finite lots sampled without replacement, the binomial distribution as an approximation when the lot size is much larger than the sample, and the Poisson distribution for cases of rare defects. In variables sampling, the normal distribution is typically used to analyze continuous measurements and estimate lot parameters like mean and variance. These distributions enable the calculation of acceptance probabilities, ensuring that sample results reliably reflect lot quality under the assumption of randomness.[8] From an economic perspective, acceptance sampling justifies partial inspection over full or no inspection by minimizing total costs in high-volume production scenarios. The total cost is formulated as the sum of inspection costs (fixed setup plus variable per-unit costs) and failure costs (penalties from accepting defective lots or rejecting good ones), with optimal plans derived to balance these through techniques like direct search optimization. This approach is particularly valuable when 100% inspection is destructive, time-consuming, or prohibitively expensive, as sampling reduces inspection efforts while maintaining quality safeguards.[26] Sampling plans incorporate risk balancing to protect both producers and consumers: the acceptable quality limit (AQL) defines the defect level at which the probability of acceptance is high (1-α, where α is the producer's risk of rejecting good lots), while the lot tolerance percent defective (LTPD) sets the threshold for low acceptance probability (β, the consumer's risk of accepting poor lots). These parameters ensure equitable protection, with plans tailored to specified α (often 0.05) and β (often 0.10) values.[9] Despite these strengths, acceptance sampling has limitations, including its reliance on random sampling and lot homogeneity for valid inferences; violations can lead to biased results. It is not suited for ongoing process improvement, as it only screens lots rather than addressing root causes of variation—for that, statistical process control tools like control charts are essential.[9]Operating Characteristic Curves
The operating characteristic (OC) curve is a graphical representation in acceptance sampling that plots the probability of acceptance (Pa) of a lot against the proportion of defects (p) in the lot, illustrating the sampling plan's ability to discriminate between acceptable and unacceptable quality levels.[13] This curve serves as the primary tool for evaluating the performance of a sampling plan, such as a single sampling plan defined by sample size n and acceptance number c, by showing how effectively it protects both producer and consumer interests.[8] For attribute sampling plans, the OC curve is constructed using the binomial distribution, assuming the lot size is large relative to the sample size. The probability of acceptance Pa(p) is the cumulative probability that the number of defects in the sample is at most c, given by the equation:
where is the binomial coefficient.[13] For variables sampling plans, the curve is derived from the normal distribution, where measurements of a quality characteristic are assumed to follow a normal distribution with known or estimated standard deviation; Pa is calculated as the probability that the sample mean falls within acceptance limits, often using z-scores standardized by the process standard deviation and sample size.[3]
Key features of the OC curve include its steepness, which measures the plan's discriminatory power—the steeper the curve, the better it distinguishes good lots (low p) from bad ones (high p).[13] The producer's risk α is defined as 1 - Pa at the acceptable quality level (AQL), typically a small value like 0.05 indicating low chance of rejecting good lots, while the consumer's risk β is Pa at the lot tolerance percent defective (LTPD), often around 0.10 to limit acceptance of poor lots.[13] In interpretation, an ideal OC curve approaches 1 for Pa at low p (accepting good lots) and 0 at high p (rejecting bad lots); for sequential sampling plans, the OC curve is complemented by an average sample number (ASN) curve, which plots the expected sample size required as a function of p to further assess efficiency.[8]