*The Business Professor*, updated March 5, 2019, last accessed August 8, 2020, https://thebusinessprofessor.com/lesson/acceptance-sampling/.

### Acceptance Sampling Definition

Acceptance Sampling is a method used in the industry for quality control. This method uses statistical sampling to inspect or test a random sample for determining whether the quality of a batch of product or service is acceptable or not.

This method is used for quality control of the products or services when the cost of 100% inspection or test is too high or time-consuming or when the test destroys the product. Acceptance sampling is considered to be an effective and efficient means to ensure the quality control of such products or services. There are mainly two different methods used for this purpose- sampling by attributes and sampling by variables.

### A Little More on What is Acceptance Sampling

The acceptance sampling method was first popularized by Dodge and Roming and it was applied by the U.S military for testing the bullets during the Second World War. During a war, each of the bullets cannot be tested beforehand as that would exhaust the bullets, on the other hand, one cannot take the risk of using bullets in the battlefield without testing it. Acceptance sampling method gives a middle ground in such a scenario. In this method, samples are picked at random from a lot and those samples are tested. Based on the result of this sample test decisions are taken whether to accept the lot or to reject it. It is the middle-road between 100% inspection and no inspection.

In general, Acceptance Sampling is done when a product is finished and leaves the production plant, but in some cases, it may also be done within the factory. It is to be remembered that this method helps in determining whether or not to accept a batch of a product, but it does not estimate the quality of the lot. Usually, the manufacturer supplies a few samples from the lot to the consumer. If the number of defects is lower than the acceptable number, the lot is accepted by the consumer.

There are two major types of acceptance sampling plans, one is by attributes and the other is by variables. Although the former is most common for acceptance sampling, the latter is also used in many cases.

**Attribute Acceptance Sampling Plan**

In this Acceptance Plan, a random sample size (which is less than the size of the lot) is selected from the lot. Then Acceptance Quality Level or AQL is determined. AQL is the maximum percent defective that can be tolerated as a process average. In other words, it is the maximum number or percentage of defective pieces in a ‘good lot’. Rejectable Quality Level or RQL signifies the percentage defectives in a lot that can be tolerated in only as a specified proportion of lots.

For example, a company receives a shipment of 10,000 pieces of the product X. The receiving company cannot afford to inspect each of the products received. They can apply an attribute sampling plan to determine how many pieces of the products need to be inspected (sample size) and how many defective products are allowed in that sample (acceptance number).

Suppose the AQL is 1.5% and the RQL is 5.0% and it is assumed that alpha=0.05 and beta=0.1. Then the sampling plan indicates 209 pieces need to be examined and if 6 or less than 6 of the examined pieces are defective, the shipment is to be accepted and if more than 6 is defective the entire shipment is to be rejected.

**Variables Acceptance Sampling Plan**

It is comparatively a complex method and requires an understanding of the statistical model of normal distribution. In this method, variables or measurement data are used instead of attribute data and the defective rate and the defective rate varies with the mean, standard deviation, and distribution shape.

### References for Acceptance Sampling

- https://www.investopedia.com/terms/a/acceptance-sampling.asp
- https://en.wikipedia.org/wiki/Acceptance_sampling
- http://www.businessdictionary.com/definition/acceptance-number.html
- https://en.wikipedia.org/wiki/Acceptance_sampling#Rationale
- https://www.isixsigma.com/dictionary/acceptance-number/

### Academic Research on Acceptance Sampling

- • Multivariate quality control, Edward Jackson, J. (1985). Communications in Statistics-Theory and Methods, 14(11), 2657-2688. The paper delves into the motivation for multivariate quality control and some of the currently available techniques. The paper primarily emphasizes on control charts and included the T2-chart. It discusses the use of principal components and talks about some recent developments, multivariate analogs of CUSUM charts and the use of the Andrews procedure. The paper highlights some of the issues associated with multivariate acceptance sampling and presents some recommendations for further research.
- • Quality control techniques for” zero defects”, Calvin, T. (1983). IEEE Transactions on Components, Hybrids, and Manufacturing Technology, 6(3), 323-328. The paper deals with the idea of “zero defects” and proposes new quality control techniques for “zero defects”. The paper argues, to achieve “zero defect” new approaches to quality control is necessary. It discusses the existing techniques and their lacking in assuring zero defects. The paper says, properly chosen specification is a way to zero defects but if it is necessary to use attributes data, the standard p and u charts are not very helpful. Rather the paper suggests using a control chart that plots the number of good items between defects on a logarithmic scale to accommodate large numbers, establishing the upper and lower limits on the number of items between defects. The number of accepted lots between rejected lots can be criterion while applying this approach on good items between defects. Here, the lot sizes may determine the sample sizes like in MIL-STD-105D. The paper argues, to achieve zero defects, a procedure can be implemented established requiring process shutdown until the problems are resolved if rejected lots come too close together.
- • Multiparameter hypothesis testing and acceptance sampling, Berger, R. L. (1982). Technometrics, 24(4), 295-300. Several parameters are to be considered while determining the quality of a product. Each of these parameters needs to meet a certain standard in order to become acceptable. This article proposes a method to determine whether all the parameters meet the required standards. In this method, each of the parameters is examined individually and decides that the product is acceptable only when all the parameters meet the required standard. The method discussed in this article has some optimal properties that include attainting exactly a prespecified consumer’s risk and uniformly minimizing the producer’s risk.
- • Probabilistic verification of discrete event systems using acceptance sampling, Younes, H. L., & Simmons, R. G. (2002, July). This paper proposes a model-independent method to verify properties of discrete event systems. In order to avoid the complexities associated with such systems, the paper resorts to a method based on Monte Carlo simulation and statistical hypothesis testing. The result of the verification as well as the properties expressed as CSL formulas both can be probabilistic. Two parameters passed to the verification procedure affect the probability of error. This paper proposes, the verification of properties can be done in an anytime manner, in the beginning with loose error bounds and as the process continues gradually tightening these bounds.
- • Acceptance sampling based on life tests: Log-logistic model, Kantam, R. R. L., Rosaiah, K., & Rao, G. S. (2001). Journal of applied statistics, 28(1), 121-128. The problem of applying acceptance sampling in the cases of truncated life test at a pre-assigned time is taken into account in this research. The paper assumes that the lifetime variate of the test products follows a distribution that belongs to Burr’s family XII of distributions also known as the log-logistic model. Under this assumption, the minimum sample size required for ensuring the specified mean life is calculated for different acceptance numbers, confidence levels and values of the ratio of the specific experimental time to the fixed mean life. The producer’s risk and operating characteristic values of the sampling plans are discussed in the paper. An example is included for illustrating the results.
- • Acceptance sampling based on the inverse Rayleigh distribution, Rosaiah, K., & Kantam, R. R. L. (2005). Economic Quality Control, 20(2), 277-286. The paper considers the inverse Rayleigh distribution as a model for a lifetime random variable. It discusses the problem of applying acceptance sampling in the cases of truncated life test at a pre-assigned time. The paper provides the detailed calculation of the minimum sample size needed for ensuring a specified average life for different acceptance numbers, different confidence levels and different values of the ratio of the fixed experimental time to the specific average life. The paper obtains the operating characteristic functions of the sampling plan. It also discusses the producer’s risk. It provides tabulation for the ratio of true average life to a specific average life ensuring acceptance with a pre-assigned probability. An example is included for illustrating the results.
- • A review of acceptance sampling schemes with emphasis on the economic aspect, Wetherill, G. B., & Chiu, W. K. (1975). Some major principles of acceptance sampling schemes are reviewed in this paper. It emphasizes the economic aspect and recent developments while reviewing the schemes. This paper also provides a classified bibliography.
- • Design of economically optimal acceptance sampling plans with inspection error, Ferrell Jr, W. G., & Chhoker, A. (2002). Computers & Operations Research, 29(10), 1283-1300. This paper provides mathematical models that can be used for designing both 100% inspection and single sampling plans. The paper argues ss human inspectors are often involved in actual implementation, the human error needs to be taken into account while developing a model and it claims to explicitly include inspection error in the model. In the 100% inspection cases, the inspection tolerance is the design parameter and resources expended for reducing inspection error. In single sampling, the suitable model can be used for determining the optimal inspection tolerance and resource disbursement in a given sampling plan or when a prescribed inspection tolerance and resource expenditure is given, the sampling plan for the sample size and inspection number can be found. All of these uses are explained in the paper with easy to understand examples that can be modified according to the need.
- • A variables sampling plan based on Cpmk for product acceptance determination, Wu, C. W., & Pearn, W. L. (2008). European Journal of Operational Research, 184(2), 549-560. This paper proposes a variable sampling plan for product sentencing (acceptance determination). The proposed plan is based on the process capability index Cpmk. The paper claims the plan will help in making more accurate and reliable decisions as it is based on the exact sampling distribution. The paper provides tables for the required sample sizes and respective critical acceptance values for various producer’s risk, the consumer’s risk, and the capability requirements acceptance quality level and the lot tolerance percent defective. It also provides a case study for explaining how to construct and apply the proposed plan in the real-life scenario.
- • Economic selection of a process level under acceptance sampling by variables, Carlsson, O. (1989). Engineering Costs and Production Economics, 16(1), 69-78. The paper argues the effect of the choice of process level for an industrial process is dependent both on internal as well as external conditions. It considers the production costs and process variability as the internal cost and prices and control plan as the external cost. The paper concludes, if the control plan is given as a tolerance interval, the exact and the approximate optimal process level and the exact and the approximate optimal expected net income per lot. In deriving this the paper assumes that the cost and price functions are linear, and the quality characteristic is normally distributed. The whole idea illustrated by an industrial example.
- • Critical acceptance values and sample sizes of a variables sampling plan for very low fraction of defectives, Pearn, W. L., & Wu, C. W. (2006). Omega, 34(1), 90-101. This paper introduces a variables sampling plan for unilateral processes that are developed on the basis of the one-sided process capability indices. It is based on the exact sampling distribution and not the approximation. This plan can be used for determining an exact number of products to be examined and the corresponding critical acceptance value to reach reliable decisions. The paper also provides tabulation for the required sample size and the corresponding critical acceptance value for various -risks and the levels of lot or process fraction of defectives corresponding to acceptable and rejecting quality levels.