In statistical quality control, acceptance sampling is one of the most important tools to control and enhance quality. This sampling is a testing procedure where samples of lots are compared with set standards. The nature of a lot can be raw material, components, or finished goods. Inspection of a lot can occur after production or before supplying to the producer. The customer’s inspection is made after receiving the product. Inspection by the producer is termed outgoing inspection, and that by the customer is known as incoming inspection [
1]. There are two primary types of the acceptance sampling plan, variable acceptance and attribute acceptance. Variable acceptance plans are based on characteristics that can be measured, while attribute acceptance plans are based on binary decisions [
1]. According to Montgomery [
1], acceptance sampling plans can be classified into two groups: variables and attributes. Variables are evaluated on mathematical scales, attributes deal with qualitative characteristics and are stated as binary results. Acceptance sampling plans for attributes can be managed easily. Attribute acceptance sampling plans are easier to execute than variable acceptance plans, therefore, their application is common in the industry to examine lots. According to Montgomery [
2], if the customer’s process identifies no faulty items, and no other valid reason for inspection exists, supplies can be accepted without inspection. If the customer observes a high percentage of failures and many defective items, then 100% inspection is proposed. Acceptance sampling is midway between these two cases and is applied when only samples of lots are examined. Acceptance sampling is beneficial in reducing the cost of inspections while ensuring that the product meets standard specifications. Acceptance sampling is a useful tool to test sensitive, expensive items or when the legal liabilities are potentially greater. In acceptance sampling, there are two undesirable possible outcomes: a good lot may be rejected or a bad lot may be accepted. The first situation, when a good lot is rejected, is termed manufacturer’s risk and denoted by α, while the second situation, when a bad lot is accepted, is called consumer’s risk and denoted by β [
3]. Schilling [
4] applied software including Excel, Minitab and Statgraphics to study acceptance sampling plans. Attribute sampling plans have different classes: single, double, multiple sequential and skip lot sampling plans. Single and double acceptance sampling plans are used mostly because they are more practical and easy.
In practice, single and double acceptance sampling plans are used due to their usefulness and ease. The single sampling plan is simple and easy to apply. The plan involves only one sample and is used to study attributes. This plan has two parameters: the number of units selected from a lot, denoted by n, and the number of defective items to be allowed, denoted by c. If more than c items are found to be faulty, the lot will be excluded. In case of rejection of the lot, corrective measures are taken, or the lot is sent back to the manufacturer. Usually, all items are in the lot are inspected, and faulty items are replaced with good items. If the number of defective items
x is less than or equal to
c, the lot will be accepted. Conventional acceptance sampling plans have been discussed by many scholars in detail [
5]. In Reference [
6], Liu developed new single sampling plans for three-class attributes. In the Reference [
7], Liu proposed new techniques to develop attribute single acceptance sampling plans. Examples are given to illustrate newly developed approaches. In conventional sampling plans, the probability
\(p\) of an item being faulty has an exact value. However, in practical decision making, the exact value of
\(p\) is typically not known, due to vagueness arising from test procedures, individual decisions, or estimation. In this situation, fuzzy set theory is applied. This concept is used to express and study vagueness emerging from uncertain and individual opinions. In the Reference [
8], Sadeghpour Gildeh proposed a single acceptance sampling plan that treats
\(p\) as a fuzzy number, using it to develop a fuzzy characteristic (FOC) curve band. According to these researchers, the width of the band depends upon the uncertainty about
\(p\), and for zero acceptance number, this band becomes convex. The degree of convexity is directly proportional to the sample size n. Single acceptance sampling plans with vague parameters that used fuzzy probability [
9] suggested a plan based on the Poisson model in a fuzzy environment. When n is large and the probability of fraction items
\(p\) is small and imprecise, a fuzzy Poisson distribution is employed. The resulting operating characteristic (OC) curve is presented with detailed examples, and the OC bands for fuzzy binomial distributions and fuzzy Poisson distributions are compared. Buckley [
10] proposed a comparison of single sampling plans with and without inspection errors. This comparison shows that there is a lower operating characteristic band to a sampling plan without inspection errors for handling quality. The effects of improper classification of good items and of faulty items on the fuzzy probability of acceptance are discussed. According to them, improper classification of good items decreases the fuzzy probability of acceptance, and improper classification of faulty items increases the fuzzy probability of acceptance. In this work [
21], Turanoğlu developed a sampling plan using the fuzzy probabilities. Further OC curve in the fuzzy environment are presented, and the concept is illustrated with examples. Acceptance sampling is a viable and inexpensive alternative to expensive complete scrutiny. This sampling provides an opportunity to measure the quality of a whole lot and to decide the lot’s acceptance or rejection. This approach saves material and time in inspection and enhances inspection quality and efficiency. Though acceptance sampling is a useful measure, it is difficult to determine its exact parameters. It is easier to define these parameters as verbal variables. In this article, they gave elastic definitions to sample size, acceptance number and probability of faulty items. They analyzed single and double acceptance sampling plans when the parameters
N,
n,
p,
c are vague, and all characteristics including operating characteristic (OC) curve, acceptance probability, average sample number (ASN), (AOQL), and average total inspection were studied with inexact parameters. These results show all options of ATI, ASN, and OC curves. In Jamkhaneh [
12], the researchers suggested reliability measures for fuzzy conditions. The fuzzy Weibull distribution is employed to describe device lifetimes. Formulas for all reliability functions, including reliability function, hazard function, and their α-cuts, are proposed. They also suggested fuzzy functions of series systems and parallel systems. They presented fuzzy reliability curves with detailed examples. Afshari [
13] recommended a fuzzy multiple deferred state sampling plan by attribute where the fraction
\(p\) of malfunctioning items is imprecise. Both probability theory and fuzzy set theory are applied to study this plan. Venkatesh [
14] suggested a sampling plan employing an exponential model in a fuzzy environment. The concept is illustrated with an example and the conclusion is made that a substantial change occurs in gastrin plasma level.
In Venkateh [
15], the authors proposed applying the fuzzy Weibull distribution to acceptance sampling. The acceptance sampling plan is illustrated by example from medical field. Acceptance probability for crisp and fuzzy parameters is calculated. The fuzzy acceptance probability is found to be more flexible and to give more information. Acceptance probability against samples to predict sample size is presented graphically. In Venkateh [
16], comparison of gamma distribution and generalized exponential distribution in the fuzzy environment is presented. The comparison showed that acceptance sampling plan results for the fuzzy gamma distribution is better than for the fuzzy generalized exponential distribution. Tong [
17] proposed a fuzzy acceptance sampling plan for the situation where sampling parameters are presented as vague numbers for quality inspection of vague quality characteristics. This sampling plan is better than the conventional fuzzy acceptance sampling plan. The sampling plan consists of three situations with different sampling parameters. The fuzzy operating characteristic (FOC) curve with this newly proposed method has two bounds and is more flexible and informative. Elango [
18] developed an acceptance sampling plan that used the fuzzy gamma distribution and found that it was better than a plan using the fuzzy generalized exponential distribution. Aryal [
20] probability distributions were applied and investigated in reliability. A neutrosophic acceptance sampling plan was explored by Aslam Aslam [
23]. Transmuted inverse Weibull distribution is applied in the acceptance sampling plan [
22].
In this study, single acceptance sampling plan based on transmuted Weibull distribution is proposed. The fuzzy proportion of faulty items and fuzzy acceptance probability is used to construct the fuzzy OC curve. The results are described by numerical examples and an application of real data is considered for illustration.