There has been increasing interest in the development and implementation of methods for improving the quality of goods and services. In the past few years, many companies have adopted the concept of total quality. Tools such as sampling inspection, statistical process control, design of experiments, and regression analysis play an important role in any program of developing and maintaining product quality. In this research, I plan to develop improvements in one of the fundamental statistical tools used for quality control: acceptance sampling.

An acceptance sampling plan is one in which a specified number of units are sampled from each lot, with the lot being accepted if a fixed number or fewer nonconformances are found in the sample. Lots that are not accepted can either be discarded or rectified. Rectification, i.e. replacing or discarding all non-conforming units after 100% inspection of rejected lots, is frequently used when manufacturing costs are high. The most common application of acceptance sampling with rectification is in semiconductor manufacturing

Estimation of the number or rate of nonconforming units in outgoing lots of a product is increasingly becoming a necessary step for suppliers. These estimates can be obtained from an independent quality audit of the outgoing product or from data that was collected during acceptance sampling. Several recent papers (e.g., Hahn (1986), Zaslavsky (1986), Brush, Hoadley, and Saperstein (1990)) have proposed estimators that could be used for estimating the number or rate of nonconformances in lots subjected to rectification acceptance sampling. Greenberg and Stokes (1992a) used the information obtained in rectification to devise a more efficient estimator than those previously proposed. Since our paper was published, 3 manufacturers (Advanced Micro Devices, Motorola, and Harris) have contacted us about the use of our estimator in the semiconductor industry.

Recently I worked on a project for Motorola concerning a situation in which the test procedure is imperfect. Because they expect that conforming items are frequently rejected by their test, they repetitively test failed devices. From the data they supplied, it is clear that their testing procedure frequently rejected devices that would later pass when they were retested. This suggests that two problems may exist in acceptance sampling. Devices that are classified as nonconforming may be conforming (false negatives) and devices that are classified as conforming may be nonconforming (false positives). This can be true in the sample as well as in the rectified units. The purpose of the research proposed here is to develop effective rectification sampling schemes for a product vulnerable to inspection error and to develop a methodology which will allow estimation of the number of undetected nonconformances remaining after a set of lots has been screened by such a plan.

None of the recent papers mentioned above, have considered the effect that inspection error would have on their estimators. On the other hand, some researchers have considered the effects of inspection error on other acceptance sampling plans. In their recent monograph, Johnson and Kotz (1991) provide tables and calculations of the average outgoing quality for many types of plans when the false positive and false negative rates are known. Lindsay (1985) described methods for estimating the probability of false positive and false negatives, and then the rates and numbers of nonconformances, when the sample is repeatedly inspected. However, these authors do not consider plans with rectification.

When inspection error is present, it is impossible to estimate the nonconformance rate without inspecting at least some of the items more than once. In a rectification sampling plan, there are a variety of ways that this can be accomplished. The sampled units could be inspected in two or more independent samples, a portion of the sample could rechecked in an error-free procedure, or if it is too expensive to provide an error free procedure, a lower cost inspection could be used.

Our goals for this research are to answer the following questions: (1) what is the best design for collecting the retest data? and (2) how can we incorporate data from retesting to estimate the nonconformance rate after acceptance sampling with rectification?