Statistically and economically based attribute acceptance sampling models with inspection errors

This dissertation examines the quality control situation where lots are either accepted or rejected on the basis of a single sample which is randomly selected from the lot. Judgment regarding the quality of an item is based on defined attributes which either conform to or deviate from prescribed standards. Traditionally, it has been assumed that no errors are introduced into the inspection process. While such as assumption eases the computational procedures involved in the evaluation of sampling schemes, it ignores considerable evidence that most inspection operations are error prone.

This research investigates the statistical and economical considerations which may be involved in an inspection process subject to such inaccuracies. In the context of this research errors are introduced into the process when an item is erroneously classified as either good or bad. Good items classified as bad items are referred to as type I errors; conversely, bad items classified as good items are referred to as type II errors. Performance measures have been identified that are meaningful in establishing trend data from which to draw inferences concerning the effects of inspection errors.

The purely statistical quality model is first evaluated for both the error free and error prone situations. The formulas for the average outgoing quality, AOQ, average total inspection, ATI, and the probability of accepting a lot, P_a are established for both situations of interest. For the error prone situation, a new probability mass function is defined which describes the conditional distribution governing the occurrence of observed defectives in a sample given the actual number of defectives in a sample. Other pertinent distributional considerations are developed and discussed. Typical numerical examples are used to illustrate the effects of both type I and type II errors which may occur either alone or in combination.

The second part of this dissertation assumes that many quality schemes originate within the structure of an economic framework; in such cases the inspection criteria should appropriately be based on economic criteria. An economic based quality control model is formulated that is applicable to either the error free environment or the error prone environment. For the purpose of this research it is assumed that the prior distribution of defectives in a lot before the sample is formed is either described by a mixed binomial distribution or a Polya distribution. The distributional considerations pertinent to the model are fully developed within the text.

Several numerical examples are evaluated to illustrate the selection of optimal sampling plans for inspection schemes subject to inaccuracies. The expected cost penalty incurred when using an optimal plan designed for an error free environment when errors actually exist was also investigated. Considerable data are presented which permits one to draw inferences pertaining to the importance of the assumed distribution. The data were presented in numerous tables and figures to aid in establishing the significant trends from which the consequences of inspection errors could be inferred.