Monitoring and detecting shifts in the mean in quality levels for a production environment with properties found in the geometric Poissin process
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For statistical process control, an important property is the underlying statistical model that is assumed to govern the defect generation process. For control of defects, the assumption is made that the defects follow a Poisson distribution. However, frequently the process is more complex and the distributions of defects are more appropriately modeled by the compound Poisson distribution. A defective item may have more than one defect that cause the item to be defective. The occurrence of defective items may follow a Poisson distribution. If an item is defective, the number of defects per item will follow another distribution. In this research, it will be assumed that given an item is defective, the number of defects follows the geometric distribution. Thus, the distribution of defects over time is the Poisson compounded with the geometric. From the viewpoint of quality control, process quality can be improved by moving special causes. Two broad types of special causes, transient and persistent special causes, are reported in the literature. Two proposed methods, the empirical Bayes control chart for the geometric Poisson random variables and the geometric Poisson CUSUM control scheme, aim at removing both transient and persistent special causes. Both proposed approaches utilize the historical information concerning the process. The former can detect shifts in much wider situations. The latter would be more sensitive to small sustained shifts caused by persistent special causes. Using the simulated data, the performance of these proposed quality control methods and classical Poisson-based control methods is compared. The test results show an underestimation of Type I error and the number of false alarms generated, if the underlying defect distribution is wrongly assumed. Although two alternatives in detecting mean shift or structure change for the geometric Poisson random variables are proposed, the relationship between these two is complementary.