Using CUSUM Control Schemes for Monitoring Quality Levels in Compound Poisson Production Environment: The Geometric Poisson Process

In contemporary modern and high volume production environments such as wafer manufacturing, a small sustained shift is not very easily detected in a short period of time, but may have a great impact on a manufacturing process. Thus, it is important to be able to detect and identify a small sustained shift of the production process in a timely manner and correct the undesired situation. The cumulative sum (CUSUM) control scheme is considered to be one of the efficient reference tools in detecting a small structure change in a process. However, for control of defects in a production process, too often the assumption is made that the defects follow a Poisson distribution. In practice, the process is more complex and the distributions of defects are more appropriately modeled by the compound Poisson distribution. In this paper, the underlying distribution is the geometric Poisson distribution, a Poisson distribution compounded by a geometric distribution, and the CUSUM control scheme based on the geometric Poisson process is addressed. An effective CUSUM control scheme can provide an adequate average run length (ARL), that can be obtained from the probability transition matrix for the Markov chain proposed by Brook and Evans (1972). With proper ARL selected, the geometric Poisson CUSUM control scheme is developed for process control.