Phase I monitoring of simple linear profiles in multistage processes with cascade property

When a multistage manufacturing process is monitored statistically, the cascade property results in a more complicated condition compared to the case when a single-stage process is controlled. The cascade property usually exists in different stages of a multistage process, where the quality of a stage influences the performance of the next stage. Moreover, sometimes the quality of a product/process is best characterized by a functional relationship. This relationship is referred to as a profile. In this paper, phase I monitoring of simple linear profile is addressed for a multistage process involving the cascade property. To aim this, the capabilities of the methods that may be used to monitor a profile in a multistage process are first assessed. Then, a statistic, named the U statistic, is introduced to provide the opportunity of removing the cascade property. This statistics provides quality engineers a way to reduce the complicated condition of monitoring a multistage process. The new approach also helps quality engineers to diagnose effectively the stage responsible for the out-of-control condition. To evaluate the effectiveness of the proposed approach, different simulated cases are analyzed numerically. In addition, a case study is provided to illustrate the applicability of the proposed method in real-world manufacturing environments.