Designing the multivariate nonconforming proportion control charts considering the measurement error

Most qualitative characteristics cannot be easily reported numerically. In such cases, each product is inspected and usually divided into two conforming and nonconforming groups based on their qualitative characteristics. Since nowadays products and processes have generally several interdependent qualitative characteristics, it is necessary to use multivariate quality control methods to make the relationship between variables and their variations. To do this, the sampling of the considered qualitative characteristic is done at the specified time intervals to check the control of the process over time after drawing the considered statistic on the control chart. A common problem while sampling is measurement error. It affects the performance of control charts, impairs their ability to detect changes in the process, and increases the cost and time to search for out-of-control situations. In this paper, the effect of measurement error on the performance of the Multivariate Nonconforming Proportion (MNP) control chart has been evaluated based on the criterion of Average Run Length (ARL) for the first time. The results imply that the measurement error has a considerable impact on the performance of this chart. Also, the results indicate that if the defective items have been wrongly considered as correct items, we would have a higher ARL compared to an ideal and accurate system. On the other hand, if the system considers right items as defective, we will have a lower ARL than the ideal and accurate system. It is proved that if both errors (considering faulty items as correct ones and vice versa) occur simultaneously, the ARL will be reduced like the previous case.