Study on Determination of Optimal Parameters for Shewhart Control Charts and Other Related Methodologies

Abstract

Taguchi’s on-line quality control methods attach importance to the reduction of variation of a quality characteristic about an engineering target. The quality characteristic can be nominal-the-best, higher-the-better, or lower-the-better. He introduced the concept of the loss function. The loss incurred to society is impacted by the variation or deviation from the target. The optimal sampling interval advocated by Taguchi in this context is important. However, Taguchi didn’t consider certain parameters like the rate of production, loss incurred due to false alarms, loss incurred due to non-detection of process abnormalities. In our prescribed methodology, these parameters have been considered along with the parameters considered by Taguchi like the cost of producing a defective item, diagnosis cost, adjustment cost, etc. to determine the optimal sampling interval. The other areas considered for determining the optimal sampling interval are control charts - both attribute and variable. These methods periodically capture the online data from the production process to identify sporadic shifts, sustained drifts, and other non-random process abnormalities that help initiate remedial measures to enable resilience to a stable state from an unstable state of a process. Shewhart invented control charts essentially to differentiate in a process between the assignable or the special causes of variation and the chance or the common causes of variation. While finding these assignable causes in a production process, there are some costs associated with the endeavor, such as the cost of sampling, testing, false alarms, removal of the assignable cause, and generating a defective item. We have considered the relevant cost components of diagnosis cost from today’s perspective of estimation along with rate of production appropriately. Due to the association of these costs, it is imperative to consider the corresponding economic consequences. In order to correctly identify and eliminate assignable or special causes of variation, appropriate determination has been made of the optimal parameters like the sample size (n), sampling interval (h), and control limits’ multiplier (k) from both economic and statistical viewpoints. This thesis focuses on determining optimal parameters for these two methodologies. For Taguchi’s online quality control methods, the sampling interval (h) has been optimized. For Shewhart’s control charts ( ¯X-chart, u-chart, p-chart), parameters such as sample size (n), sampling interval (h), and control limits’ multiplier (k) have been optimized based on economic and statistical considerations. Apart from the Shewhart control chart, we have also optimized for the control chart with memory (CUSUM) parameters like the decision interval (H) along with the sample size (n) and the sampling interval (h).