Control Charts For Monitoring The Depth Ratio Of Normal Variables

In statistical process control,control chart is an important tool to monitor product quality.Traditional control charts are mostly used to monitor single or multiple quality characteristics.In some cases,however,there are many quality assessment scenarios in production and manufacturing where practitioners are interested in monitoring the stability of the ratio of normal random variables more closely than their mean vector or covariance matrix.For example,the proportion of the components in the food and drugs industry.For more general cases,the ratio to be monitored may not be the division of two values.This is a challenge to the design of the control charts.The depth ratio has wide applications in natural science,production and engineering and is defined with slightly different terminology in various contexts.In determining diamond quality,depth percentage or ratio,defined as the ratio of total depth(height)to the average length and width,is often considered an essential parameter.European commission usually measures the shape of various agricultural or process products using a similar concept of the ratio between length and the average of large and small equatorial diameters.Therefore,it is very important to design control charts for monitoring the depth ratio of normal variables.Firstly,this paper studies the distribution function of depth ratio and an approximation of the distribution which can be used in most practical situations.Based on the distribution function,the Shewhart control chart for monitoring depth ratio with exact control limits and approximate control limits is designed.Compared with the Shewhart-RZ control chart using approximate control limits,the control chart proposed in this research can be widely used in many scenarios.In order to further improve the monitoring efficiency of the control chart to the small shift of the depth ratio,this study proposes the two one-sided EWMA and modified EWMA schemes.Considering that the control limits of the Shewhart control chart,which are based on different distribution functions of the depth ratio,may have an impact on the performance of the charting scheme,the performance of Shewhart-type schemes with exact and approximate control limits is studied by simulation in this research.The average run length of the monitoring process is calculated by Monte Carlo simulation.It is used to evaluate the charting parameters and compare the capabilities of the proposed control charts in detecting shifts.In the comparison study,since a shift of correlation coefficient may lead the process to an out-of-control state,the shift of depth ratio and/or the correlation coefficients are considered.Finally,based on the filtering and analysis of diamond data,an illustration of the proposed charting schemes is provided with the problem of monitoring the diamond cut quality.The real-life data analysis shows the efficacy of the proposed monitoring schemes in practice.