While the charting parameters of the basic charts are determined by common wisdoms, the parameters of the optimal and adaptive charts are designed optimally in order to minimize an index average extra quadratic loss for the best overall performance. The nine charts are categorized into three types (the type, CUSUM type and SPRT type) and three versions (the basic version, optimal version and adaptive version). This article compares the effectiveness and robustness of nine typical control charts for monitoring both process mean and variance, including the most effective optimal and adaptive sequential probability ratio test (SPRT) charts. A numerical example is given for illustration. The influence of the reference sample size is examined. The in-control and the out-of-control performance properties of the chart are investigated in simulation studies in terms of the mean, the standard deviation, the median, and some percentiles of the run length distribution. Control limits are tabulated for implementation in practice. Being nonparametric, all in-control properties of the proposed chart remain the same and known for all continuous distributions. The plotting statistic combines two popular nonparametric test statistics: the Wilcoxon rank sum test for location and the Ansari–Bradley test for scale. In this article, a single distribution-free Shewhart-type chart is proposed for monitoring the location and the scale parameters of a continuous distribution when both of these parameters are unknown. The assumption of normality is crucial for the validity of these charts. Several proposals have been published recently, where a single (combination) chart that is simpler and may have performance advantages, is used. Traditional statistical process control for variables data often involves the use of a separate mean and a standard deviation chart. In addition, the proposed model is evaluated by comparing performances of the joint X-bar and R charts, and X-bar and s charts for different sample sizes.
![xbar r charts xbar r charts](https://www.isixsigma.com/wp-content/uploads/2013/02/Examples-of-Xbar-and-Range-Xbar-R-Chart.gif)
Furthermore, the performance of the proposed method is examined and compared with that of Shewhart Control Charts by evaluating Type II error.
![xbar r charts xbar r charts](https://www.leansigmacorporation.com/wp/wp-content/uploads/2016/02/Xbar-R-chart-JMP_1.4.png)
The Fuzzy Inference Control System includes four stages to detect and distinguish mean and/or variance shifts in the quality characteristic. This paper presents a new method based on a fuzzy inference system for determining shifts in the process. Considering the cost that is caused by delay in defining the variability, it is important to determine the variation correctly and quickly in a production process. The determination of variability affects the cost and the quality in a process. Control charts are to detect the occurrence of the shifts in a process rapidly so that their causes can be found and the necessary corrective action can be taken before a large quantity of nonconforming products are manufactured. In a production environment, control charts are the most important tool to determine whether a process is in-control or out-of-control. Statistical process control is a very useful method to improve the product quality and reduce reworks and scraps. The simplicity in the design of the modified joint and R chart makes it suitable for the industries where the data are positively correlated.
![xbar r charts xbar r charts](https://www.researchoptimus.com/images/charts8.jpg)
The performance of modified joint X-bar and R chart (with correlation) is compared with the performance of modified joint and R chart (without correlation) for sample size of 5, suggested by Prajapati & Mahapatra (2007) and it is observed that the performance of joint chart deteriorates as the level of correlation increases. It is found that the joint modified X-bar and R chart outperforms the joint Shewhart and R chart at all the levels of correlation and process shifts in the mean and standard deviation. The performance of joint chart is measured in terms of average run lengths (ARLs) and compared with joint Shewhart chart for sample sizes of 5. The design of modified joint X-bar and R chart is based upon the sum of chi-squares theory. But, when it is required to monitor the changes in both the mean and standard deviation of the process, the joint X-bar and R charts should be used simultaneously for better results. The chart is used to monitor the process mean while range (R) and S charts are used to monitor the process standard deviation.