R2R controllers combine statistical process control and engineering process control. These techniques based on exponentially weighted moving average (EWMA) scheme and adjust input variable based on difference between forecasted and real output. This type of control consist some cost as difference inventory or overtime and etc.In this paper we suggest that first output controls by residual control chart, and if there is out of control case then the process is adjusted, also in this paper with considering cost parameters in adjustment of process, an economic model to optimum design of combination of R2R control and residual control chart is developed.

We introduce a method for the statistical design of a depth-based control chart, using the percentile-based approach. The proposed control chart is affine invariant and is asymptotically distribution-free. Generally, the performance of a control chart is evaluated with the average run length metric. The average run length metric has a geometric distribution skewed to the right with a large standard deviation and may not be a proper measure for evaluating the control chart. Therefore, we use the statistical design method of control charts with the PL approach, which is an improvement and development on classical statistical design. By employing constraints on average run length, the length of in-control and out-of-control performances are guaranteed with predetermined probabilities and we can ensure that the in-control run length exceeds the desired value and the out-of-control run length is less than the desired value. Simulation studies show that the proposed control chart is more efficient than the average run length approach.

This article presents a new multivariate non-parametric control chart in Phase II based on depth functions. The proposed statistic is an affine invariant and asymptotically free distribution. Indeed, the asymptotic distribution of the proposed statistic is derived under the in-control process. Based on simulation studies, the performance of the proposed statistic, using several depth functions, has been evaluated and compared with three competing statistics. The results show that the introduced statistic performs adequately and, in some cases, outperforms the competing statistics. A real dataset has been analyzed based on the proposed chart.

Control charts are one of the most effective tools used in quality control to monitor various quality characteristics in a process with the aim to improve quality of the product. Usually, in Shewhart control charts, the normality assumption met for the data, but sometimes there is lack of information regarding the statistical distribution of the observations. For this reason, non-parametric control charts are used in this situation. In this research, non-parametric sign charts are introduced to deal with the lack of information regarding observations’ statistical distribution. Nonparametric Generalized Weighted Moving Average Sign Control Chart (NS GWMA) designed using statistical design and average run length (ARL) and its statistical performance was studied. But statistical design is not enough to ensure the performance of a control chart, so in the next steps, economic design (ED) and economic-statistical design (ESD) were applied using cost model of Lorenzen and Vance, in order to optimize both statistical and economical characteristics of the control chart. The results show that the non-parametric sign-generalized weighted moving average sign control chart performed well in detecting small shifts in the process and also had optimized cost and time to perform.

Control charts are widely used to identify changes in the production process, and when it comes to identifying small changes in the process, non-Shewhart control charts such as exponential weighted moving average (EWMA) and generalized weighted moving average (GWMA) are better alternatives to control charts than Shewhart 𝑋 control chart. In this context, nonparametric control charts are used when the distribution of quality characteristic of the process is unknown, in which the Sign control chart is one of the most popular nonparametric control charts. In this paper, for the first time, a generalized weighted moving average sign control chart using a ranked set sampling (RSS) design is introduced. The performance of the proposed control chart is evaluated using simulated data according to the average run length evaluation criterion, and simulation studies showed that the GWMA sign control chart under the RSS design is better at detecting small process changes than the EWMA sign control chart under the RSS design. The performance of the proposed control chart has been evaluated using simulated data according to the average execution length evaluation criteria in Julia software, and the simulation studies show that due to changes in the regulatory parameter α, the 𝐺𝑊𝑀𝐴𝑆𝑅𝑆𝑆 + control chart is more sensitive than the 𝐸𝑊𝑀𝐴𝑆𝑅𝑆𝑆 + control chart in discovering small process changes, also the 𝐺𝑊𝑀𝐴𝑆𝑅𝑆𝑆 + control chart is more sensitive than the 𝐺𝑊𝑀𝐴𝑆𝑆𝑅𝑆 + and 𝐷𝐺𝑊𝑀𝐴𝑆𝑆𝑅𝑆 + control charts are more efficient and sensitive in identifying small process changes, which shows the advantage of RSS scheme over SRS scheme. In addition, it was proved that increasing k and the number of cycles and large values of q decrease 𝐴𝑅𝐿1 and increase the power of the control chart in identifying process changes.

Considering profiles as functional variables, two control charts are proposed for their monitoring in phase II. Due to its conformity with the nature of real-world profiles, applying functional model leads to proposed control charts obtained through functional data analysis techniques with desired features. These include simplicity in calculation and possibility of using them for different profiles such as linear and non-linear (even in the presence of complex within-profile Autocorrelation forms). These features distinguish the functional model from the regression models common in profile monitoring. Simulated computer simulations show that, in different states, the proposed control charts have a lower average run length than other methods, which indicates the desired performance of the proposed functional approach. Morevere, in some non-linear cases with complex autocorrelation, other methods completely fail, and only the proposed control charts are able to detect the occurring deviation, and even in these cases, the average run length of these control charts is highly desirable.