The synthetic EWMA chart consists of the combination of EWMA and conformal run length (CRL) charts. Since the shape of the run length distribution changes with displacement, the percentiles of the run length distribution are used to evaluate the performance of the EWMA synthetic average chart based on the RSS scheme.
A Short Overview of Quality Control Charts
To meet the conditions of today's industrial environment, much research has been conducted to improve the current design techniques by introducing more effective and powerful control charts. Moreover, control charts have not only been widely used in industrial process recently, but also gained its wide applications in other sectors, such as health care, signal segmentation, fishing, nuclear engineering and so on. (Haq, 2014).
An Outline for Statistical Process Control (SPC)
If there are special cause variations in a process, the process can be interpreted as out of control. Constant observation and monitoring of process quality using an SPC chart is required to ensure that the process continues to operate stably.
Types of Univariate Control Charts
For cases with a focus on detecting small average process shifts, the CUSUM chart and the EWMA chart are chosen as an effective replacement for the traditional control chart. For detecting gradations in the non-conforming fraction, a synthetic np table for attribute data is suggested by Wu et al.
Objectives of the Study
The final part of Chapter 1 highlighted the objectives of the study and the arrangement of the thesis. The first part of Chapter 2 explained a number of common performance evaluation tools for a control chart.
Measures of Performance of a Control Chart
Average Run Length (ARL)
According to Gan (1993), MRL is a more credible measure to assess the effectiveness of a quality map, since the disturbance of the skewness of the run length distribution is less likely to affect the control scheme. From the study, Lee and Khoo (2017b) revealed that the effectiveness of the synthetic np-diagram is greater than the classical np-diagram for detecting process change using MRL.
Percentiles of the Run Length Distribution
Shu et al. 2013) investigates the run length distribution of CUSUM scale diagrams using the piecewise collocation method. In addition, a Progressive Mean control plot using ARL and percentage points of the run length distribution is introduced by Abbas et al.
Control Charts available in literature
Median chart
Since the mean and median behave similarly for a normal distribution, the median graph and mean graph perform equally well (Khoo, 2005). For the case when the mean, and standard deviation, are known in an in-control process, then the control limits of the median map.
Exponentially Weighted Moving Average (EWMA) median chart
In case the mean and standard deviation are known in a stable process, the control limits of the median diagram can be calculated as follows. Some modification is required in the calculation of the estimators for the order statistics, which are demonstrated as follows.
Conforming Run Length chart
Synthetic Exponentially Weighted Moving Average (EWMA) control chart
Scariano and Calzada (2009) showed that the sensitivity of EWMA chart can be increased by including the CRL scheme in the EWMA chart. For the construction of Synthetic EWMA chart, assume the individual observations, i.e. F F1, 2, .., Fn , are simple random sample of size n normally distributed with mean F , and standard deviation F. Then the sample mean based on the simple random sample of size n can be calculated as F1 F2.
The construction of the control chart is continued by making the F F1, 2, .. designators consist of a series of independent random variables that follow the normal distribution. For the calculation of the control limits for the Synthetic EWMA card, it is clear to display that. After obtaining the UCL and LCL from the map, we can begin the monitoring process by plotting the EWMA test statistics, Zt, against time, t.
However, the process terminates when the enumerated CRL value is less than or equal to L.
A review of Ranked Set Sampling (RSS) charts
- Ranked set sampling (RSS)
- Quality control charts using ranked set sampling (RSS)
- Synthetic EWMA chart under Ranked Set Sampling (RSS)
From the comparison result, it is noted that the effectiveness of the perfect or imperfect ranking-based control charts is greater than the conventional chart. Moreover, in the circumstances when the mean in a process is unfixed and the mean and variance in a process are not independent of each other, Abbasi et al. 2019) used the coefficient of variation (CV) maps under various RSS schemes such as RSS, MRSS and extreme RSS (ERSS) to improve the detection capabilities of the existing CV map using SRS. Takahasi and Wakimoto (1968) suggested that the mean of the ordered set sample, XRSS is an unbiased estimator of mean, X and it is more precise than the mean of the simple random sample, XSRS.XRSS is calculated as.
The solid concept of Shewhart's control chart lead to the creation of modern improved process inspection techniques such as the EWMA control chart. To improve the weakness of the EWMA chart, Haq, et al. 2016) implement an RSS approach to the EWMA synthetic map to monitor the process average. Their findings show that the performance of the proposed Synthetic EWMA table is higher than other quality scales based on SRS.
The asymptotic control limits of the EWMA synthetic control chart at time t are presented as follows.
Introduction
Implementation of Ranked Set Sampling Scheme on Synthetic EWMA median chart
The following steps describe the step-by-step procedure for forming the EWMA Synthetic average chart using perfect RSS schemes:. Set the sample size, n. Initialize the smoothing constant, λ, and the lower bound of the CRL graph, L. Next, adjust the j-th subset. Since the control chart structure for the proposed RSS-based chart is complex, the specific distribution of the sample mean of subset j is challenging to determine.
The value of L is always positive and symbolizes the lower limit of the CRL chart. To assess the run length profiles of the RSS-based proposed map, the step-by-step procedures are explained as follows: ii). Set the magnitude of the process offset, δ=0. iii) Calculate the EWMA series based on . iv) Plot the test statistic, Zj,RSS, against the control limits of the RSS-based proposed chart.
The program for estimating the run length profiles of the proposed RSS-based chart for an unstable process also works in the same way as the process in control with an alternating value of δ and shown in Appendices D to F.
An evaluation of the performance of the Synthetic EWMA median chart
- Average Run Length (ARL)
- Percentiles of Run Length Distribution
Median Run Length (MRL) of the synthetic EWMA median chart under RSS with sample size, n3,5,9 are shown in bold text in Table 3.3 to Table 3.5 respectively. Since the mean is greater than the median, this shows that the run length distribution is not symmetric and it changes with δ. The run length distribution is noted to be highly skewed to the right, especially when δ is small.
This is because the profile of the sequence length distribution shifts to nearly symmetric when the process is unstable. Using a simulation program written in SAS software, the (n, λ, L, K) values are used to calculate the series length percentiles of the EWMA synthetic median plot under RSS. Additionally, additional detail can be obtained from higher length percentiles such as the 90th or 95th percentile for a process.
To illustrate the dispersion of the run length distribution from the quality graph, the 95th run length percentiles will be used to subtract the 5th run length percentiles.
Discussion and Comparison of proposed chart with EWMA median – RSS chart
From the data set, our main concern is to establish the statistical control of the flow width of the resistance in the process (measured in microns) using the proposed average synthetic EWMA chart under the RSS. Note that the RSS-based average EWMA competitive chart has also been created for the sake of comparison. The (n, λ, L, K) values for the proposed synthetic EWMA average chart are taken from Table 3.1 while the (λ, K) values for the RSS-based EWMA average chart are taken from Table 3.6.
In our study, only two selected situations are considered to evaluate the sensitivity of the proposed RSS-based EWMA synthetic average chart and EWMA average chart. In the first case, from Figure 3.1, the 42nd sample exceeds the control limits of the average synthetic EWMA chart under the RSS. On the other hand, from Figure 3.2, it is observed that all the samples stayed between the control limits of the average EWMA chart under the RSS.
It is clearly seen in Figure 3.4 that all the test statistics lay between the control limits of the EWMA median plot under the RSS, which means that an uncontrolled event cannot be detected when the process is unstable.
Introduction
The study of the proposed synthetic EWMA median chart under ranked set sampling (RSS) contributed to the improvement of statistical quality control practice, especially in the area of variable control charts. In this thesis, step-by-step procedures for calculating the parameters of the synthetic EWMA median chart under RSS are presented to facilitate the practitioner's use of the charts. SAS programs that contained the step-by-step procedures are provided in the appendices to make the calculations of the parameters in the proposed diagram under RSS simple and fast.
Some parameters of the proposed chart in the RSS scheme are created for selected situations and displayed in tables for immediate reference. The presented method for generating the proposed chart under RSS complements the works of Haq et al. 2016) who suggest making a synthetic EWMA average chart under RSS. Note that the sensitivity of the proposed chart under the RSS scheme is higher than its competing counterparts in terms of series length profiles.
In order to provide a clearer insight into the application of the parameters that were obtained through the procedures presented in the creation of the proposed graph according to the RSS scheme, an example of the application is demonstrated.
Suggestions for further research
The findings of Abid et al. 2017) revealed that the proposed non-parametric Cumulative Sum (CUSUM) sign control chart under RSS outperforms the non-parametric CUSUM sign chart under SRS with respect to process average monitoring efficiency. Abbasi et al. 2019) illustrated that the detection capability of the CV graphs under numerous ranked sampling schemes such as RSS, median RSS (MRSS) or extreme RSS (ERSS) is higher than the existing CV graph based on SRS. Optimal design of the process average synthetic graph based on average run length.
EWMA control chart for coefficient of variation using log-normal transformation under ranked sampling. On unbiased estimates of the population mean based on the sample stratified by ordination. Percentiles of the run-length distribution of the exponentially weighted moving average (EWMA) median chart.
Optimal design of the adaptive EWMA chart for the mean based on median run length and expected median run length. Optimal design of the chart with variable sample size based on median run length and expected median run length. A synthetic control chart for monitoring the small shifts in a process asset based on a property inspection.
