Small Shifts In Process Mean Research Articles

THE PERFORMANCES OF MIXED EWMA-CUSUM CONTROL CHARTS BASED ON MEDIAN-BASED ESTIMATORS UNDER NON-NORMALITY

Exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts have been regularly used to monitor small process mean shifts. More recently, a mixture of EWMA and CUSUM charts known as mixed EWMA-CUSUM (MEC) control chart has been introduced for better small shift detection. However, like its predecessor, the MEC chart requires the normality assumption to ensure optimal performances. In the presence of outliers, which is the cause of non-normality, the parameters of the chart may be overestimated, leading to an unreliable monitoring process. To mitigate this problem, this paper employed median-based estimators namely, the median and modified one-step M-estimator (MOM), to control the location parameter via the MEC control chart. In this study, the performance of robust MEC charts for Phase II monitoring of location was compared with the standard MEC chart that is based on the sample mean. The performance of the robust MEC charts in terms of the average run length (ARL) on various g-and-h distributions clearly shows that a robust MEC chart based on the MOM estimator performs well regardless of the distributional shapes.

Robust Point M-Scale Estimators for Monitoring the Mean of EWMA Control Chart under the Normal Process

The Exponentially Weighted Moving Average (EWMA) control chart is well known in statistical process control (SPC) for identifying small shifts in process parameters. In general, the EWMA control chart is focused on data set estimates of the process characteristics under study. In this research, for observing the process mean, we modify the control limits of the EWMA control chart by using robust point M-scale estimators under the normal process. We compare five robust point M-scale estimators proposed in SPC literature, namely Sn, Qn, MAD, Tauτ^ and FQn. The traditional control limits are evaluated and distinctions are also made based on true process control limits. The control charts performances under study are estimated via a Monte-Carlo simulation analysis by calculating the interval widths (IWs) and optimal out-of-control average run length (ARL1). From the simulation studies results, it is seen that although all proposed and traditional control limits are closely approximating the true limits of the process but the proposed control limits based on robust FQn, Tauτ^ and MAD estimators are comparatively more stringent for detecting the small shifts in process mean. Moreover the results demonstrates that at optimal size of interval L, the control charts based on MAD, Tauτ^ as well as FQn estimators demonstrate higher sensitivity in comparison to the traditional control chart based on sample standard deviation (S) and proposed control charts based on QnandSn estimators in terms of having smaller ARL1 values. The OC curves of ARL1 indicate similar findings. For the purposes of illustration, a real-life data set is analyzed that also confirms our conclusions from the simulation analysis to a certain degree.

A trimmed EWMA control charts for non-normal processes

The traditional exponentially weighted moving average (EWMA) control chart is very popular in statistical process control (SPC) for detecting small shifts in process mean and variance. It performs well under the assumption of normality but when data violate this assumption, a modified approach is needed. In this paper, we proposed a variable control chart for monitoring the process mean by modifying the traditional EWMA control chart for non-normal processes based on the modified trimmed standard deviation (MTSD). The proposed EWMA control charts are based on three different heuristic methods, namely, the Trimmed Exponentially Weighted Moving Average (T-EWMA) control chart, the Trimmed Weighted Standard Deviation Exponentially Weighted Moving Average (TWSD-EWMA) control chart and the Trimmed Weighted Variance Exponentially Weighted Moving Average (TWV-EWMA) control chart. The performances of the control charts under study are evaluated via a Monte-Carlo simulation study by calculating expected out-of-control points and expected widths under non-normal distributions (i.e. Gamma and Exponential). Also, since the EWMA statistic $${Z}_{i}$$ is sensitive to detect mean shifts in the process, the out-of-control average run length ( $${ARL}_{1}$$ ) is considered in the simulation as a best tool to measure the sensitivity. The simulation studies are being carried out for the purpose and results showed that the proposed trimmed control charts performed well for non-normal processes in terms of their shorter expected widths, more number of expected out-of-control points and quick out-of-control signals as compared to non-trimmed versions of control charts. Moreover it is also observed that as weighing constant $$\lambda$$ increases, the sensitivity of the proposed trimmed control charts increases. For illustration purposes, a real-life data set from a public service industry is analyzed which confirming our conclusions from the simulation analysis. Consequently, for estimating of the process mean when the distribution of the data is non-normal, we recommend the trimmed versions of EWMA control charts for the practitioners to be used.

An Enhanced Auxiliary Information-Based EWMA-t Chart for Monitoring the Process Mean

The exponentially weighted moving average t chart using auxiliary information (AIB-EWMA-t chart) is an effective approach for monitoring small process mean shifts when the process standard deviation is unstable or poorly estimated. To further enhance the sensitivity of the AIB-EWMA-t chart, in this study, we propose an AIB generally weighted moving average (GWMA) t chart (AIB-GWMA-t chart) to monitor the process mean. The existing EWMA-t, GWMA-t, and AIB-EWMA-t charts are special cases of the AIB-GWMA-t chart. Numerical simulation studies indicate that the AIB-GWMA-t chart performs uniformly and substantially better than the EWMA-t and GWMA-t charts in terms of average run length. Moreover, the AIB-GWMA-t chart with large design and adjustment parameters also outperforms the AIB-EWMA-t chart when the correlation coefficients are within a certain range. An illustrative example is provided to highlight the efficiency of the proposed AIB-GWMA-t chart in detecting small process mean shifts.

A new double exponentially weighted moving average control chart using repetitive sampling

PurposeThe purpose of this paper is to propose a double exponentially weighted moving average control chart using repetitive sampling (RS-DEWMA) for a normally distributed process variable to improve the efficiency of detecting small process mean shift.Design/methodology/approachThe algorithm for the implementation of the proposed chart is developed and the formulae for the in-control and out-of-control average run lengths (ARLs) are derived. Tables of ARLs are presented for various process mean shift. The performance of the proposed chart is investigated in terms of the average run-length for small process mean shift and compared with the existing DEWMA control chart. Numerical examples are given as illustration of the design and implementation of the proposed chart.FindingsThe proposed control chart is more efficient than the existing DEWMA control chart in the detection of small process mean shifts as it consistently gives smaller ARL values and quickly detects the process shift. However, the performance of the proposed chart relatively deteriorates for large smoothing constants.Practical implicationsThe application of repetitive sampling in the control chart literature is gaining wide acceptability. The design and implementation of the RS-DEWMA control chart offers a new approach in the detection of small process mean shift by process control personnel.Originality/valueThis paper fills a gap in the literature by examining the performance of the repetitive sampling DEWMA control chart. The use of repetitive sampling technique in the control chart is discussed in the literature, however, its use based on the DEWMA statistic has not been considered in this context.

Capability index-based control chart for monitoring process mean using repetitive sampling

ABSTRACTA new process capability index (PCI) control chart for monitoring the process mean using repetitive sampling is presented. The design of the proposed control chart is based on an unbiased estimator of the process standard deviation for a normally distributed quality characteristic. The formulae for the in-control and out-of-control average run length and the standard deviation of the run length (SDRL) are derived. Tables of in-control and out-of-control average run length (ARL) and SDRL for various shifts, δ are presented. The proposed control chart outperforms existing control chart in detecting relatively small process mean shifts in terms of ARLs and SDRLs. Numerical example is presented to demonstrate the application of the proposed chart.

The EWMA control chart based on robust scale estimators

The exponentially weighted moving average (EWMA) chart is very popular in statistical process control for detecting the small shifts in process mean and variance. This chart performs well under the assumption of normality but when data violate the assumption of normality, the robust approaches needed. We have developed the EWMA charts under different robust scale estimators available in literature and also compared the performance of these charts by calculating expected out-of-control points and expected widths under non-symmetric distributions (i.e. gamma and exponential). The simulation studies are being carried out for the purpose and results showed that amongst six robust estimators, the chart based on estimator Q_n relatively performed well for non-normal processes in terms of its shorter expected width and more number of expected out-of-control points which shows its sensitivity to detect the out of control signal.

Shewhart控制图与CUSUM控制图的比较研究

本文首先简单介绍了统计过程控制图的产生及其发展，分析了一般的Shewhart控制图和CUSUM控制图的优缺点，其次给出了Shewhart控制图的估计方法和CUSUM控制图的理论基础及其构造方法。最后，我们比较了在过程受控与失控两种情况下的一般的Shewhart控制图和CUSUM控制图的平均运行长度，得出了CUSUM控制图对监测过程均值小漂移比一般的Shewhart控制图更灵敏的结论。 This paper firstly presents a brief introduction to the emergence and development for the figure of statistical process control, and analyzes the advantages and disadvantages between Shewhart control chart and the CUSUM control chart, and then gives the Shewhart control chart estimation method and the theoretical basis and construction method for the CUSUM control chart. Finally, we compare the average run length in the process of controlled and uncontrolled under the two kinds of situations with Shewhart control chart and the CUSUM control chart. It is concluded that the CUSUM control chart for monitoring small process mean shifts is more sensitive than the Shewhart control chart.

Designing of a hybrid exponentially weighted moving average control chart using repetitive sampling

A hybrid exponentially weighted moving average (HEWMA) control chart using repetitive sampling is presented in this manuscript. The designing of the proposed control chart is developed for a normally distributed quality characteristic. The formulas for the in-control and the out-of-control average run lengths (ARLs) are derived. Control chart coefficients are determined by considering the target in-control ARL. The tables for ARLs are presented for various target in-control ARLs and process shift parameters. The performance of the proposed control chart is compared with the existing control chart in terms of ARLs, which shows that the proposed chart performs better in detecting relatively small process mean shifts. An industrial example is given for the illustration purpose.

The economic design of multivariate binomial EWMA VSSI control charts

Since multi-attribute control charts have received little attention compared with multivariate variable control charts, this research is concerned with developing a new methodology to employ the multivariate exponentially weighted moving average (MEWMA) charts for m-attribute binomial processes; the attributes being the number of nonconforming items. Moreover, since the variable sample size and sampling interval (VSSI) MEWMA charts detect small process mean shifts faster than the traditional MEWMA, an economic design of the VSSI MEWMA chart is proposed to obtain the optimum design parameters of the chart. The sample size, the sampling interval, and the warning/action limit coefficients are obtained using a genetic algorithm such that the expected total cost per hour is minimized. At the end, a sensitivity analysis has been carried out to investigate the effects of the cost and the model parameters on the solution of the economic design of the VSSI MEWMA chart.

Minimal Euclidean distance chart based on support vector regression for monitoring mean shifts of auto-correlated processes

Though traditional control charts have been widely used as effective tools in statistical process control (SPC), they are not applicable in many industrial applications where the process variables are highly auto-correlated. In this study, one new minimal Euclidean distance (MED) based monitoring approach is proposed for enhancing the monitoring mean shifts of auto-correlated processes. Support vector regression (SVR) is used to predict the values of a variable in time series. Through calculating minimal Euclidean distance (MED) values over time series, a novel MED chart is developed for monitoring mean shifts, and it can provide a comprehensive and quantitative assessment for the current process state. The performance of the proposed MED control chart is evaluated based on average run length (ARL). Simulation experiments are conducted and one industrial case is illustrated to validate the effectiveness of the developed MED control chart. The analysis results indicate that the developed MED control chart is more effective than other control charts for small process mean shifts in auto-correlated processes, and it can be used as a promising tool for SPC.

A new non‐parametric CUSUM mean chart

AbstractNot all data in practice came from a process with normal distribution. When the process distribution is non‐normal or unknown, the commonly used Shewhart control charts are not suitable. In this paper, a new non‐parametric CUSUM Mean Chart is proposed to monitor the possible small mean shifts in the process. The sampling properties of the new monitoring statistics are examined and the average run lengths of the proposed chart are examined. Two numerical examples are used to illustrate the proposed chart and compare with the two existing charts, assuming normality and Beta distribution, respectively. The CUSUM Mean Chart showed better detection ability than those two charts in monitoring and detecting small process mean shifts. Copyright © 2010 John Wiley & Sons, Ltd.

Non-normality and combined double sampling and variable sampling interval control charts

A combination of double sampling the chart and the variable sampling interval of the chart (DSVSI chart) increases the sensitivity in the small shift detection. A usual assumption of control charts is that the process observations are normally distributed. However, this assumption may not be true for some processes in the real industry. This paper presented the performance of DSVSI chart under non-normality and compared it with the Shewhart chart and the variable parameters chart. The compared results show that the DSVSI chart has the best performance in detecting small process mean shifts.

The Performance of Double Sampling Control Charts Under Non Normality

In this study, the performances of double sampling (DS) chart under non normality are presented and compared with Shewhart chart and VP chart. The comparison result shows the DS chart has performance as good as the VP chart, but it is more effective than the Shewhart chart in detecting small process mean shifts.

A new <span style="text-decoration: overline">X</span> chart comparable to CUSUM and EWMA charts

The most popular and efficient charts to catch small shifts in process mean are CUSUM and EWMA charts. However, they catch the shift on process average if there is a single and sustained shift, and are unable to detect the changes when samples are not taken from same stream. In such situation, the new chart is proposed in the present work and compared with conventional CUSUM and EWMA charts. It is found that both CUSUM and EWMA schemes are ineffective to catch the process shifts in such cases. It can be noted that the new (proposed) chart provides the highest percentage of correct signals compared to CUSUM and EWMA schemes for all process shifts. It is also clear from the study that for all the process shifts, the EWMA scheme is most ineffective out of the three schemes. The average run length (ARLs) of proposed scheme is much lower than both CUSUM and EWMA schemes for all the shifts in the process average.

Capability adjustment for gamma processes with mean shift consideration in implementing Six Sigma program

Abstract In the 1980s, Motorola, Inc. introduced its Six Sigma quality program to the world. Some quality practitioners questioned why the Six Sigma advocates claim it is necessary to add a 1.5σ shift to the process mean when estimating process capability. Bothe [Bothe, D.R., 2002. Statistical reason for the 1.5σ shift. Quality Engineering 14 (3), 479–487] provided a statistical reason for considering such a shift in the process mean for normal processes. In this paper, we consider gamma processes which cover a wide class of applications. For fixed sample size n, the detection power of the control chart can be computed. For small process mean shifts, it is beyond the control chart detection power, which results in overestimating process capability. To resolve the problem, we first examine Bothe’s approach and find the detection power is less than 0.5 when data comes from gamma distribution, showing that Bothe’s adjustments are inadequate when we have gamma processes. We then calculate adjustments under various sample sizes n and gamma parameter N, with power fixed to 0.5. At the end, we adjust the formula of process capability to accommodate those shifts which can not be detected. Consequently, our adjustments provide much more accurate capability calculation for gamma processes. For illustration purpose, an application example is presented.

Exponentially Weighted Moving Average (EWMA) Control Charts for Monitoring an Analytical Process

The exponentially weighted moving average (EWMA) control chart is very effective in detecting small shifts in process mean or variance, but so far has not been well presented in the field of analytical chemistry. The main difference from the Shewhart chart is that the EWMA chart combines current data with historical observations by essentially taking a weighted average with weighting factor w of the most current sample observations and historical observations. We show that the EWMA chart with 0.05 < w < 0.20 is more effective in detecting small shifts in mean and variance than the Shewhart chart. In addition, the EWMA chart can also be used to forecast the observation in the next period, which can help analysts take preventive actions before process departures to the out-of-control state. Another advantage of using the EWMA chart is its good performance for observations that are not normally distributed or are autocorrelated.

Non-normality and the variable parameters [formula omitted] control charts

Recent studies has demonstrated the adaptive (i.e., variable sample sizes, sampling intervals, and/or action limit coefficients) X ¯ charts are quicker than standard Shewhart (SS) X ¯ control chart in detecting process mean shifts. The usual assumption for designing a control chart is that the data or measurements are normally distributed. However, this assumption may not be true for some processes. In this paper, the performance of the adaptive X ¯ chart is evaluated under non-normality. Based on the study, it is shown that the variable parameters (VP) X ¯ chart is more effective than the other adaptive control charts in detecting small process mean shifts under non-normality. More importantly, the risk of false alarm of the VP X ¯ chart can be substantially decreased.

Robustness of the Variable Sample Size and Control Limit ¯X Chart to Non Normality

Recent studies demonstrated that the variable sample size (VSS) ¯X chart is quicker than the standard Shewhart (SS) ¯X chart in detecting small process mean shifts. The usual assumption for designing a control chart is that the data or measurements are normally distributed. However, this assumption may not be tenable in some production processes. The Burr distribution has been used in the literature to represent various non normal distributions. In this article, the Burr distribution will be employed to evaluate the control charts for non normal populations. We first show the VSS and Shewhart ¯X charts are sensitive to non normality. Then we propose a method of varying the sample size and the control limits simultaneously. The variable sample size and control limit VSSCL ¯X chart is shown to be quicker than the VSS ¯X chart in detecting small and moderate shifts in the process. Most importantly, the risk of false alarm for the VSSCL ¯X chart can be substantially decreased. In addition, with proper selection of chart parameters, the VSSCL ¯X chart is more robust to non normality than the VSS and Shewhart ¯X charts.

On the design of variable sample size and sampling intervals [formula omitted] charts under non-normality

Recent theoretical studies demonstrated the variable sampling rate (variable sample size and/or sampling interval) X¯ charts are quicker than standard Shewhart (SS) X¯ control charts in detecting process mean shifts. The usual assumption for designing a control chart is that the data or measurements are normally distributed. However, this assumption may not be true. The Burr distribution has been used in the literature to represent various non-normal distributions. In this paper, the Burr distribution will be employed to design the variable sample size and sampling intervals (VSSI) X¯ charts under non-normality. Two types of X¯ charts are proposed and studied: the symmetric-limit chart and the asymmetric-limit chart. Based on the study, it is shown that (a) the variable sample size (VSS) and VSSI X¯ charts are more effective than the SS and variable sampling intervals (VSI) X¯ charts in detecting small process mean shifts, (b) the VSS and VSSI X¯ charts are more robust against non-normality. In addition, under consideration of the false alarm rate and detection ability, the asymmetric-limit chart is a better choice.