Control Charts for Joint Monitoring of Mean and Variance: An Overview

Abstract In the control chart literature, a number of one-and two-chart schemes has been developed to simultaneously monitor the mean and variance parameters of normally distributed processes. These “joint” monitoring schemes are useful for situations in which special causes can result in a change in both the mean and the variance, and they allow practitioners to avoid the inflated false alarm rate which results from simply using two independent control charts (one each for mean and variance) without adjusting for multiple testing. We present an overview of this literature covering some of the one-and two-chart schemes, including those that are appropriate in parameters known (standards known) and unknown (standards unknown) situations. We also discuss some of the joint monitoring schemes for multivariate processes, autocorrelated data, and individual observations. In addition, noting that normality is often an elusive assumption, we discuss some available nonparametric schemes for jointly monitoring location and scale. We end with a conclusion and some recommendations for areas of further research.

[1]  Smiley W. Cheng,et al.  On 2-D Control Charts , 2008 .

[2]  Chung-Ho Chen,et al.  Using one EWMA chart to jointly monitor the process mean and variance , 2010, Comput. Stat..

[3]  Fugee Tsung,et al.  Likelihood Ratio-Based Distribution-Free EWMA Control Charts , 2010 .

[4]  Antonio Fernando Branco Costa,et al.  A Single EWMA Chart for Monitoring Process Mean and Process Variance , 2006 .

[5]  Marion R. Reynolds,et al.  Monitoring the Process Mean and Variance Using Individual Observations and Variable Sampling Intervals , 2001 .

[6]  Amitava Mukherjee,et al.  A Distribution‐free Control Chart for the Joint Monitoring of Location and Scale , 2012, Qual. Reliab. Eng. Int..

[7]  M. A. Graham,et al.  Phase I Statistical Process Control Charts: An Overview and Some Results , 2008 .

[8]  Zhang Wu,et al.  A single CUSUM chart using a single observation to monitor a variable , 2007 .

[9]  Marion R. Reynolds,et al.  Combinations of Multivariate Shewhart and MEWMA Control Charts for Monitoring the Mean Vector and Covariance Matrix , 2008 .

[10]  Marion R. Reynolds,et al.  Control Charts for Monitoring the Mean and Variance of Autocorrelated Processes , 1999 .

[11]  Zhonghua Li,et al.  Self-starting control chart for simultaneously monitoring process mean and variance , 2010 .

[12]  D. Hawkins.  Self‐Starting Cusum Charts for Location and Scale , 1987 .

[13]  Antonio Fernando Branco Costa,et al.  Joint economic design of x and R charts under Weibull shock models , 2000 .

[14]  Y. Lepage.  A combination of Wilcoxon's and Ansari-Bradley's statistics , 1971 .

[15]  Smiley W. Cheng,et al.  Monitoring Process Mean and Variability with One EWMA Chart , 2001 .

[16]  Eugenio K. Epprecht,et al.  Estimating the Standard Deviation in Quality-Control Applications , 2010 .

[17]  Smiley W. Cheng,et al.  A New Chart for Monitoring Service Process Mean , 2012, Qual. Reliab. Eng. Int..

[18]  F. Gan.  Joint monitoring of process mean and variance using exponentially weighted moving average control charts , 1995 .

[19]  Arthur B. Yeh,et al.  Unified CUSUM Charts for Monitoring Process Mean and Variability , 2004 .

[20]  Changliang Zou,et al.  A control chart based on likelihood ratio test for monitoring process mean and variability , 2010, Qual. Reliab. Eng. Int..

[21]  Douglas M. Hawkins,et al.  Self-Starting Multivariate Control Charts for Location and Scale , 2011 .

[22]  Herbert Moskowitz,et al.  Joint economic design of EWMA control charts for mean and variance , 2008, Eur. J. Oper. Res..

[23]  Jin Zhang.  Powerful goodness‐of‐fit tests based on the likelihood ratio , 2002 .

[24]  Yu Tian,et al.  Weighted-loss-function CUSUM chart for monitoring mean and variance of a production process , 2005 .

[25]  Douglas M. Hawkins,et al.  Statistical Process Control for Shifts in Mean or Variance Using a Changepoint Formulation , 2005, Technometrics.

[26]  Marion R. Reynolds,et al.  Robust CUSUM charts for monitoring the process mean and variance , 2009, Qual. Reliab. Eng. Int..

[27]  Charles P. Quesenberry,et al.  DPV Q charts for start-up processes and short or long runs , 1991 .

[28]  Zachary G. Stoumbos,et al.  Economic statistical design of adaptive control schemes for monitoring the mean and variance: An application to analyzers , 2005 .

[29]  Seyed Taghi Akhavan Niaki,et al.  The Max EWMAMS control chart for joint monitoring of process mean and variance with individual observations , 2011, Qual. Reliab. Eng. Int..

[30]  Smiley W. Cheng,et al.  Semicircle Control Chart for Variables Data , 1996 .

[31]  Sheng Zhang,et al.  A CUSUM scheme with variable sample sizes and sampling intervals for monitoring the process mean and variance , 2007, Qual. Reliab. Eng. Int..

[32]  Michael B. C. Khoo,et al.  A New Bivariate Control Chart to Monitor the Multivariate Process Mean and Variance Simultaneously , 2004 .

[33]  Smiley W. Cheng,et al.  Single Variables Control Charts: an Overview , 2006, Qual. Reliab. Eng. Int..

[34]  Antonio Fernando Branco Costa,et al.  An adaptive chart for monitoring the process mean and variance , 2007, Qual. Reliab. Eng. Int..

[35]  Smiley W. Cheng,et al.  A New EWMA Control Chart for Monitoring Both Location and Dispersion , 2004 .

[36]  Marion R. Reynolds,et al.  Variable sampling intervals for multiparameter shewhart charts , 1989 .

[37]  Fei-Long Chen,et al.  Economic Design of Max Charts , 2010 .

[38]  Marion R. Reynolds,et al.  Comparisons of Some Exponentially Weighted Moving Average Control Charts for Monitoring the Process Mean and Variance , 2006, Technometrics.

[39]  Smiley W. Cheng,et al.  MAX CHART: COMBINING X-BAR CHART AND S CHART , 1998 .

[40]  Marion R. Reynolds,et al.  Control Charts and the Efficient Allocation of Sampling Resources , 2004, Technometrics.

[41]  Douglas M. Hawkins,et al.  Combined Charts for Mean and Variance Information , 2009 .

[42]  Mao Hong,et al.  The Economic Design of Multivariate MSE Control Chart , 2011 .

[43]  Zhang Wu,et al.  Monitoring Process Mean and Variability with One Double EWMA Chart , 2010 .

[44]  Gyo-Young Cho,et al.  Multivariate Control Charts for Monitoring the Mean Vector and Covariance Matrix , 2006 .

[45]  Qin Zhou,et al.  A control chart based on likelihood ratio test for detecting patterned mean and variance shifts , 2010, Comput. Stat. Data Anal..

[46]  Zhonghua Li,et al.  A multivariate control chart for simultaneously monitoring process mean and variability , 2010, Comput. Stat. Data Anal..

[47]  Changliang Zou,et al.  A New Chart for Detecting the Process Mean and Variability , 2011, Commun. Stat. Simul. Comput..

[48]  Marion R. Reynolds,et al.  Multivariate Control Charts for Monitoring the Process Mean and Variability Using Sequential Sampling , 2007 .

[49]  Zachary G. Stoumbos,et al.  Should Exponentially Weighted Moving Average and Cumulative Sum Charts Be Used With Shewhart Limits? , 2005, Technometrics.

[50]  Charles W. Champ,et al.  Effects of Parameter Estimation on Control Chart Properties: A Literature Review , 2006 .

[51]  Antonio Fernando Branco Costa,et al.  Monitoring Process Mean and Variability with One Non-central Chi-square Chart , 2004 .

[52]  Fah Fatt Gan,et al.  Interval Charting Schemes for Joint Monitoring of Process Mean and Variance , 2004 .

[53]  Fah Fatt Gan,et al.  Joint monitoring of process mean and variance , 1997 .

[54]  Marion R. Reynolds,et al.  Individuals control schemes for monitoring the mean and variance of processes subject to drifts , 2001 .

[55]  Rickie J. Domangue,et al.  Some omnibus exponentially weighted moving average statistical process monitoring schemes , 1991 .