The synthetic [Xbar] chart with estimated parameters

A synthetic [Xbar] chart consists of an integration of a Shewhart [Xbar] chart and a conforming run length chart. This type of chart has been extensively used to detect a process mean shift under the assumption of known process parameters. However, in practice, the process parameters are rarely known and are usually estimated from an in-control Phase I data set. The goals of this article are to (i) evaluate (using a Markov chain model) the performances of the synthetic [Xbar] chart when the process parameters are estimated; (ii) compare it with the case where the process parameters are assumed known to demonstrate that these performances are quite different when the number of samples used during Phase I is small; and (iii) suggest guidelines concerning the choice of the number of Phase I samples and to provide new optimal constants, especially dedicated to the number of samples used in practice.

[1]  S. W. Roberts.  Control chart tests based on geometric moving averages , 2000 .

[2]  Charles W. Champ,et al.  Exact results for shewhart control charts with supplementary runs rules , 1987 .

[3]  H. J. Huang,et al.  A synthetic control chart for monitoring process dispersion with sample standard deviation , 2005, Comput. Ind. Eng..

[4]  Tom Burr,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 2001, Technometrics.

[5]  Maria E. Calzada,et al.  The Generalized Synthetic Chart , 2009 .

[6]  F L Chen,et al.  VARIABLE SAMPLING INTERVAL SYNTHETIC CONTROL CHARTS FOR JOINTLY MONITORING PROCESS MEAN AND STANDARD DEVIATION , 2006 .

[7]  Philippe Castagliola,et al.  Computational Statistics and Data Analysis an Ewma Chart for Monitoring the Process Standard Deviation When Parameters Are Estimated , 2022 .

[8]  Trevor A Spedding,et al.  Implementing Synthetic Control Charts , 2000 .

[9]  Giovanni Celano,et al.  THE EXACT RUN LENGTH DISTRIBUTION AND DESIGN OF THE S2 CHART WHEN THE IN-CONTROL VARIANCE IS ESTIMATED , 2009 .

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

[11]  Z. Yanga,et al.  On the Performance of Geometric Charts with Estimated Control Limits , 2011 .

[12]  Gemai Chen,et al.  THE MEAN AND STANDARD DEVIATION OF THE RUN LENGTH DISTRIBUTION OF X̄ CHARTS WHEN CONTROL LIMITS ARE ESTIMATED Gemai Chen , 2003 .

[13]  Vaidyanathan Ramaswami,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 1999, ASA-SIAM Series on Statistics and Applied Mathematics.

[14]  Charles P. Quesenberry,et al.  The Effect of Sample Size on Estimated Limits for and X Control Charts , 1993 .

[15]  Charles W. Champ,et al.  The Performance of Exponentially Weighted Moving Average Charts With Estimated Parameters , 2001, Technometrics.

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

[17]  Gemai Chen,et al.  The run length distributions of the R, s and s2 control charts when is estimated , 1998 .

[18]  Zhang Wu,et al.  A SYNTHETIC CONTROL CHART FOR MONITORING THE PROCESS MEAN OF SKEWED POPULATIONS BASED ON THE WEIGHTED VARIANCE METHOD , 2008 .

[19]  H.-J. Huang,et al.  A synthetic control chart for monitoring process dispersion with sample range , 2005 .

[20]  Philippe Castagliola,et al.  A Synthetic Scaled Weighted Variance Control Chart for Monitoring the Process Mean of Skewed Populations , 2009, Commun. Stat. Simul. Comput..

[21]  Frederick S. Hillier,et al.  X-Bar- and R-Chart Control Limits Based on A Small Number of Subgroups , 1969 .

[22]  C. H. Sim.  Combined X-bar and CRL Charts for the Gamma Process , 2003, Comput. Stat..

[23]  Patrick D. Bourke,et al.  Performance Comparisons for the Synthetic Control Chart for Detecting Increases in Fraction Nonconforming , 2008 .

[24]  David A. Wood,et al.  What a performance , 2004 .

[25]  Francisco Aparisi,et al.  Synthetic-X control charts optimized for in-control and out-of-control regions , 2009, Comput. Oper. Res..

[26]  S. W. Roberts,et al.  Control Chart Tests Based on Geometric Moving Averages , 2000, Technometrics.

[27]  Maria E. Calzada,et al.  A Note on the Lower-Sided Synthetic Chart for Exponentials , 2003 .

[28]  Maria E. Calzada,et al.  THE ROBUSTNESS OF THE SYNTHETIC CONTROL CHART TO NON-NORMALITY , 2001 .

[29]  James M. Lucas,et al.  Exponentially weighted moving average control schemes: Properties and enhancements , 1990 .

[30]  Eugenio K. Epprecht,et al.  Synthetic control chart for monitoring the pprocess mean and variance , 2006 .

[31]  Trevor A Spedding,et al.  A Synthetic Control Chart for Detecting Small Shifts in the Process Mean , 2000 .

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

[33]  Marion R. Reynolds,et al.  Shewhart x-charts with estimated process variance , 1981 .

[34]  L. Allison Jones,et al.  The Statistical Design of EWMA Control Charts with Estimated Parameters , 2002 .

[35]  S. Psarakis,et al.  EFFECT OF ESTIMATION OF THE PROCESS PARAMETERS ON THE CONTROL LIMITS OF THE UNIVARIATE CONTROL CHARTS FOR PROCESS DISPERSION , 2002 .

[36]  William H. Woodall,et al.  Evaluating and Improving the Synthetic Control Chart , 2002 .

[37]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[38]  E. Castillo.  Computer Programs -- Evaluation of Run Length Distribution for X Charts with Unknown Variance , 1996 .