Optimal design of the synthetic chart for the process mean based on median run length

Control charts are usually designed using average run length as the criterion to be optimized. The shape of the run length distribution changes according to the shift in the mean, from highly skewed for an in-control process to approximately symmetric when the mean shift is large. Since the shape of the run length distribution changes with the mean shift, the Median Run Length (MRL) provides a more meaningful interpretation for in-control and out-of-control performances of the charts, and it is readily understood by practitioners. This article presents an optimal design procedure for a synthetic chart able to monitor the mean based on the MRL under the zero- and steady-state modes. The synthetic chart integrates and conforming run length charts. Pseudocodes and Mathematica programs are presented for the computation of the optimal parameters of the synthetic chart based on a desired in-control MRL (MRL(0)), a given sample size, and a specified mean shift for which a quick detection is needed. [Supplementary materials are available for this article. Go to the publisher's online edition of IIE Transactions for additional example, additional performance study, proof, tables, and figures.]

[1]  K. Waldmann,et al.  Bounds for the Distribution of the Run Length of Geometric Moving Average Charts , 1986 .

[2]  K. Waldmann,et al.  Bounds for the Distribution of the Run Length of One-Sided and Two-Sided CUSUM Quality Control Schemes , 1986 .

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

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

[5]  Cheng-Chang Chang,et al.  Economic design of the VSSI control charts for correlated data , 2007 .

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

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

[8]  Fah Fatt Gan,et al.  An optimal design of ewma control charts based on median run length , 1993 .

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

[10]  Ronald B. Crosier,et al.  Fast Initial Response for CUSUM Quality-Control Schemes: Give Your CUSUM A Head Start.: Give Your CUSUM A Head Start. , 2000 .

[11]  Douglas C. Montgomery,et al.  A FAST INITIAL RESPONSE SCHEME FOR THE EXPONENTIALLY WEIGHTED MOVING AVERAGE CONTROL CHART , 1996 .

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

[13]  Li Yue.  The synthetic control chart , 2001 .

[14]  William H. Woodall,et al.  The Distribution of the Run Length of One-Sided CUSUM Procedures for Continuous Random Variables , 1983 .

[15]  Andrew C. Palm,et al.  Tables of Run Length Percentiles for Determining the Sensitivity of Shewhart Control Charts for Averages with Supplementary Runs Rules , 1990 .

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

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

[18]  Linda Lee Ho,et al.  Economic design of an X chart for short-run production , 2009 .

[19]  G. Barnard.  Control Charts and Stochastic Processes , 1959 .

[20]  Trevor A Spedding,et al.  A Synthetic Control Chart for Detecting Fraction Nonconforming Increases , 2001 .

[21]  Seven Knoth,et al.  Fast initial response features for EWMA control charts , 2005 .

[22]  Fah Fatt Gan,et al.  An optimal design of cumulative sum control chart based on median run length , 1994 .

[23]  Ronald B. Crosier,et al.  A new two-sided cumulative sum quality control scheme , 1986 .

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

[25]  M. S. de Magalhães,et al.  A hierarchy of adaptive control charts , 2009 .

[26]  Philippe Castagliola,et al.  On t and EWMA t charts for monitoring changes in the process mean , 2009, Qual. Reliab. Eng. Int..

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

[28]  Alireza Faraz,et al.  A unification and some corrections to Markov chain approaches to develop variable ratio sampling scheme control charts , 2011 .

[29]  Michael B. C. Khoo,et al.  Performance Measures for the Shewhart Control Chart , 2004 .

[30]  Douglas C. Montgomery,et al.  Statistical quality control : a modern introduction , 2009 .

[31]  Wei Jiang,et al.  Adaptive CUSUM procedures with Markovian mean estimation , 2008, Comput. Stat. Data Anal..

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

[33]  Wei Jiang,et al.  A Markov Chain Model for the Adaptive CUSUM Control Chart , 2006 .

[34]  A. F. Bissell,et al.  Cusum Techniques for Quality Control , 1969 .

[35]  Antonio Fernando Branco Costa,et al.  Adaptive control charts: A Markovian approach for processes subject to independent disturbances , 2006 .

[36]  Takayuki Kotani,et al.  A Synthetic Exponentially Weighted Moving-average Chart for High-yield Processes , 2005 .

[37]  Antonio Fernando Branco Costa,et al.  The double sampling and the EWMA charts based on the sample variances , 2008 .

[38]  James M. Lucas,et al.  Fast Initial Response for CUSUM Quality-Control Schemes: Give Your CUSUM A Head Start , 2000, Technometrics.

[39]  Fah Fatt Gan,et al.  The run length distribution of a cumulative sum control chart , 1993 .

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

[41]  D. A. Evans,et al.  An approach to the probability distribution of cusum run length , 1972 .

[42]  Charles W. Champ.  Steady-state run length analysis of a Shewhart quality control chart with supplementary runs rules , 1992 .

[43]  Eugenio K. Epprecht,et al.  Monitoring the process mean and variance using a synthetic control chart with two-stage testing , 2009 .