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**The optimal control chart procedure.**
*(English)*
Zbl 1249.93178

Summary: The Moving Average (MA) chart, the Exponentially Weighted Moving Average (EWMA) chart and the CUmulative SUM (CUSUM) chart are the most popular schemes for detecting shifts in a relevant process parameter. Any control chart system of span \(k\) is specified by a partition of the space \({\mathbb R}^k\) into three disjoint parts. We call this partition as the control chart frame of span \(k\). A shift in the process parameter is signalled at time \(t\) by having the vector of the last \(k\) sample characteristics fall out of the central part of this frame. The optimal frame of span \(k\) is selected in order to maximize the Average Run Length (ARL) if shift in the relevant process parameter is on an acceptable level and to minimize it on a rejectable level. We have proved in this article that the set of all frames of span \(k\) with an appropriate metric is a compact space and that the ARL for continuously distributed sample characteristics is continuous as a function of the frame. Consequently, there exists the optimal frame among systems of span \(k\). General attitude to control chart systems is the common platform for universal control charts with the particular point for each sample and variable control limits plotted one step ahead.

### MSC:

93E20 | Optimal stochastic control |

49J30 | Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |

62F15 | Bayesian inference |

62P30 | Applications of statistics in engineering and industry; control charts |

### Keywords:

control chart; frame of span \(k\); average run length; probability distribution; compact metric space### References:

[1] | Atienza O. O., Ang B. W., Tang L. C.: Statistical process control and forecasting. Internat. J. Quality Science 1 (1997), 37-51 |

[2] | Engelking R.: General Topology. PWN, Warszawa 1977 · Zbl 0684.54001 |

[3] | Feigenbaum A. V.: Total Quality Control. McGraw-Hill, New York 1991 |

[4] | Gitlow H., Gitlow S., Oppenheim, A., Oppenheim R.: Tools and Methods for the Improvement of Quality. Irwin, Boston 1989 · Zbl 0713.62102 |

[5] | James P. T. J.: Total Quality Management: An Introductory Text. Prentice Hall, London 1996 |

[6] | Arquardt D. W.: Twin metric control - CUSUM simplified in a Shewhart framework. Internat. J. Quality & Reliability Management 3 1997), 220-233 |

[7] | Ncube M. M.: Cumulative score quality control procedures for process variability. Internat. J. Quality & Reliability Management 5 (1994), 38-45 |

[8] | Quesenberry C. P.: SPC Methods for Quality Improvement. Wiley, New York 1997 |

[9] | Roberts S. W.: A comparison of some control chart procedures. Technometrics 1 (1966), 239-250 |

[10] | Srivastava M. S., Wu Y.: Economical quality control procedures based on symmetric random walk model. Statistica Sinica 6 (1996), 389-402 · Zbl 0843.62100 |

[11] | Taguchi G.: Quality engineering in Japan. Commentaries in Statistics, Series A 14 (1985), 2785-2801 |

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