Optimal statistical designs of multivariate EWMA and multivariate CUSUM charts based on average run length and median run length

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Date
2006-11
Authors
Ming Ha, Lee
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Abstract
A multivariate control chart is a powerful tool in process control involving a simultaneous monitoring of several correlated quality characteristics. The multivariate cumulative sum (MCUSUM) and multivariate exponentially weighted moving average (MEWMA) charts are often recommended in process monitoring when a quick detection of small or moderate sustained shifts in the mean vector is desired. This thesis develops a graphical method to determine the optimal choices of the parameters of the MEWMA and MCUSUM charts for a quick detection of a desired size of a shift. These control charts are statistically designed and the method used follows the general approach that is currently available only for univariate EWMA and CUSUM charts. The performances of the control charts are measured by the average run length (ARL) that is derived using a Markov chain approach or a simulation method. The design based on the median run length (MRL) is also given as the MRL is a more meanin~ful measure of centrality with respect to the highly skewed run length distribution. The design based on MRL is made using the Markov chain approach or the simulation method. Besides providing a graphical approach to simplify the existing procedure in the design of optimal MEWMA and MCUSUM charts which are based mainly on the ARL, this thesis introduces the optimal design strategy of the two charts using the MRL, which are currently available for the univariate EWMA and CUSUM charts only. Each of the design schemes is based on the aspects of the run length distribution derived using independent observations from the multivariate normal distribution. The number of quality characteristics considered. are p = 2. 3, 4. 5, 8 and 10. The procedure for the optimal design of the MEWMA and MCUSUM charts consists of spedfying the desired in-control ARL or MRL and the magnitude of the shift (i.e., the square root of the noncentr~lity parameter, o) in the process mean vector to be detected quickly. This is MCUSUM terdiri daripada penetapan ARL atau MRL dalam kawalan yang diingini dan magnitud anjakan dalam vektor min proses (iaitu, punca kuasa dua parameter tak memusat, o) yang akan dikesan dengan cepat. lni diikuti dengan pemilihan kombinasi parameter-parameter carta, iaitu pemalar peficinan, r untuk carta MEWMA (atau nilai rujukan, k untuk carta MCUSUM) dan had kawalannya yang sepadan, H. Kombinasi parameter-parameter carta yang diperoleh adalah optima dengan kenyataan bahawa bagi ARL atau MRL dalam kawalan yang ditetapkan, ia menghasilkan ARL atau MRL luar kawalan yang terendah untuk magnitud anjakan dalam vektor min yang ditetapkan untuk pengesanan cepat. Graf-graf parameter carta optima bagi carta MEWMA dan MCUSUM untuk pelbagai ARL atau MRL dafam kawalan berdasarkan magnitud anjakan dalam vektor min yang benainan adalah diberikan. Contoh-contoh yang menunjukkan apfikasi graf-graf ini turut diberikan.
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Quality control
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