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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Keywords
Quality control