Abstract
Control charts act as the most effective statistical process control (SPC) tools for the monitoring of manufacturing processes. In this study, we propose and investigate a set Shewhart-type variability control chart based on the utilization of auxiliary information for efficient phase II process monitoring. The design parameters of the proposals are derived under correlated setups for the monitoring of variability parameter. The properties of these charting structures are evaluated in terms of average run length and some other related measures. The performance abilities of these charts are compared with each other and also with some existing counterparts. The comparisons revealed that the proposed charts are very efficient at detecting shifts in the variability parameter and have the ability to perform better than the competing charts in terms of run length characteristics. We have also used real datasets to illustrate the application of the proposed structures in practical situations.
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Appendix
Appendix
Some distributional results: We define M = s 2 y /s 2 x and Q = σ 2 y /σ 2 x ; then, the joint probability density function of the random variables M and Q (the bivariate chi-square distribution obtained from the bivariate Wishart distribution) is given by:
where n > 3, − 1 < ρ < 1 and \( {}_0F_1\left[\frac{n-1}{2};\frac{\rho^2 mq}{{\left(2-2{\rho}^2\right)}^2}\right] \) is the hyper-geometric function as defined in Omar and Joarder [30]. Some more details in this regard may be seen in Gradshteyn and Ryzhik [18], Gunst and Webster [19], and Joarder [22]. For the case where ρ = 0, the abovementioned joint density function of M and Q may be written as the product of two independent chi-square variables with n − 1 degrees of freedom each. The probability density function of G r = M/Q is given as:
where n > 3, − 1 < ρ < 1 and \( B\left(\frac{1}{2},\frac{n-1}{2}\right) \) represents the well known beta function. Omar and Joarder [30] named it as correlated F distribution which may have symbolic representation as F(n − 1, n − 1, ρ). More details in this regard may be seen in Bose [10] and Finney [14].
The cumulative distribution function (ϒ) of G r is given as:
where Г represents the gamma function, n > 3 and − 1 < ρ < 1. (cf. Omar et al. [31]).
The median point of the correlated F(n − 1, n − 1, ρ) distribution given in Eq. (1) turns out to be 0.5 (cf. Omar and Joarder [30]). Moreover, there are three limiting distributional forms of \( {f}_{G_r}\left({g}_r\right) \) given as: when n → ∞, \( {f}_{G_r}\left({g}_r\right)\to \mathrm{Normal}\left({m}_r,{d}_r\right) \); when ρ → 0, \( {f}_{G_r}\left({g}_r\right)\to F\left(n-1,n-1\right) \); when ρ → 1, \( {f}_{G_r}\left({g}_r\right)\to {t}_{n-1} \). (cf. Omar and Joarder [30]).
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Riaz, M., Abbasi, S.A., Ahmad, S. et al. On efficient phase II process monitoring charts. Int J Adv Manuf Technol 70, 2263–2274 (2014). https://doi.org/10.1007/s00170-013-5418-7
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DOI: https://doi.org/10.1007/s00170-013-5418-7