Monitoring multivariate–attribute processes based on transformation techniques

Abstract

Control charts are widely used in monitoring the quality of a product or a process. In most of the cases, quality of a product or a process can be characterized by two or more correlated quality characteristics. Many control charts have been proposed for monitoring multivariate or multi-attribute quality characteristics, separately, but sometimes the correlated variables and attribute quality characteristics represents the quality of a process. In this paper, the use of four transformation methods is proposed to monitor the multivariate–attribute processes. In the first one, the distribution of correlated variables and attribute quality characteristics are transformed to approximate multivariate normal distribution, and then the transformed data are monitored by multivariate control charts including T ² and MEWMA. Based on the second transformation method, the correlated variables and attribute quality characteristics are transformed, such that the correlation between the quality characteristics approaches to zero, then univariate control charts are used in monitoring the transformed data. In the third and fourth proposed methods, a combination of two transformation methods is used to make the quality characteristics independent and to transform them to normal distribution. The difference between the third and fourth method is the order of using the transformation techniques. The performance of the proposed methods is evaluated by using simulation studies in terms of average run length criterion. Finally, the proposed approach is applied to a real dataset.

Appendices

Appendix A: Proof that the transformed data are uncorrelated after symmetric root transformation

$$ E\left({\displaystyle {\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}\left(\mathbf{x}-{\boldsymbol{\upmu}}_{\mathbf{x}}\right)}\right)={\displaystyle {\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}E\left(\mathbf{x}-{\boldsymbol{\upmu}}_{\mathbf{x}}\right)}={\displaystyle {\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}\left(E\left(\mathbf{x}\right)-{\boldsymbol{\upmu}}_{\mathbf{x}}\right)=}{\displaystyle {\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}\left({\boldsymbol{\upmu}}_{\mathbf{x}}-{\boldsymbol{\upmu}}_{\mathbf{x}}\right)=}{\displaystyle {\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}(0)=}0 $$

$$ \begin{array}{l} Cov\left({\displaystyle {\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}\left(\mathbf{x}-{\boldsymbol{\upmu}}_{\mathbf{x}}\right)}\right)= Cov\left({\displaystyle {\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}\mathbf{x}-{\displaystyle {\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}{\boldsymbol{\upmu}}_{\mathbf{x}}}}\right)= Cov\left({\displaystyle {\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}\mathbf{x}}\right)=\left({\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}\right)\left({\boldsymbol{\Sigma}}_{\mathbf{x}}\right){\left({\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}\right)}^{\prime }=\hfill \\ {}=\left({\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}\right)\left({\boldsymbol{\Sigma}}_{\mathbf{x}}^{\frac{1}{2}}{\boldsymbol{\Sigma}}_{\mathbf{x}}^{\frac{1}{2}}\right)\left({\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}\right)=\left({\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}{\boldsymbol{\Sigma}}_{\mathbf{x}}^{\frac{1}{2}}\right)\left({\boldsymbol{\Sigma}}_{\mathbf{x}}^{\frac{1}{2}}{\boldsymbol{\Sigma}}_{\mathbf{x}}^{-\frac{1}{2}}\right)=\mathbf{I}*\mathbf{I}=\mathbf{I}\hfill \end{array} $$

Appendix B: The dataset for real case in Sect. 5

Table 5 The real dataset