Design of a Phase II Control Chart for Monitoring the Ratio of two Normal Variables

In many industrial scenarios, on-line monitoring of quality characteristics computed as the ratio of two normal random variables can be required. Potential industrial applications can include monitoring of processes where the correct proportion of a property between two ingredients or elements within a product should be maintained under statistical control; implementation of quality control procedures where the performance of a product is measured as a ratio before and after some specific operation, for example a chemical reaction following the introduction of an additive in a product and monitoring of a chemical or physical property of a product, which is itself defined and computed as a ratio. This paper considers Phase II Shewhart control charts with each subgroup consisting of n > 1 sample units. From one subgroup to another, the size of each sample unit, upon which a single measurement is made, can be changed. An approximation based on the normal distribution is used to efficiently handle the ratio distribution. Several tables are generated and commented to show the statistical performance of the investigated chart for known and random shift sizes affecting the in-control ratio. An illustrative example from the food industry is given for illustration. Copyright © 2014 John Wiley & Sons, Ltd.

[1]  Josep Antoni Martín Fernández,et al.  Out-of-control signals in three-part compositional T2 control chart , 2014 .

[2]  John C. Young,et al.  Hotelling's T 2 Statistic , 2011, International Encyclopedia of Statistical Science.

[3]  G. Marsaglia.  Ratios of Normal Variables and Ratios of Sums of Uniform Variables , 1965 .

[4]  J. C. Hayya,et al.  A Note on the Ratio of Two Normally Distributed Variables , 1975 .

[5]  Douglas C. Montgomery,et al.  A review of multivariate control charts , 1995 .

[6]  Frank B. Alt.  Multivariate Quality Control , 1984 .

[7]  L MasonR,et al.  ホテリングのT2統計量を利用した多変量統計的工程管理(MVSPC)を実装する , 2001 .

[8]  Marianne Frisén,et al.  On multivariate control charts , 2011 .

[9]  Douglas C. Montgomery,et al.  Statistical quality control : a modern introduction , 2009 .

[10]  R. C. Geary.  The Frequency Distribution of the Quotient of Two Normal Variates , 2022 .

[11]  Dolun Oksoy,et al.  The quotient of two correlated normal variables with applications , 1994 .

[12]  H. Hotelling.  Multivariate Quality Control-illustrated by the air testing of sample bombsights , 1947 .

[13]  Charles W. Champ.  Multivariate Statistical Process Control with Industrial Applications , 2004 .

[14]  Andrej Blejec,et al.  distribution of the ratio of jointly normal variables , 2004, Advances in Methodology and Statistics.

[15]  Giovanni Celano,et al.  Statistical Performance of a Control Chart for Individual Observations Monitoring the Ratio of Two Normal Variables , 2014, Qual. Reliab. Eng. Int..

[16]  D. Hinkley.  On the ratio of two correlated normal random variables , 1969 .

[17]  Dolun Oksoy,et al.  STATISTICAL PROCESS CONTROL BY THE QUOTIENT OF TWO CORRELATED NORMAL VARIABLES , 1993 .

[18]  Stelios Psarakis,et al.  Multivariate statistical process control charts: an overview , 2007, Qual. Reliab. Eng. Int..

[19]  Andrew W. Spisak,et al.  A control chart for ratios , 1990 .

[20]  Giovanni Celano,et al.  Performance of t control charts in short runs with unknown shift sizes , 2013, Comput. Ind. Eng..

[21]  Sujin Kim,et al.  Optimal Design of a CUSUM Chart for a Mean Shift of Unknown Size , 2010 .

[22]  E. C. Fieller.  THE DISTRIBUTION OF THE INDEX IN A NORMAL BIVARIATE POPULATION , 1932 .

[23]  Giovanni Celano,et al.  Monitoring the Coefficient of Variation Using EWMA Charts , 2011 .

[24]  Arthur B. Yeh,et al.  Multivariate Control Charts for Monitoring Covariance Matrix: A Review , 2006 .