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

Abstract

International audienc

The performance of the Shewhart-RZ control chart in the presence of measurement error

“…In the literature, there are some studies about the distribution of the ratio of two normal variables, for further details see for instance: Geary (1930) and Hayya, Armstrong, and Gressis (1975), Cedilnik, Kosmelj, and Blejec (2004) and Pham-Gia, Turkkan, and Marchand (2006). As developed in Celano and Castagliola (2016a), approximations for the probability density function (…”

“…A typical example would be a linear dimension of a mechanical part, where the dispersion of observations should be significantly smaller than the nominal value to meet the fixed tolerance limits. In this case, according to Celano and Castagliola (2016a), the normal approximation for F Z (z|γ X , γ Y , ω, ρ) is excellent and it can be written as…”

“…Celano and Castagliola (2016b) demonstrated that the Phase II synthetic control chart for monitoring the ratio of two normal variables has always a better statistical sensitivity than Shewhart-RZ control chart. Tran, Castagliola, and Celano (2016) extended Celano and Castagliola (2016a)'s paper by incorporating Run Rules to the Shewhart-RZ control chart making it more sensitive to small changes in the ratio. Finally, the properties and design strategies of Exponentially Weighted Moving Average (EWMA) control charts and Cumulative Sum (CUSUM) control charts for this ratio have been investigated by Tran, Castagliola, and Celano (forthcoming-a) and Tran, Castagliola, and Celano (forthcoming-b), respectively.…”

“…Recently, Celano et al (2014) studied the statistical properties of a Shewhart chart for individual measurements (the Shewhart-RZ control chart), monitoring the ratio of two normal variables. Celano and Castagliola (2016a) extended this work by assuming subgroups consisting of n > 1 sample units and each of these sample units are allowed to change in size from one sample to another. Celano and Castagliola (2016b) demonstrated that the Phase II synthetic control chart for monitoring the ratio of two normal variables has always a better statistical sensitivity than Shewhart-RZ control chart.…”

“…The goal of this paper is to investigate the performance of the Shewhart-RZ control chart proposed by Celano and Castagliola (2016a) in the presence of measurement errors by assuming the same measurement error model as in Linna, Woodall, and Busby (2001). The study is organised as follows: in Section 2, the ratio Z distribution is introduced; in Section 3, the linear covariate error model for the ratio Z is defined; in Section 4, the formulas of the control limits and the performance metrics of the Shewhart-RZ control chart are discussed; in Section 5, the effect of measurement error on the Shewhart-RZ control chart performance is investigated; in Section 6, two illustrative examples are provided to show the implementation of the Shewhart-RZ control chart in the presence of measurement errors.…”