Process Capability Indices under Non-Normality Conditions using Johnson Systems

Abstract

Process capability indices (PCIs) quantify the ability of a process to produce on target and within specifications performances. Basic indices designed for normal processes gives flawed results for non-normal process. Numerous methods have been proposed for non-normal processes to estimate PCIs in which some of them are based on transformation methods. The Johnson system comprising three types that translate a continuous non-normal distribution to normal. The aim of this paper is to estimate four basic indices for non-normal process using Johnson system with single straightforward procedure. The efficacy of the proposed approach can be assessed for all three

A power transformation for non-normal processes capability estimation

“…where parameter λ is estimated by employing the maximum likelihood estimator (MLE) [16]. Johnson system of distribution uses Johnson curves (S B , S U , S L ) to transform the data [17]. S B covers bounded distributions, S U unbounded and S L covers distributions which are bounded only on the lower side [17].…”

“…S B covers bounded distributions, S U unbounded and S L covers distributions which are bounded only on the lower side [17]. These three distributions are generated by transformations of the form [17]:…”

“…In this study, parameters of Johnson sytem will be estimated by quantile estimation method [17]. Niaki and Abbiasi [15] first mentioned root transformation technique for skewness reduction in multi-attribute process monitoring.…”

“…For example, in case of Johnoson transformation for 76.5 % of samples USL and LSL couldn't be transformed by equation ( 8) as x > λ+ϵ, which made most of samples fitted to S B not succesfully transformed. For Johnson transformation, authors in [17] used percentiles approach to estimate C p and C pk , which won't be discussed in this paper. In order to identify the method which gives the most accurate capability estimation, differences C p − C ′ p and C pk − C ′ pk were observed.…”

“…16 and Fig. 17 • To identify which method performes best, differences between target capability indices and estimated capability indices after transformation were compared for each distribution separately by calculating relative bias (RB) according to [17]:…”