Combined X-bar and CRL Charts for the Gamma Process

Abstract

This paper (i) discusses the combined $\bar x$ and conforming run length (CRL) charts under the assumption that the quality characteristic under study follows a Gamma(?, ?, β) distribution with known parameters ?, ? and β, (ii) examines the performance of the combined $\bar x$ and CRL charts for the Exp(?, β) process, when its parameters ? and β are unknown and must be estimated from preliminary samples taken during the trial period, and (iii) obtains the adjusted control limits of the combined $\bar x$ and CRL charts that have a specified in-control average run length value for samples taken from the Exp(?, β) process with unknown parameters. Studies of the average run lengths reveal that the combined $\bar x$ and CRL charts outperform the Shewhart $\bar x$ chart with either asymmetric probability limits or 3-sigma control limits.