Combined X-bar and CRL Charts for the Gamma Process

This paper (i) discusses the combined <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\bar x$</EquationSource> </InlineEquation> and conforming run length (CRL) charts under the assumption that the quality characteristic under study follows a Gamma(ν, γ, ß) distribution with known parameters ν, γ and ß, (ii) examines the performance of the combined <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\bar x$</EquationSource> </InlineEquation> and CRL charts for the Exp(γ, ß) process, when its parameters γ and ß are unknown and must be estimated from preliminary samples taken during the trial period, and (iii) obtains the adjusted control limits of the combined <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\bar x$</EquationSource> </InlineEquation> and CRL charts that have a specified in-control average run length value for samples taken from the Exp(γ, β) process with unknown parameters. Studies of the average run lengths reveal that the combined <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\bar x$</EquationSource> </InlineEquation> and CRL charts outperform the Shewhart <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\bar x$</EquationSource> </InlineEquation> chart with either asymmetric probability limits or 3-sigma control limits. Copyright Physica-Verlag 2003