Phase-I Design Scheme for -chart Based on Posterior Distribution

This article develops a Phase I design structure of X‾$\bar X $ -Chart, namely Bayesian X‾$\bar X $ -Chart, based on Bayesian (posterior distribution) framework assuming the normality of the quality characteristic to incorporate parameter uncertainty. Our approach consists of two stages: (i) construction of the control limits for X‾$\bar X $ -Chart based on posterior distribution of unknown mean μ and (ii) evaluation of the performance of the proposed design structure. The proposed design structure of X‾$\bar X $ -Chart is compared with the frequents design structure of X‾$\bar X $ - Chart in terms of (i) width of the control region and (ii) power of detecting a shift in the location parameter of the process. It has been observed that the proposed design structure of X‾$\bar X $ -Chart is performs better than the usual design structure to detecting shifts in the parameter of the process when the prior mean is close to the unknown target value.