Optimal designs of the double sampling X¯ chart with estimated parameters

The double sampling (DS) X¯ chart detects small and moderate mean shifts quickly. Furthermore, this chart can reduce the sample size. The DS X¯ chart is usually investigated assuming that the process parameters are known. Nevertheless, the process parameters are usually unknown and are estimated from an in-control Phase-I dataset. This paper (i) evaluates the performances of the DS X¯ chart when process parameters are estimated by means of a new proposed theoretical method, (ii) shows that performances with estimated parameters are different from that with known parameters, and (iii) proposes three optimal design procedures: the first design minimizes the out-of-control average run length, the second design minimizes the in-control average sample size and the third design minimizes the average extra quadratic loss, by considering the number of Phase-I samples in these three designs. Additionally, for ease of implementation, this paper provides the new optimal parameters specially computed for the DS X¯ chart with estimated parameters, based on the number of Phase-I samples used in practice. These findings will lead to a more economically feasible process monitoring situation, especially when the process parameters are unknown.