Optimal designs of the variable sample size and sampling interval X¯ chart when process parameters are estimated

The idea of varying the X¯ chart's parameters has been explored extensively by many researchers. The variable sample size and sampling interval (VSSI) X¯ chart is among the adaptive control charts which improves the diagnostic abilities of the standard X¯ chart for a quick detection of small and moderate shifts in the process mean. The VSSI X¯ chart is usually investigated under the assumption of known process parameters. In practice, process parameters are rarely known and they need to be estimated from an in-control historical Phase-I dataset. Therefore, in this paper, the Markov chain approach for the VSSI X¯ chart with estimated parameters is developed to facilitate process monitoring in manufacturing and service industries. The performance of the VSSI X¯ chart is examined and evaluated when process parameters are estimated and is compared with the case where process parameters are known. The new optimal design strategies for the VSSI X¯ chart with estimated process parameters, for minimizing the out-of-control average time to signal and the average extra quadratic loss are developed so that the chart's optimization results and charting parameters can be compared with its known process parameters counterpart. By considering the number of Phase-I samples used by practitioners in manufacturing, new optimal charting parameters computed from the proposed optimal design procedures are provided. By taking into account of the impact of parameter estimation on the properties of a control chart, the quality and productivity of manufacturing processes in an industry will be enhanced.