Simultaneously guaranteeing the in-control and out-of-control performances of the S2 control chart with estimated variance

Recent studies on the effects of parameter estimation on control charts have focused on their conditional in-control (IC) performance and recommended either the minimum number of Phase I samples (m) or adjustments to the control limit factor (L) that guarantee a desired IC performance with a high probability. In most cases, the numbers of samples required are prohibitively large in practice, and the adjustments for smaller numbers of samples entail as a counterpart a deterioration of the chart's out-of-control (OOC) performance. This presents the user with a hard decision, in which he or she will have difficulty in finding the best compromise between the objectives of good (or acceptable) IC performance, OOC performance, and a practicable number of Phase I samples. Therefore, in the context of the S-2 chart, we propose a new approach that takes both the desired IC and OOC performances (that should be within specified tolerances with a specified high joint probability) as constraints for the optimization of the pair (L, m). This is the first work that simultaneously treats the choice of m and the control limit adjustment in the framework of an optimization problem. With our model, the user can automatically obtain the most feasible (minimum m) solution that satisfies his/her requirements on both the IC and OOC performances.