Simultaneously guaranteeing the in‐control and out‐of‐control performances of the S 2 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.