Economic and economic statistical designs of the synthetic X  chart using loss functions
This paper proposes the economic and economic statistical designs of the synthetic X  chart. In the economic design, the optimal chart parameters that minimize the expected cost function are obtained, while in the economic statistical design, the optimal chart parameters are obtained by minimizing the expected cost function, subject to constraints on the in-control average run length (ARL0) and the out-of-control average run length (ARL1). A small increase in cost is incurred when the statistical constraints are added to the economic design, however, a significant improvement in statistical performance is attained. The sensitivity of the optimal cost and the chart parameters for different loss functions and input parameters is investigated. The effects of misspecification of the type of the loss function, and the Taguchi loss coefficient, as well as the risk aversion coefficient of the loss function, are also investigated. In addition, effects of the process capability index are studied. Based on numerical studies, comparisons are made between the synthetic X  , Shewhart X  , and EWMA charts.
loss functions quality control robustness and sensitivity analysis constraints satisfaction economic and economic statistical models