Efficient Design and Sensitivity Analysis of Control Charts Using Monte Carlo Simulation

The design of control charts in statistical quality control addresses the optimal selection of the design parameters (such as the sampling frequency and the control limits) and includes sensitivity analysis with respect to system parameters (such as the various process parameters and the economic costs of sampling). The advent of more complicated control chart schemes has necessitated the use of Monte Carlo simulation in the design process, especially in the evaluation of performance measures such as average run length. In this paper, we apply two gradient estimation procedures---perturbation analysis and the likelihood ratio/score function method---to derive estimators that can be used in gradient-based optimization algorithms and in sensitivity analysis when Monte Carlo simulation is employed. We illustrate the techniques on a general control chart that includes the Shewhart chart and the exponentially-weighted moving average chart as special cases. Simulation examples comparing the estimators with each other and with "brute force" finite differences demonstrate the possibility of significant variance reduction in settings of practical interest.