A Gradient Approach to the Optimal Design of CUSUM Charts Under Unknown Mean-Shift Sizes

Abstract

Traditional cumulative sum (CUSUM) control charts are often designed to optimize the detection performance for a prescribed magnitude of mean shift when monitoring the mean level of a process. However, the shift to occur in the future is often unknown. To account for the uncertainty of the shift size, different design criteria have been proposed. However, all the design methods discussed in the literature are based on Monte Carlo simulations, and the optimal parameters are obtained in a trial-and-error manner (e.g., Ryu et al. (2010)). Notice that the average-run length (ARL) of the CUSUM chart can be formulated as an integral equation based on the recurrence relationship of the CUSUM charting statistics; this paper proposes a gradient-based approach for efficient design and analysis of CUSUM charts under unknown mean-shift sizes. The proposed method is shown to be more accurate and efficient than Monte Carlo simulations.

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