The economically designed two-sided Bayesian X¯ control chart

This paper studies a model for the economic design of an adaptive X¯ chart for short production runs that are subject to the occurrence of assignable causes, which may either increase or decrease the mean of the quality characteristic. At each sampling instance, the probabilities that the process operates under the effect of an assignable cause are updated using Bayes’ theorem. All three chart parameters, i.e., the time until the next sampling instance, the sample size and the control limit are adaptive and depend on these probabilities. We derive properties that facilitate the cost computation and eventually the optimization of the proposed scheme. Then, we evaluate the effectiveness of this chart by comparing its cost against the expected cost of (a) a fixed-parameter Shewhart chart optimized for short runs and (b) a variable-parameter Shewhart chart, optimized for an infinite process. The numerical results indicate that the potential savings from using the Bayesian scheme are significant.