Economic design of variable sampling interval T2 control charts—A hybrid Markov Chain approach with genetic algorithms

Hotelling’s T2 chart is the most widely used multivariate procedure for two or more related quality characteristics, but it is slow in detecting small process shifts. Recently, the variable sampling interval (VSI) control scheme in which the sampling interval between two success sampling points is varied based on the preceding T2 value has been shown to provide an increase to the detecting efficiency of the original T2 chart. In this paper a method is proposed to conduct the economic design of the VSI T2 chart when the in-control process mean vector and covariance matrix are unknown. It is assumed that only one assignable cause of variation exists and the time between occurrences is exponentially distributed. Accordingly, the Markov Chain approach is allowable to develop a cost model. In applying genetic algorithms to minimize the cost function, the optimal values of sample size, sampling interval lengths, upper control limit and warning limit used to choose one of the sampling interval lengths can be determined. Variable sampling interval and original T2 charts are compared with respect to the expect cost per unit time. Sensitivity analysis on the quantity of initial samples to estimate for in-control process mean vector and covariance matrix is also presented.