An economic evaluation of variable sampling interval Shewhart control charts for means

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Virginia Polytechnic Institute and State University

A model is developed to describe the total expected cost per unit time of monitoring the mean of a production process with a Variable Sampling Interval control chart. This chart has the feature that the time delay between successive samples is not constant but depends on the result of the most recent sample observation.

The model assumes that the quality characteristic of interest is distributed normally with known, constant variance. Process states are defined to derive a Markov chain model. The solution to its steady state probability equations is obtained in terms of decision variables pertinent to the economic design of control charts. Each entry in the resulting probability vector is interpreted as the average proportion of time spent in the corresponding process state. The cost of occupying each state is also defined in terms of the model parameters, and the total expected cost is given as the product of the cost and probability vectors. This product is then divided by the product of the probability vector and the vector comprising the time durations of each state to obtain a total expected cost per unit time model.

Model behavior is analyzed to determine that Variable Sampling Interval control charts for means are economically attractive when sampling/inspection costs and nonconforming product costs as defined in this paper are of similar magnitude. Variable Sampling Interval charts are not attractive economically when either of those cost categories dominates the other.