An Economic Study of X̄-Charts with Warning Limits

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Abstract
This paper investigates the economic design of X̄-charts with both warning and action limits, based on a widely studied process model. The optimization is accomplished by a three-dimensional pattern search. Various comparisons are made among the minimum cost designs of X̄-charts with and without warning limits and of Cusum charts. No appreciable differences are found. X̄-charts with warning limits are thus recommended for practical application as they are much easier to handle than Cusum charts, and they provide more psychological protection than X̄-charts with action limits only. It is proposed that in application the band enclosed by the warning limits should be 0.85 times as wide as the band enclosed by the action limits. A simplified economic scheme for the design of control parameter values is also suggested.

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