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Optimizing Defect Levels and Losses from Gage Errors

Defect Levels and Gage Losses for Systems of Gages with Guard Bands

Defect Leyels for a Single Gage System FigurÄ™ 8 describes the probability of having product fali in a rangÄ™ above the upper specification limit of the product distribution as well as the probability of the gage measuring the units with readings below the guard band and passing the units as good product. Therefore, by multiplying the probability of having reject units in the dx rangÄ™ by the associated probability of those units being accepted by the gage as good, then integrating from the upper specification limit to positive infinity, the defect level of product being sent on to the customer can be determined. This is often measured in parts per million (ppm).

By using the equality shown in Equation (3), it can be shown mathematically that the defect level can be calculated from

P(good|reject) = J [f(z)J g'(y)dy]dz    (8)

USL z

given that g'(y) is N(gb,cjg) such that the gage distribution is placed on top of the guard band. By doing this, the mean of the gage distribution remains the guard band limit while the starting point of the integration increases as the value of zj increases for the product. The calculation is madÄ™ easier by maintaining the same distribution throughout the calculation, and therefore, the right side of Equation (3) is used. If a guard band is used, then the mean of the gage distribution becomes the guard band rather than the upper specification limit. Equation (8) calculates the probability of a defective unit being accepted by the gage as a good unit in a single gage system with a guard band employed. This is also equivalent to the defective level being sent on to the customer.

Defect Leyels for Multiple Gage Systems Often, multiple gages are used simultaneously to reduce the defect level being shipped to a customer while attempting to keep the gage losses at a reasonable level. For two gages, the defect level is

P(good|reject) = J[f(z)Jg', (w)dwJg'₂ (x)dx]dz

USL z    z

(9)