Encyclopedia and Handbook of Process Capability Indices: A Comprehensive Exposition of Quality Control Measures

Chapter 5: The Loss Indices

5.1 Process Loss and the L e Index

One of the most commonly understood basic primitive criterion for interpretation of the process capability is the so-called yield index, which is simply defined as the proportion of conforming items. However, the yield measure does not distinguish among the product that fall inside of the specification limits. To remedy this situation, a quadratic loss function is usually employed to distinguish between the products that fall inside of the specification limits by increasing "the penalty" as the departure from the target value increases. Hsiang and Taguchi (1985) were the first to introduce the loss function approach to quality improvement focussing on the reduction of variation around the target value. However, a quadratic loss function by itself does not provide a comparison with the specification limits and also depends on the unit of the characteristic involved. To address these issues, Johnson (1992) proposed the concept of relative expected squared error loss L e for symmetrical cases, by approaching capability in terms of loss functions. Actually, here the opposite concept of worth is used. It was postulated that a characteristic achieves its maximal worth W T, when X = T, following by decreasing values of the worth as X moves away from the target value T (eventually the worth becomes zero, and then negative). The worth function can be described by W T ? k(X ? T) 2, for W T ? k(X ?

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