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Regret Indices and Capability Quantification

Given finite values of (J. and ct, the Bienayme-Tchebysheff inequality States that the probability of an absolute deviation from p equaling or exceeding k -ct is less than or equal to 1 over k squared for any positive, fixed k. And the bound is "sharp" in the sense that a distribution exists that attains equality for any such given k factor. Anyway, the implied lower bound on process yield is thus

Prob( LTL < X < UTL) ;> 1 - l/(9 Cj)    [26]

Rodriguez (1992) stresses that yield calculations from Cp indices assuming normality can be ąuite misleading. This point is certainly driven home by the results listed in Table 4, where normal-theory process yields are compared with the distribution-free lower bound given by [26],

Peam, Kotz and Johnson (1992) suggest that capability indices of the form C₀ =(UTL-LTL)/(0-cr) with 0 somewhat less than 6 lead to conformance fractions within (p-0<y,p+0a) that [at least for distributions reasonably approximated by a member of the Pearson system] can be much less sensitive to distributional form than when 0=6, as in Cp . On the other hand, [26] simply becomes Yield £ lOO[l-4/(0² -C₀)]% in this case.

Taguchi and Wu (1979, page 16) wam against attempts to artificially inflate Cp_m by simply widening tolerance limits. And Taguchi, Elsayed, and Hsiang (1989), page 12, say "we introduce a monetary evaluation of the quality of products, assuming tolerances are correct.

It is also interesting to notę that Taguchi, Elsayed, and Hsiang (1989) State on pages 11-12 that Cpm "is a poor measure of ąuality level because management and engineers cannot comprehend the actual significance of its values..." and "Percent defectives and warranty costs are understandable because they are monetary-related measures." Then, on pages 16-17, they add "losses caused by deviation are reciprocally proportional to the square of the Cp_m indices." In other words, the cost/loss they recommend is simply expected regret, ER.

In summary then, Cp-type statistics can be criticized for being too arbitrary in the sense that their numerator tolerance could be somewhat artificial and their denominator VER assumes that regret is quadratic. Cp. type statistics end up depending only on summary statistics like means and variances that cannot accurately predict process yields for non-normal distributions. Finally, although useful in quality improvement programs for a