Interval estimation of capability index Cpmk for manufacturing processes with asymmetric tolerances

The index Cpmk combines the merits of the three earlier indices Cp, Cpk, Cpm and alerts the user if the process variance increases and/or the process mean deviates from its target value. In practice, treat the calculated estimate Cˆpmk as true value and ignore the effect on asymmetric tolerances may lead to misinterpretation of process capability. Pearn et al. [Pearn, W. L., Chen, K. S. & Lin, P. C. (1999). On the generalizations of the capability index Cpmk for asymmetric tolerances. Far East Journal of Theoretical Statistics, 3(1), 47-66.] introduced a generalization of Cpmk, which referred to as Cpmk″, to handle processes with asymmetric tolerances. However, the sampling distribution of the estimator Cˆpmk″ is exceedingly complex and the derivation of an interval estimation of Cˆpmk″ is mathematically intractable. In this paper, we reformulate the explicit formulas and propose a heuristic algorithm to compute a lower confidence bound on Cpmk″, which presents a measure on the minimum capability of the process, to enhance the applicability of the theoretical results. Tables are provided to assist the practitioners for a wide range of real world situation involving processes capability analysis. Equations and tables to estimate approximate sample size necessary to achieve a desired confidence limit with a specified confidence level are also developed. An application example on the trench capacitor etch process is also presented for illustrating the applicability of the generalization.