Measuring process performance based on expected loss with asymmetric tolerances

Abstract

By approaching capability from the point of view of process loss similar to C-pm, Johnson (1992) provided the expected relative loss Le to consider the proximity of the target value. Putting the loss in relative terms, a user needs only to specify the target and the distance from the target at which the product would have zero worth to quantify the process loss. Tsui (1997) expressed the index L-e as Le = L-ot + L-pe, which provides an uncontaminated separation between information concerning the process relative off-target loss (L-ot) and the process relative inconsistency loss (L-ot). Unfortunately, the index L-e inconsistently measures process capability in many cases, particularly for processes with asymmetric tolerances, and thus reflects process potential and performance inaccurately. In this paper, we consider a generalization, which we refer to as L-e", to deal with processes with asymmetric tolerances. The generalization is shown to be superior to the original index L-e. In the cases of symmetric tolerances, the new generalization of process loss indices L-e", L-ot" and L-pe" reduces to the original index L-e, L-ot, and L-pe, respectively. We investigate the statistical Properties of a natural estimator of L-e" L-ot" and L-pe" when the underlying process is normally distributed. We obtained the rth moment, expected value, and the variance of the natural estimator (L) over cap (e)", (L) over cap (ot)", and (L) over cap (pe)". We also analyzed the bias and the mean squared error in each case. The new generalization V measures process loss more accurately than the original index L-e.