The C sub pk index and its confidence intervals. (process capability indicators)

Introduction

There have been, over the years, a number of process capability indices proposed in order to assess the capability of a process for meeting certain specifications. Kane (1986) comprehensively investigated the [C.sub.p] and [C.sub.pk] indices and their estimators. Generally, the [C.sub.pk] index takes into account the process variation as well as the location (mean) of the process relative to the specification limits, while the [C.sub.p] index reflects only the magnitude of the process variation.

Chan et al. (1988) developed the [Mathematical Expression Omitted] index in order to take into account the departure of the process mean from the target (nominal) value. This process capability index is also discussed in Spiring's article (1991). It was shown by Boyles (1991) that the [Mathematical Expression Omitted] index is identical to the process capability index proposed by Taguchi (1985). Pearn et al. (1992) introduced a new (so-called "third generation") process capability index, [C.sub.pmk], in an attempt to detect a smaller departure of the process mean from the target value. Recently, Greenwich and Jahr-Schaffrath (1995) introduced the process incapability index, [C.sub.pp], which is a transformation of the [Mathematical Expression Omitted]. This [C.sub.pp] index provides an uncontaminated separation between information concerning the process inaccuracy (departure of the process mean from the target value) and information concerning the process imprecision (the magnitude of the process variation). This kind of information separation is not available with the [Mathematical Expression Omitted] index.

In spite of all the introductions of new process capability indices, the [C.sub.pk] index remains most prevalent in practice. As often is the case, when no target value is specified, the [C.sub.pk] index is just as effective and informative as the other new indices, particularly when used with its sub-indices. These sub-indices are explained and their confidence intervals are presented in this article. These confidence intervals are constructed based on the asymptotic normality of the estimators for the sub-indices of the [C.sub.pk] index, and the process distribution need not be normal nor be known.

The [C.sub.pk] index and its sub-indices

The [C.sub.pk] index is defined as

[C.sub.pk] = minimum{[C.sub.pl], [C.sub.pu]} (1)

where [C.sub.pl] and [C.sub.pu] are the sub-indices of [C.sub.pk] such that

[C.sub.pl] = [Mu] - LSL / 3[Sigma] (2)

and

[C.sub.pu] = USL - [MU] / 3[Sigma], (3)

where the [Mu] and [Sigma] are the mean ([Mu] = E[x]) and standard deviation ([Sigma] = [square root of E[[(X - [Mu]).sup.2]])] of the quality characteristic X of interest, and LSL and USL are the lower and upper specification limits respectively These are widely accepted definitions and commonly used; for example, see Juran and Gryna (1988), Montgomery (1991) and Ryan (1989).

Definition (1) indicates the use of the value of [C.sub.pl] or [C.sub.pu], whichever is smaller, for [C.sub.pk]. If the value of the C is larger than some number (for instance, 1.00 conventionally, or 1.33), then the process is judged to be meeting the specifications. Otherwise, it is judged to be not meeting the specifications. That is, this [C.sub.pk] index indicates whether the production process in question is actually meeting its given specifications or not, as opposed to whether it would be capable of meeting its specification or not. This is the reason why the [C.sub.pk] index is sometimes referred to as the "performance index".

Confidence intervals for [C.sub.pk]

Developed and presented in this section are estimators and confidence intervals for [C.sub.pu], [C.sub.pl] and [C.sub.pk]. It is assumed that the process quality characteristic of interest X has the fourth moment (that is, E[[X.sup.4]] [less than] …