Page 123 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

Char Count= 0
August 31, 2006
2:57
JWBK119-09
108 Computing Process Capability Indices for Nonnormal Data
and C pk , are defined as follows:
specification width USL − LSL
C p = = , (9.1)
process width 6σ

USL − μ μ − LSL

C pk = min C pu , C pl = min , , (9.2)
3σ 3σ
where USL and LSL denote the upper and lower specification limits, respectively. 1
2
Since the process mean μ and the process variance σ are unknown, they are usually
2
estimated from the sample statistics ¯x and s .
2
English and Taylor examined the effect of the nonnormality assumption on PCIs
and concluded that C pk is more sensitive to departures from normality than C p . Kotz
3
and Johnson provided a survey of work on the properties of PCIs and their esti-
mators when the distribution is nonnormal. Among them are Clements’ method, 4
6
5
the Johnson--Kotz--Pearn method, and ‘distribution-free’ PCIs. A new index C s ,
7
proposed by Wright, incorporates an additional skewness correction factor in the
5
8
denominator of the C pmk index developed by Johnson et al. Choi and Bai proposed
a heuristic weighted variance method to adjust the value of PCIs according to the de-
gree of skewness by considering the standard deviations above and below the process
mean separately.
Several authors have proposed a new generation of PCIs based on assumptions
9
about the underlying population. Pearn and Kotz based their new PCIs on a Pear-
5
sonian distribution. Johnson et al. suggested replacing 6σ in the denominator of
equation (9.1) by 6θ, where θ is chosen so that the “capability” is not greatly affected
10
by the shape of the distribution. Castagliola introduced a nonnormal PCI calculation
method by estimating the proportion of nonconforming items using Burr’s distribu-
11
tion. V¨annman proposed a new family of indices C p (u, v), parameterized by (u,v),
which includes many other indices as special cases. Deleryd 12 investigated suitable
values for u and v when the process distribution is skewed. C p (1,1), which is equiva-
lent to C pmk , is recommended as being most suited to handling nonnormality in PCIs.
While it is well known that C p and C pk are not indicative of the process capability for
nonnormal process characteristics and that various methods are available to compute
some surrogate PCIs, there is a lack of a comprehensive performance comparison
of these methods. In particular, practitioners are most interested in knowing which
methods will give a PCI that can be compared, on the same scale, with C pk , and how
well the various methods perform under slight, moderate and severe departures from
normality. In this chapter we examine the performance of seven different methods in
reflecting the true capability of the process by comparing the C pu value generated via
simulations with a target C pu value.

9.2 SURROGATE PCIs FOR NONNORMAL DATA

In this section a summary of the seven different methods included for comparison in
this chapter is presented.