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Capability indices for processes with asymmetric
tolerances
Kuen‐Suan Chen a & Wen‐Lee Pearn b
a Department of Industrial Engineering and Management , National Chin‐Yi Institute of Technology , Taichung, Taiwan 411, R.O.C.
b
Department of Industrial Engineering and Management , National Chiao Tung University , Hsinchu, Taiwan 300, R.O.C.
Published online: 03 Mar 2011.
To cite this article: Kuen‐Suan Chen & Wen‐Lee Pearn (2001) Capability indices for processes with asymmetric tolerances,
Journal of the Chinese Institute of Engineers, 24:5, 559-568, DOI: 10.1080/02533839.2001.9670652 To link to this article: http://dx.doi.org/10.1080/02533839.2001.9670652
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Journal of the Chinese Institute of Engineers, Vol. 24, No. 5, pp. 559-568 (2001) 559
CAPABILITY INDICES FOR PROCESSES WITH ASYMMETRIC
TOLERANCES
Kuen-Suan Chen*
Department of Industrial Engineering and Management National Chin-Yi Institute of Technology
Taichung, Taiwan 411, R.O.C. Wen-Lee Pearn
Department of Industrial Engineering and Management National Chiao Tung University
Hsinchu, Taiwan 300, R.O.C.
Key Words: process capability indices, process yield, process centering,
target value.
ABSTRACT
Process capability indices (PCIs) for processes with symmetric tolerances have received substantial research attention. But, PCIs for processes with asymmetric tolerances have been comparatively neglected. Recently, Boyles (1994) reviewed the existing PCI litera-ture and proposed several new indices to handle processes with asym-metric tolerances. In this paper we analyze PCIs based on various pro-cess characteristics, then introduce a new class of capability indices to handle processes with asymmetric tolerances. The proposed new indi-ces are compared with existing PCIs in terms of proindi-cess yield, proindi-cess centering, and process characteristic related to loss functions. The re-sults indicate that the new indices are superior to the existing capabil-ity indices, and provide greater accuracy in current applications using PCIs to measure process potential and performance.
*Correspondence addressee
I. INTRODUCTION
Process capability indices (PCIs), whose pur-pose is to provide a numerical measure on whether a production process is capable of producing items meeting the quality requirement preset by the customers, have received substantial attention in the quality control and statistical literature. Examples include Kane (1986), Chan, Cheng and Spiring (1988), Choi and Owen (1990), Boyles (1991), Pearn, Kotz and Johnson (1992), Franklin and Wasserman (1992), Johnson (1992), Kushler and Hurley (1992), Boyles (1994), Vannman (1995), Pearn and Chen
(1996), and many others. Most research work, however, has focused on developing and investigat-ing PCIs for processes with symmetric tolerances. A process is said to have a symmetric tolerance if the target value T is the midpoint of the specification in-terval (LSL, USL). That is, T=M=(USL+LSL)/2, where
USL and LSL are the upper and the lower
specifica-tion limits.
For processes with symmetric tolerances, sev-eral capability indices have been proposed to provide u n i t l e s s m e a s u r e s o f p r o c e s s p o t e n t i a l a n d performance. These include Cp, Cpk, Cpm, and Cpmk
(see Kane (1986), Chan, Cheng and Spiring (1988),
and Pearn, Kotz and Johnson (1992)). A superstruc-ture containing these four basic indices may be written as (see Vännman (1995)):
Cp(u, v) =
d – u µ– M
3 σ2+ v(µ– T)2 (1)
Where µ is the process mean, σ is the process stan-dard deviation, d=(USL−LSL)/2, M=(USL+LSL)/2, T
is the target value, and u, v≥0. It is easy to verify that Cp(0, 0)=Cp, Cp(1, 0) =Cpk, Cp(0, 1)=Cpm, and Cp(1, 1)=Cpmk.
As noted by Boyles (1991), Cp and Cpk are
yield-based indices which are independent of the target T, which may fail to account for process centering (the ability to cluster around the target) with symmetric tolerances, but have an even greater problem with asymmetric tolerances: process yield is maximized (for fixed σ) by µ=M, but T≠M. In this case, process
yield and centering are conflicting criteria. For Cpm,
Pearn, Kotz and Johnson (1993) considered the fol-lowing example (see Fig. 1) with asymmetric toler-ance (LSL, T, USL), where T={3(USL)+(LSL)}/4, and σ=d/3. Then, for processes A and B with µA=T− d/2=M and µB=T+d/2=USL both have the index value
of Cpm=0.555 and equal degrees of clustering around
the target (as |µ−T|=d/2 for both processes A and B). However, the expected proportions nonconforming are approximately 0.27% for process A and 50% for process B. Clearly, Cpm inconsistently measures
pro-cess capability in this case and is inappropriate for asymmetric cases. These problems call for a need to generalize the four basic indices to cover cases with asymmetric tolerances so that positive use of PCIs can be continued.
II. EXISTING PCIS′ FOR ASYMMETRIC TOLERANCES
There are several generalizations of Eq. (1) pro-p o s e d t o h a n d l e pro-p r o c e s s e s w i t h a s y m m e t r i c tolerances, which overcome some problems of Cpk
and Cpm. The first generalization proposed for
pro-cesses with asymmetric tolerances shifts one of the two specification limits, so that the new (shifted)
specification limits are symmetric to the target value (see Kane (1986), and Chan, Cheng and Spiring (1988)). That is, the generalization replaces the true specification limits (T−Dl, T+Du) with the new
sym-metric limits (unjustified sometimes) T±d*
, where d*
= min{Dl, Dm}, Du=USL−T and Dl=T−LSL, then applies
the standard definitions of Cp, Cpk, Cpm, and Cpmk.
With this generalization, the indices defined in (1) can be rewritten as the following:
Cp * (u, v) = d * – u µ– T 3 σ2+ v(µ– T)2 (2)
This approach yields the following generalized indices Cp*, Cpk* , Cpm* , and Cpmk* . Unfortunately, these generalized indices can understate process capabil-ity by restricting the process to a proper subset of the actual specification range, as observed by Boyles (1994). For example, consider a process with mean µ=T−d/2=M, and standard deviation σ=d/3, where the target value T = {3(USL) + (LSL)}/4 (see Fig. 2). Then, we have Cpk
*
=Cpmk
*
=0. The expected propor-tions nonconforming, however, are approximately 0.27%. Both indices Cpk
*
and Cpmk
*
severely under-state process capability in this case. It is clear that if
Du= Dl, then the production tolerance becomes
sym-metric and the generalized indices defined in (2) re-duce to those basic ones defined in (1).
Another generalization proposed for processes with asymmetric tolerances shifts both specification limits (see Fig. 3) to obtain one that is symmetric (Kushler and Hurley (1992), and Franklin and Wasserman (1992)). That is, the generalization re-places the true specification limits (T−Dl, T+Du) with
the new symmetric limits (unjustified sometimes)
T±(Dl+Du)/2, then applies the standard definitions of Cp, Cpk, Cpm, and Cpmk. With this generalization, the
indices defined in (1) can be rewritten as the following:
Cp′(u, v) =
d – u µ– T
3 σ2+ v(µ– T)2 (3)
This approach yields the generalized indices Cp′, Fig. 1 Process A and B both have Cpm=0.55. But the expected
proportions non-conforming are 0.27% for A and 50% for B
Fig. 2 The process has index values Cpk*=Cpmk* =0. But the
ex-pected proportion non-comforming is no greater than 0.27%
K.S. Chen & W.L. Pearn: Capability Indices for Processes with Asymmetric Tolerances 561
Cpk′ , Cpm′ , and Cpmk′ , which can either under-or
over-state process capability, depending on the position of µ relative to T as noted by Boyles (1994). For example, consider the following two processes with µA=T−d, µB=T+d/2=USL, σ=d/6, and T={3(USL)+
(LSL)}/4 (see Fig. 4). Then, for process A we have
Cpk′ =Cpmk′ =0 , and for process B we have Cpk′ =1.0 and Cpmk′ =0.32. For both indices Cpk′ and Cpmk′ , the
val-ues given to process B are higher than those given to process A. But, the expected proportion nonconform-ing, of process B, is approximately 50%, which is sig-nificantly greater than that (approximately 0.135%) of process A. Obviously, both indices Cpk′ and Cpmk′
understate or overstate process capability in this case. We note that if Du=Dl. Then the specification
toler-ance becomes symmetric and the generalized indices defined in (3) reduce to those basic ones defined in (1).
To overcome the problems with asymmetric tolerances, Boyles (1994) Defined a smooth function
S(x, y)=Φ−1
{Φ(x)/2+Φ(y)/2}/3, where Φ(x) is the cu-mulative function of the standard Normal distribution. Based on this smooth function, Boyles (1994) con-sidered a new index Spk generalized from Cpk. The
index is defined as Spk=S((USL−µ)/σ, (µ−LSL)/σ. We
note that given Spk=c, we can calculate the process
yield as % yield=Φ((USL−µ)/σ)−Φ((LSL−µ)/σ)=2Φ (3c)−1 for arbitrary values of c. Therefore, Spk
rep-resents the actual process yield unlike Cpk which is
only approximately related to process yield (Boyles (1994)). Extending this generalization to the index
Cpmk, we obtain Spmk=S((USL−µ)/τ, (µ−LSL)/τ), where
τ=[σ2
+(µ−T)2
]−1/2
(see Boyles (1994)). A superstruc-ture for this generalization may be written as the following: Sp(v) = S( USL –µ) σ2+ v(µ– T)2 , µ– LSL σ2+ v(µ– T)2) (4)
Where v≥0. It is easy to verify that Sp(0)=Spk, and Sp(1)=Spmk.
In a recent paper, Johnson, Kotz and Pearn (1994) proposed a flexible Capability index called
Cjkp to handle non-normal populations. Since the
in-dex Cjkp can be expressed as (3 2 )−1 min {Du/τu,
Dl/τl}, where (τu)2=σ2{(1−Φ(ζ))(1+ζ2)−ζφ(ζ)},
(τl)2=σ2{Φ(ζ)(1+ζ2)+ζφ(ζ)} with φ(•) representing the
density function of the standard Normal distribution and ζ=(µ−T)/σ, the corresponding Boyles’ generali-zation for the Cjkp index then becomes Sjkp=Φ−1{Φ(Du/
2τu)/2+Φ(Dl/ 2τl)/2}/3. In addition to the above
generalizations, Boyles (1994) also considered the following two indices: Cm
✩
={3(λ1)1/2}−1, and Cpm
+
= {3(λr)1/2}−1 for asymmetric tolerances, where λ1=
(τu/Du)2+(τu/Du)2, λr=2λ1/(1+min{(r)2, (r)−2}), and
r=Dl/Du. It is clear that if r=1 (or equivalently, Du=Dl), then we have λr=λ1, and both generalizations
Cm
✩
and Cpm+ reduce to the basic index Cpm.
Boyles (1994) analyzed these six capability indices, Cpmk, Spmk, Cjpk, Sjpk, Cm
✩
, and Cpm+ , and pro-vided a comparison in order to assess their accuracy in measuring process potential and performance. The comparison is based on several process characteris-tics including (a) process yield, and (b) process centering. Boyles (1994) pointed out that:
(A) Cpmk and Spmk are superior to the other four
indices in terms of process yield. Spmk is
closely reated to actual process yield, while
Cpmk is only related to approximate process
yield. Thus, Cpmk may be viewed as an
ap-proximation to Spmk;
(B)Cm
✩
, Cjpk and Sjpk, provide no protection at all
with respect to process yield, and therefore should not be considered further;
(C) Spmk guarantees levels of process yield
con-ventionally associated with given index levels,
c, across all r valued, while Cpm+ provides such guarantees only for ranges of (r, c); Cpm+ places bounds on µ proportional to tolerance, while the bounds given to Spmk are disproportionately
sharp on the “long” side of the specification ; Boyles (1994) concluded that Spmk is well-calibrated
with respect to process yield, and is most appropriate for general use.
We point out, however, that for fixed standard deviation σ, these six indices (including the most ap-propriate index Spmk) obtain their maximal values Fig. 3 Cp′(u, v) shifts the given specification limits (LSL, USL)
to the new specification limits (LSL, USL) which are sym-metric to the target T
Fig. 4 Process A has Cpk′ =0, and Cpmk′ =0. Process B has Cpk′ =1,
and Cpm′ =0.32. But, the expected proportions
non-con-forming are 0.135% for A, and 50% for B
not at µ=T, but at some µ*
which is between the target value T and M=(USL+LSL)/2 (see Fig. 5). The value of µ*
relative to T and M reflects the compromise established by each of the six indices be-t w e e n p r o c e s s c e n be-t e r i n g a n d p r o c e s s y i e l d . Consequently, these six indices may conflict or show inconsistent results in terms of process yield, process centering, and other process characteristics, and thus, reflect process capability inaccurately. For example, consider the following case with asymmetric toler-ance (LSL, T, USL)=(26, 50, 58). Assume we have t w o p r o c e s s e s A a n d B w i t h µA= 4 9 , µB= 5 0
respectively, and standard deviation σA=σB=5.33. It
is easy to verify that the index values of Cpmk, Spmk, Cjpk, Sjpk, Cm
✩
, and Cpm+ for process A are higher than those for process B in this case. While process B is on-target, process A is off-target. None of the six indices discussed in Boyles (1994) reflect process ca-pability accurately enough in this case.
III. NEW PCIS′ FOR ASYMMETRIC TOLERANCES
In this section, we consider a new class of gen-eralized capability indices. The design of the new PCIs is based on the following criteria used in Chio and Owen (1990), Pearn, Kotz and Johnson (1992), and Boyles (1994) in analyzing and comparing the existing capability indices: (a) process yield, (b) pro-cess centering, and (c) a propro-cess characteristic related to loss functions. The new indices may be defined as:
Cp′′(u, v) =
d*– uF*
3 σ2+ vF2 (5)
Where F=max{d(µ−T)/Du, d(µ−T)/Dl}, F*=max
{d*
(µ−T)/Du, d(µ−T)/Dl}, and u, v≥0. This
generali-zation yields the following new indices Cp′′(0, 0)=Cp′′, Cp′′(1, 0)=Cpk′′ , Cp′′(0, 1)=Cpm′′ , and Cp′′(1, 1)=Cpmk′′ . We
note that if T=M (tolerance is symmetric), then F=
F*
=|µ−T| and generalized indices Cp′′(u, v) reduce to
the basic indices Cp(u, v) defined in Eq. (1). Further,
if µ=T (process is on target), then Cpmk′′ = Cpm′′ = Cpk′′ =Cp′′=d*/3σ. But, in general, the relationships
among the four new indices Cp′′, Cpk′′ , Cpm′′ , and Cpmk′′
can be established as the following:
Cpk′′ = Cp′′(1 – k) , Cpm′′ = Cp′′(1 + dK/σ] 2 )1/2, Cpmk′′ = Cp′′(1 – k)(1 + [dK/σ] 2 )1/2,
Where K=max{(µ−T)/Du, d(T−µ)/Dl}=F*/d*. Thus,
in developing the new indices we have replaced |µ−T| with F*
, and (µ−T)2
with F2
in (2). This en-sures that the new indices Cp′′(u, v) obtain the
maxi-mal values at µ=T regardless of whether the toler-ances are symmetric or asymmetric.
For processes with asymmetric tolerances, the corresponding loss function is also asymmetric to T. We take into account the asymmetry of the loss func-tion by adding the factors d*
/Du and -d*/Du to µ−T
according to whether µ is greater, or less, than T. The factors d*
/Du and -d*/Dl ensures that if processes A
and B with µA>T and µB<T satisfy (µA−T)/Du=(T−µB)/ Dl, then the index values given to A and B are the
same (Fig. 6). It is easy to verify that if the process is on the specification limits (µ=LSL, or µ=USL), then
Cpk′′ =Cpmk′′ = 0. On the other hand, if LSL<µ<USL, then
we have Cp′′(u, v)>0.
In Figs. 7(A), 7(B), 7(C), we plot contours of
Cp′′(u, v) (dashed) and Spk (solid) for the standard
in-dex values; 1/3, 2/3, 1, 4/3, 5/3, and, 2, with Fig. 7(A) for Cpk′′ , Fig. 7(B) for Cpm′′ , and Fig. 7(C) for Cpmk′′ . In all three cases, we have Cp′′(u, v)<Spk for all
values of µ. Thus, given a process with Cp′′(u, v)=c
which is 2{1−Φ(3c)}. Further, given Cp′′(u, v)>c, we
can calculate the bounds on |µ−T| as:
T – (1 – R)Dl
3c v + u(1 – R)<µ< T +
(1 – R)Du
3c v + u(1 – R), Where R=|1−r|/(1+r), and r=Dl/Du. Therefore, the
bounds on |µ−T| corresponding to Cpk ′′
>c would be
T−Dl<µ<T+Du (equivalently, LSL<µ<USL). The
bounds on |µ−T| corresponding to Cpm ′′ >c would be T−{(1−R)/3c}Dl<µ<T+{(1−R)/3c}Du, and the bounds on |µ−T| corresponding to Cpmk ′′ >c would be
Fig. 5 For fixed σ, the existing indices obtain maximal values not at T, but at some µ* that is between the target T and
midpoint m
Fig. 6 For fixed σ, the new indices obtain maximal values at T, and give same index values to processes A, and B, satisfy-ing (µA−T)/Du=(T−µB)/Dl
K.S. Chen & W.L. Pearn: Capability Indices for Processes with Asymmetric Tolerances 563
T−{(1−R)/(3c+ 1−R)}Dl<µ<T+{(1−R)/(3c+1−R)}Du.
IV. COMPARISONS
In this section, we compare the new generaliza-tions, Cp′′(u, v), with the existing generalizations
described in section 2. We note that Boyles (1994) has provided a comparison among the existing indices, and made the conclusion that Spmk is the most
appropriate index for general use. We will provide the same comparison which is based on the criteria used by Boyles (1994) including (1) process yield, (2) process centering, adding another criterion (3) a process characteristic (relationships to loss function) considered by Choi and Owen (1990). We first focus on the relationships to the yield-based index Spk, and
second on process centering (the ability to cluster around the target) in the form of bounds placed on |µ−T|, and last on a process characteristic related to loss functions.
1. Process Yield
Inspection of the contour plots in section 3 of Boyles (1994) reveals that with asymmetric tolerances the existing indices are maximized (for fixed σ) not by µ=T, but by a value µ*
between T and M (see Figs. 4(b), 5(b), 6(b), 6(c), 7(b), and 8(b) in Boyles (1994)), as we indicated earlier. On the other hand, we ob-served that in Figs. 7(A), 7(B), 7(C), the proposed generalizations Cp′′(u, v) are maximal (for fixed σ) by
µ*
=T which occurs when the contours of Cp′′(u, v)
reach their maximal height at σ=σ*
. Assume the pro-cess corresponding to µ=µ*
and σ=σ*
has Spk value
denoted as Spk
*
, a function of r and c, where r=Dl/Du.
Since Cp′′(u, v)<Spk, (the contours of Cp′′(u, v) are
undercovered by Spk contours for the same level c),
we conclude that if Cp′′(u, v)=c, then the process yield
must be no less than that corresponding to Spk=c. It
can be easily seen that the condition Cp′′(u, v) with
µ*
=T implies σ*
=(d−|M−T|)/3c.
For the six indices, Cpmk, Spmk, Cjkp, Sjkp, Cpm
✩ , and Cpm
+
discussed in Boyles (1994), a comparison based on process yield in terms of Spk for a continuum Fig. 7 (A) Contours of Cpk′′ (dashed) and Spk (solid) for the
stan-dard values 1/3, 2/3, 1, 4/3, 5/3, and 2 (top to bottom in plot). (B) Contours of Cpm′′ (dashed) and Spk (solid) for the
standard values 1/3, 2/3, 1, 4/3, 5/3, and 2 ( top to bottom in plot). (C) Contours of Cpmk′′ (dashed) and Spk (solid) for
the standard values 1/3, 2/3, 1, 4/3, 5/3, and 2 (top to bot-tom in plot)
Fig. 8 Process yield in terms of Spk for a continuum of processes
with constant values c=1 and c=5/3 for the four indices shown. All the curves are symmetric about r=Dl/Du=1 on
a logarithmic scale (curves labeled ‘1’ for Cpk′′, Cpm′′ , Cpmk′′ ,
and curves labeled ‘2’ for Cpmk)
of processes with constant values c=1 and c=5/3 is provided (see Fig. 9 in Boyles (1994)). Boyles (1994) noted that only Cpmk and Spmk can assure that the
pro-cess yield is at or above nominal index levels for all values of r. In Fig. 8, we plot Spk
*
curves for the new generalizations Cp′′(u, v) and Cpmk at index levels c=1 and c=5/3. It can be seen that the Spk
*
curves for
Cp′′(u, v) are bounded by the straight line c and the Spk
*
curve for Cpmk. We note that for any level c the Spk
*
curve for Spmk is also bounded by the straight line c and the Spk
*
curve for Cpmk although it is omitted
from Fig. 8 (see also Fig. 9 in Boyles (1994)). Thus,
Cp′′(u, v) (except for (u, v)=(0, 0)) guarantees process
yield at or above nominal index levels for all values of r (like Cpmk and Spmk).
2. Process Centering
Process centering is defined as the ability of the process to cluster around the target value T. In most
cases, process centering can be measured by the de-parture of process mean µ from the target value T, |µ−T|. If we impose the condition that the index value is no less than a given level c, then we can calculate the bounds on |µ−T| for the existing indices as well as the new generalizations Cp′′(u, v), which can be
ex-pressed in the form:
T−klDl<µ<T+kuDu
For unitless functions (kl, ku) of r and c. The (kl, ku)
values for the six indices discussed by Boyles (1994) as well as the proposed new generalizations are dis-played in Table 1, where c′=Φ−1
{2Φ(3c)−1}/3<c. From Table 1, we can see that:
(a) The bounds for Spmk and Sjkp are very close
to but slightly greater than those for Cpmk and Cjkp respectively. Thus, Cpmk and Cikp are
su-perior to Spmk and Sjkp in terms of process
centering.
(b) The bounds for Cpmk and Cjkp are tighter than
those for Cpm
*
and Cpm+ . Thus, Cpmk and Cjkp
are superior to Cpm
*
and Cpm+ in terms of pro-cess centering.
(c) Since f (r)/(3c+f(r))<1, and f(r)/(3c+f(r))<
f(r)/3c, the bounds for Cpmk′′ are tighter than
those for Cpk′′ and Cpm′′ .
Therefore, in Fig. 9 we only plot (-kl, ku) curves
(as a function of r) for indices Cpmk′′ , Cpmk and Cjkp with c=4/3. In Fig. 9, we note that the bound for Cpmk′′ (curves labeled “1”) is significantly tighter than
that of Cpmk (curves labeled “2”) for all values of r.
The bound for Cpmk′′ is also tighter than that of Cjkp (curves labeled “3”) except for r≅1. Therefore, Cpmk′′ is considered to be superior to Cpmk and Cjkp
(and hence superior to Spmk) in terms of process
centering.
Fig. 9 Bounds placed on |µ−T| for c=4/3 by the three indices shown. Positive values represent fractions of Du,
nega-tive values represent fractions of Dl, and 0 represents T
(curves labeled ‘1’ for Cpmk′′ , curves labeled ‘2’ for Cpmk,
and curves labeled ‘3’ for Cjpk
Table 1 Constants for bounds on |m−T| implied by various indices
kl ku Cpk′′ 1 1 Cpm′′ f(r)/3c f(r)/3c Cpmk′′ f(r)/(3c+f(r)) f(r)/(3c+f(r)) Cpmk min{1/(3c+1), 1/r(3c−1)} min{1/(3c−1), 1/r(3c+1)} Spmk min{1/(3c′+1), 1/r(3c′−1)} min{1/(3c′−1), 1/r(3c′+1)} Cpm ✩ 1/(3c) 1/(3c) Cpm + 1/{3c[A(r)]1/2 } 1/{3c[A(r)]1/2 } Cjkp 1/{3c(2)1/2} 1/{3c(2)1/2} Sjkp 1/{3c′(2)1/2} 1/{3c′(2)1/2} Note: f(r)=1−|1−r|/(1+r)>0, where r=Dl/Du, A(r)=2/(1+min{(r)2 , (r)−2 }), and c′=Φ−1 {2Φ(3c)−1}/3
K.S. Chen & W.L. Pearn: Capability Indices for Processes with Asymmetric Tolerances 565
3. A process Characteristic Related to Loss Func-tions
I n t h e f o l l o w i n g , w e c o m p a r e t h e n e w generalizations Cp
′′
(u, v) with the six indices discussed in Boyles (1994) based on a process characteristic dis-cussed in Choi and Owen (1990), which is related to loss functions. As we discussed earlier, the new in-dices Cp
′′
(u, v) obtain the maximal values when the process is on-target (µ*
=T). On the other hand, the six indices Cpmk, Spmk, Cjkp, Sjkp, Cpm
✩
, and Cpm+ obtain the maximal values when the process is off- target (M<µ*
<T). To illustrate this point, we consider the following example with specifications (LSL, T, USL) =(26, 50, 58). Since Du=USL−T=8, and Dl=T− LST=24, the process has an asymmetric tolerance.
Table 2 displays the values of the six indices
discussed in Boyles (1994) as well as the proposed new indices Cpk
′′
, Cpm ′′
, Cpmk′′ for various values of µ,
with fixed standard deviation σ=8/3. We note that in Table 2, Cpmk, Spmk are maximized by µ*=49, and the
other four indices Cpmk, Cpm
+
, Cjkp, and Sjkp are
maxi-mized by µ*
=48. In all cases, we have M<µ*
<T. On the other hand, the new generalizations Cp
′′
(u, v) are maximized by µ*
=50=T, and the index values are 1.00 for all three new indices Cpk
′′ , Cpm ′′ , and Cpmk ′′ . Further, the new indices have taken into account the asymmetry of the loss function. Thus, given two processes A and B with µA>T and µB<T satisfying
(µA−T)/Du=(T−µB)/Dl, the (new) index values given
to A and B are the same. Table 3 is a summary of processes (taken from Table 2) satisfying (µA−T)/Du=
Table 2 A comparison among the new indices and existing ones for various of µ and fixed σ=8/3, (LSL, T,
USL)=(26, 50, 58) µ Cpk′′ Cpm′′ Cpmk′′ Cpmk Spmk Cpm ✩ Cpm + Cjkp Sjkp 26 0.000 0.164 0.000 0.000 0.178 0.331 0.247 0.234 0.391 27 0.042 0.171 0.007 0.014 0.188 0.346 0.258 0.244 0.399 28 0.083 0.179 0.015 0.030 0.198 0.361 0.269 0.255 0.407 29 0.125 0.187 0.023 0.047 0.210 0.378 0.282 0.267 0.417 30 0.167 0.196 0.033 0.066 0.233 0.396 0.296 0.280 0.427 31 0.208 0.206 0.043 0.087 0.237 0.417 0.311 0.295 0.439 32 0.250 0.217 0.054 0.110 0.253 0.440 0.328 0.311 0.452 33 0.292 0.229 0.067 0.136 0.272 0.465 0.347 0.329 0.466 34 0.333 0.243 0.081 0.164 0.292 0.493 0.368 0.349 0.483 35 0.375 0.258 0.097 0.197 0.316 0.525 0.391 0.371 0.501 36 0.417 0.275 0.114 0.234 0.343 0.261 0.418 0.397 0.523 37 0.458 0.294 0.135 0.276 0.375 0.603 0.449 0.426 0.548 38 0.500 0.316 0.158 0.325 0.412 0.651 0.485 0.460 0.577 39 0.542 0.342 0.185 0.383 0.455 0.707 0.527 0.500 0.611 40 0.583 0.371 0.217 0.451 0.506 0.773 0.576 0.547 0.652 41 0.625 0.406 0.254 0.533 0.565 0.853 0.635 0.603 0.702 42 0.667 0.447 0.298 0.632 0.632 0.950 0.708 0.672 0.764 43 0.708 0.496 0.351 0.667 0.706 1.072 0.799 0.758 0.843 44 0.750 0.555 0.416 0.711 0.778 1.231 0.917 0.872 0.948 45 0.792 0.625 0.495 0.765 0.845 1.443 1.075 1.029 1.096 46 0.833 0.707 0.589 0.832 0.911 1.719 1.282 1.259 1.316 47 0.875 0.800 0.700 0.914 0.987 2.003 1.493 1.612 1.657 48 0.917 0.894 0.820 1.000 1.086 2.052 1.530 1.947 2.010 49 0.958 0.970 0.930 1.053 1.119 1.739 1.296 1.376 1.482 50 1.000 1.000 1.000 1.000 1.068 1.340 1.000 1.000 1.068 51 0.875 0.800 0.700 0.819 0.899 1.033 0.770 0.755 0.840 52 0.750 0.555 0.416 0.600 0.699 0.817 0.609 0.590 0.691 53 0.625 0.406 0.254 0.415 0.538 0.664 0.495 0.476 0.590 54 0.500 0.316 0.158 0.277 0.425 0.553 0.412 0.394 0.520 55 0.375 0.258 0.097 0.176 0.347 0.469 0.350 0.333 0.470 56 0.250 0.217 0.054 0.102 0.292 0.406 0.302 0.287 0.433 57 0.125 0.187 0.023 0.044 0.254 0.356 0.265 0.252 0.404 58 0.000 0.164 0.000 0.000 0.225 0.316 0.236 0.224 0.383
(T−µB)/Dl. For example, consider processes A and B
with µA=51>T, and µB=47<T. Clearly, we have
(µA−T) /Du=1/8, and (T−µB)/Dl=3/24=1/8. Thus,
qual-ity loss for processes A and B are the same. Check-ing Table 3 for the index values correspondCheck-ing to µA=51 and µB=47, we have Cpk
′′
=0.875, Cpm ′′
=0.800, and Cpmk′′ =0.700 for both processes A and B. On the
other hand, the values the other six indices give to process B are considerably higher than those given to process A. In particular, for indices Cpm
✩ , Cpm+ ,
Cjkp, and Sjkp the values given to process B are roughly
twice those given to process A. V. ESTIMATION OF Cp
′′′′
(u, v) To estimate the new indices Cp
′′
(u, v), Pearn and Chen (1995) considered the natural estimators which can be defined as the following:
Cp′′(u, v) = d*– uF* 3 S2+ vF2 , W h e r e F*= max{d*( X – T)/Du, d * (T – X )/Dl} a n d F = max{d( X – T)/Du, d(T – X )/Dl} with X =(
Σ
Xi i = 1 n )/ n, and S = {(n – 1) – 1 (Xi– X ) 2Σ
i = 1 n }1/2, the conventional estimators of µ and σ which may be obtained from a process that is demonstrably stable (in-control).As an example, we consider the following pro-cess with asymmetric specification tolerances
USL=16, T=13.5, and LSL=10. Suppose the sample
mean X =14, and the sample standard deviation S=
1. Then, we can calculate d=(USL−LSL)/2=3, d*
=min {Du, Dl}=min{2.5, 3.5}=2.5, F =max{d(X −T)/Du, d ( T− X ) / D l} = 0 . 6 , a n d F * = m a x { d* (X −T ) / Du, d*
(T−X )/D l}=0.5. Thus, we may obtain Cp ′′ (u, v)= (2.5−0.5u){3(1+0.36v)1/2 }−1 . By setting (u, v)=(0, 0), (1, 0), (0, 1), (1, 1), we obtain Cp ′′ =0.83, Cpk ′′ =0.67, Cpm ′′ =0.71, and Cpmk ′′ =0.57.
Pearn and Chen (1995) investigated the statisti-cal properties of the estimators Cp
′′
(u, v) and obtained the exact distributions of Cp
′′
(u, v) although the deri-vations were cumbersome. Pearn and Chen (1995) also derived the formulas for the exact r-th moment (about zero) of the estimators Cp
′′
(u, v). Expressions of the r-th moment, the expected value, and the vari-ance formulas as well as other inferential properties are as complicated as those which appeared in Vännman and Kotz (1995). Further, Pearn and Chen (1995) showed that in special cases where the speci-fication tolerances are symmetric (Du=Dl), their
re-sults are identical to (reduce to) those obtained by Vännman and Kotz (1995).
VI. CONCLUSIONS
In this paper, we first reviewed the existing gen-eralizations of the basic capability indices Cp(u, v)
including Cp
*
(u, v), Cp′(u, v), Sp(v) and many others,
which have been proposed to handle processes with asymmetric tolerances. Then, we introduced a new class of generalizations which we referred to as
Cp′′(u, v). The new generalizations Cp′′(u, v) are
de-veloped from the basic indices Cp(u, v) by taking into
account the asymmetry of the specification tolerance (loss function).
The proposed new generalizations are compared Table 3 The corresponding index values for processes satisfying (µA−T)/Dµ=(T−µB)/Dl
µ Cpk′′ Cpm′′ Cpmk′′ Cpmk Spmk Cpm ✩ Cpm + Cjkp Sjkp 47 0.875 0.800 0.700 0.914 0.987 2.003 1.493 1.612 1.657 51 0.875 0.800 0.700 0.819 0.899 1.033 0.770 0.755 0.840 44 0.750 0.555 0.416 0.711 0.778 1.231 0.917 0.872 0.948 52 0.750 0.555 0.416 0.600 0.699 0.817 0.609 0.590 0.691 41 0.625 0.406 254 0.533 0.565 0.853 0.635 0.603 0.702 53 0.625 0.406 0.254 0.415 0.538 0.664 0.495 0.476 0.590 38 0.500 0.316 0.158 0.325 0.412 0.651 0.485 0.460 0.577 54 0.500 0.316 0.158 0.277 0.425 0.553 0.412 0.394 0.520 35 0.375 0.258 0.097 0.197 0.316 0.525 0.391 0.371 0.501 55 0.375 0.258 0.097 0.176 0.347 0.469 0.350 0.333 0.470 32 0.250 0.217 0.054 0.110 0.253 0.440 0.328 0.311 0.452 56 0.250 0.217 0.054 0.102 0.292 0.406 0.302 0.287 0.433 29 0.125 0.187 0.023 0.047 0.210 0.378 0.282 0.267 0.417 57 0.125 0.187 0.023 0.044 0.254 0.356 0.265 0.252 0.404 26 0.000 0.164 0.000 0.000 0.178 0.331 0.247 0.234 0.391 58 0.000 0.164 0.000 0.000 0.225 0.316 0.236 0.224 0.383
K.S. Chen & W.L. Pearn: Capability Indices for Processes with Asymmetric Tolerances 567
with existing ones in terms of process yield, process centering (the ability to cluster around the target), and a process characteristic related to loss functions. The results indicate that: (1) the new generalizations, particularly, Cpk′′ , Cpm′′ , and Cpmk′′ guarantee process
yield at or above nominal index levels for all given index values (like Spmk, the index recommended by
Boyles (1994)), (2) Cpmk′′ is superior to Spmk in terms
of process centering, and (3) Cpk′′ , Cpm′′ , and Cpmk′′
ob-tain the maximal values (for fixed σ) at µ*
=T (on-target), while the others (including Spmk) obtain
the maximal values at some µ*
with M<µ*
<T (off-target). In practical application, process engineers can set their machine parameter as target value when
Cp(u, v) is applied to evaluate process capability.
Large Cp(u, v) insures high process yield and small Cp(u, v) indicates the chance of process improvement.
Thus, the proposed new generalizations are superior to existing ones, which provide a greater accuracy in current practice of using PCIs to monitor process po-tential and performance.
ACKNOWLEDGMENT
This work was supported by the National Sci-ence Council of Republic of China under Grant NSC89-2213-E-167-004.
NOMENCLATURE µ the process mean
σ the process standard deviation
USL upper specifications limit
LSL lower specifications limit
d (USL−LSL)/2
M (USL+LSL)/2
T the target value
Du T−LSL
Dl USL−T
d*
min{Dl, Du}
Φ(x) the cumulative function of the standard Nor-mal distribution Spk S((USL−µ)/σ, (µ−LSL)/σ Spmk S((USL−µ)/τ, (µ−LSL)/τ) τ [σ2 +(µ−T)2 ]−1/2 (τu)2 σ2{(1−Φ(ζ))(1+ζ2)−ζφ(ζ)}, (τl)2=σ2{Φ(ζ) (1+ζ2 )+ζφ(ζ)}
φ(•) representing the density function of the stan-dard Normal distribution
(τl)2 σ2{Φ(ζ)(1+ζ2)+ζφ(ζ)} ζ (µ−T)/σ Sjkp Φ−1{Φ(Du/ 2τu)/2+Φ(Dl/ 2τl)/2}/3 Cpm ✩ {3(λ1)1/2}−1 Cpm + {3(λr)1/2}−1 λ1 (τu/Du)2+(τl/Dl)2 λr 2λ1/(1+min{(r)2, (r)−2}) r Dl/Du F max{d(µ−T)/Du, d(T−µ)/Dl} F* max{d* (µ−T)/Du, d*(µ−T)/Dl} K max{(µ−T)/Du, dT(−µ)/Dl}=F*/d* R |1−r|/(1+r) σ* (d−|M−T|)/3c c′ Φ−1 {2Φ(3c)−1}/3 F* max{d* (X −T)/Du, d*(T−X )/D l} F max{d(X −T)/Du, d(T−X )/D l} X (
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!"#$%&'()*+
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Manuscript Received: Sep. 22, 2000 Revision Received: Mar. 16, 2001 and Accepted: Mar. 30, 2001
