The C pk index for asymmetric tolerances

To account for the asymmetry, Kane (1986) seems to be the first to modify Cp , Cpk , Cpu and Cpt indices for the situations

PCIs with Asymmetric Specification Limits 183

where USL — T ^ T — LSL . These modifications are denoted by Cp , C*pk , Cpu and Gv\, and define to be

Cp = min Cpk -

USL-T T - LSL 3a ' 3 a

= min[C*pu,C*pi } ,

(10.1) (10.2) where

r —

^USL-T _3a

respectively.

1 - I V-T\ ) USL-T ^{> Cpi —}

T -LSL 3a

\H-T\

T -LSL,

Observe that the index C^p represents the relative size of the smaller (one-sided) semi-tolerance, rather than the relative size of the entire specification range involved in Cp . The semi-tolerances are, by definition, the distances between the target and the specification limits. In these cases, the potential of a process is interpreted in terms of the ability of the distribution's half-width to fit within the (smaller) semi-tolerance. These indices attain their maximum at [i = T . For example, the upper limits for Cpt

a n d o pU are (T — LSL) / 3a and (USL — T) / 3a , respectively, while the original indices Cpu and Cpi do not possess upper bounds. The values (T - LSL) / 3a and (USL - T) / 3a are the arguments of the min function in Cp (equation (10.1)). As the mean deviates from the target, the value of Cp decreases linearly.

When the mean shifts by a distance equal to or exceeding the corresponding semi-tolerance, the index value reduces to zero.

Consequently C*pt = 0 if | ji - T \ > T - LSL , while C*pu = 0 if | /i — T | > USL — T . These properties can be used for an one- sided specification limit with a target value. The index C ^ is also defined to be zero for | /x — T | > T — LSL or | \i - T \ >

USL — T and can be represented as:

184 Encyclopedia and Handbook of Process Capability Indices

Cpk — Cp (1 — k J ,

where k* =\fi-T |/min{T - LSL,USL - T} . Therefore C*pk has a maximum value equal to Cp , which is attained when fj, = T corresponding to k = 0 . Also, C*pk will have zero value whenever the mean is shifted away from the target by an amount equal to or exceeding the smaller semi-tolerance (i.e. Cpk = 0 for k > 1 ).

Recall that the index Cpk vanishes only when the mean is located at or outside the specification limits.

The initial generalization developed for processes with asymmetric tolerances simply shifts one of the two specification limits, so that the new (shifted) specification limits become symmetric with respect to the target value (Kane (1986); Chan et al. (1988)). In other words, this generalization replaces the original specification limits (T — DhT + Du) by the new symmetric limits (on occasion admittedly unjustified) T ± d , where d =

min{Du,D,} , Du =USL - T , Dl =T - LSL , in order to apply the standard definition of Cpk . The generalization may be rewritten as:

c;

^k

=

^d

*~

^l

£

^a

~

^Tl

. (io.3)

Evidently, if Du = Dt , the specification tolerance becomes symmetric and the generalization as expressed by equation (10.3) reduces to the original Cpk defined in (3.1). Boyles (1994) notes that this generalization can underestimate the process capability by restricting the process to a proper subset of the actual specification range. Indeed consider a process with n = T — d / 2 and <T = d / 3 , where the target value is T = (3USL + LSL) j4 . For this process, we have Cpk = 0 . However, the expected proportion nonconforming is approximately 0.27%. Therefore, the index C*pk does understate the capability of the process in this case. An example with USL = 50 , LSL = 10 , T = 40 ,

PCIs with Asymmetric Specification Limits 185

l_i = T - d/2 = 30, and a = d/3 = 2 0 / 3 is depicted in Figure 10.2.

Figure 10.2. An example for USL = 50, LSL = 10 , T = 40 , fi = 30 , ]and cr = 2 0 / 3 .

An alternative generalization suggested for processes with asymmetric tolerances shifts both specification limits to arrive at one that is symmetric (Franklin and Wasserman (1992), Kushler and Hurley (1992)). This modification replaces the original specification limits (T — DhT + Du) with the new symmetric limits T ± (D[ + Du)/2 (which, as in the previous situation, may sometimes be unjustified), and next applies the standard definition of Cpk . For this generalization, the index Cpfc (defined in equation

(3.1)) can be rewritten as:

CL _pk d'-\n-T

3a ^(10.4)

where d' = (Dt + Du) / 2. Evidently if Du = Dt , the specification tolerance becomes symmetric and C'pk defined in equation (10.4) reduces to Cp£ . Boyles (1994) also observes that this approach can

186 Encyclopedia and Handbook of Process Capability Indices

either understate or overstate the process capability, depending on the position of /x relative to T . Indeed, consider two processes A and B with fiA = T - d , aA = d/6 , fiB = T - 3d / 4 ,

aB=d/U and T = (3USL + LSL)/4 . For the process A we have A Cpk = 0 . While the expected proportion nonconforming is approximately 0.135%. Consequently, in this case C'pk understates the capability of the process A. In contrast, it is straightforward to be verity that for the process B, BC'pk = 1. Here however, the expected proportion nonconforming is approximately 99.865%

rather than (See Fig 10.3). Obviously g C ^ overstates the process capability.

To remedy the situation, Pearn and Chen (1998) proposed the index Cpk , - another generalization of Cpk - for processes with asymmetric tolerances. The motivation for the new index Cpk is based on the general criteria stipulated by Boyles (1994), Choi and Owen (1990) and Pearn et al. (1992) when analyzing and comparing the existing capability indices dealing with (a) process yield; (b) process centering; (c) other process characteristics. The generalization Cpk (the Pearn-Chen index) is defined formally as

d* — A*

c;

^k

=

^L

-^-, (io-5)

where A* = m a x ^ V - T)/Du, d*(T - n)/Dt} , DU=USL-T , Dt =T - LSL, d* = min{A,,I>,} . Observe that d*(/z - T)/Du =

\mm{Du,Dl}(n-T)]/(USL-T). Obviously, if T = M (a symmetric tolerance), then d = Du = Dt = d , A =\ fj, — M \ and Cpk reduces to the original index Cpfc . We note that the index Cpk attains the maximal values at fj, = T , regardless of whether the preset specification tolerances are symmetric or not.

Table 10.1 provides numerical values of Cpk , C'pk and Cpk for several selected values of the "parameters": T, USL, LSL, ji and

PCIs with Asymmetric Specification Limits 187

a. Pearn and Chen (1998) provide a thorough comparison among the three indices, Cpk , C'pk and Cpk .

Table 10.1. Numerical values of Cpk , C'pk and Cpk for various values of ix and fixed a = 1 0 / 3 , with {USL, T, LSL) = (10, 40, 50).

V 10 13 16 19 22 25 28 31 34 37 41 42 43 44 45 46 47 48 49 50

Cpk 0.000 0.300 0.600 0.900 1.200 1.500 1.800 1.900 1.600 1.300 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000

Cpk 0.000 0.000 0.000 0.000 0.200 0.500 0.800 1.100 1.400 1.700 1.900 1.800 1.700 1.600 1.500 1.400 1.300 1.200 1.100 1.000

Cpk 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000

For processes with asymmetric tolerances, the corresponding loss function is also asymmetric with respect to T , which can be defined as

Loss(x) —

[(T - x)/(T - LSL)]2, LSL <x<T, [(x - T)/(USL -T)f, T < x < USL,

1 otherwise.