Specialized Capability Indices

The CAPABILITY Procedure

Specialized Capability Indices

This section describes new specialized capability indices that are computed when you specify the SPECIALINDICES option on the PROC CAPABILITY statement.

The Index S_jkp

Boyles (1994) proposed a smooth version of C_jkp defined as

$S_{jkp}=S ( \frac{{USL} - T}{ \sqrt{ 2 E_{X\gt T}[(X-T)^2] } } , \frac{T - {LSL}}{ \sqrt{ 2 E_{X\lt T}[(X-T)^2] } } )$

The CAPABILITY procedure estimates S_jkp as

$\hat{S}_{jkp}=S ( \frac{ {USL} - T }{ \sqrt{ 2 \sum_{X_{i} \gt T} (X_{i} - T... ... } } , \frac{ T - {LSL} }{ \sqrt{ 2 \sum_{X_{i} \lt T} (X_{i} - T)^2 / n } } )$

where $S(x,y)=\Phi^{-1}[\{\Phi(x) + \Phi(y)\}/2]/3$ .

The Index C_pp

Chen (1998) devised a process incapability index based on the C_pm^* index. The first term measures inaccuracy and the second measures imprecision. The C_pp index is estimated as

$\hat{C}_{pp}=( \frac{\bar{X} - T}{d^{*} / 3} )^2 + ( \frac{s}{d^{*} / 3} )^2$

where d^* = min( USL - T, T - LSL ).

The Index C_pp^''

The index C_pp does not handle asymmetric tolerances well, as discussed by Kotz and Lovelace (1998). To address that shortcoming, Chen (1998) defined the index C_pp^'', which is estimated by

$\hat{C}_{pp}^{''}=( \frac{\hat{A}}{d^{*} / 3} )^2 + ( \frac{s}{d^{*} / 3} )$

where

$\hat{A}=\max \{ \frac{(\bar{X} - T)d}{T - {LSL}} , \frac{(T - \bar{X})d}{{USL} - T} \}$

and d = (USL - LSL) / 2.

The Index C_pg

Marcucci and Beazley (1988) defined the index

$C_{pg}=\frac{1}{C_{pm}^2}$

which is estimated as

$\hat{C}_{pg}=\frac{1}{\hat{C}_{pm}^2}$

The Index C_pq

Gupta and Kotz (1997) introduced the index C_pq, which is estimated by

$\hat{C}_{pq}=\hat{C}_p [ 1 - \frac{1}2 ( \frac{\bar{X} - T}s )^2 ]$

The Index C_p^W

Bai and Choi (1997) defined the index

$C_p^W=\frac{C_p}{\sqrt{ 1 + | 1 - 2 P_x | }}$

where $P_x={Pr} (X \leq \mu)$ . It is estimated by

$\hat{C}_p^W=\frac{\hat{C}_p}{\sqrt{ 1 + | 1 - 2 \hat{P}_x | }}$

where $\hat{P}_x$ is the fraction of observations less than or equal to $\bar{X}$ .For more information on C_p^W, see Kotz and Lovelace (1998).

The Index C_pk^W

Bai and Choi (1997) also proposed the index

$C_{pk}^W=\min \{ \frac{{USL} - \mu}{3 \sigma \sqrt{2 P_x}} , \frac{\mu - {LSL}}{3 \sigma \sqrt{2 (1 - P_x)}} \}$

It is estimated by

$\hat{C}_{pk}^W=\min \{ \frac{{USL} - \bar{X}}{3 s \sqrt{2 \hat{P}_x}} , \frac{\bar{X} - {LSL}}{3 s \sqrt{2 (1 - \hat{P}_x)}} \}$

where $\hat{P}_x$ is the fraction of observations less than or equal to $\bar{X}$ .For more information on C_pk^W, see Kotz and Lovelace (1998).

The Index C_pm^W

The index C_pm^W, also introduced by Bai and Choi (1997), is defined as

$C_{pm}^W=\frac{C_{pm}}{\sqrt{1 + | 1 - 2P_T |}}$

where $P_T={Pr}(X \leq T)$ . It is estimated by

$\hat{C}_{pm}^W=\frac{\hat{C}_{pm}}{\sqrt{1 + | 1 - 2 \hat{P}_T | }}$

where $\hat{P}_T$ is the fraction of observations less than or equal to T. For more information on C_pm^W, see Kotz and Lovelace (1998).

The Index C_pc

Luceño (1996) proposed the index

$C_{pc}=\frac{{USL} - {LSL}}{6 \sqrt{\frac{\pi}2 E | X - M|}}$

where M = ( USL + LSL) / 2. It is estimated by

$\hat{C}_{pc}=\frac{{USL} - {LSL}}{6 \sqrt{\frac{\pi}2 c}}$

where

$c=\frac{1}n \sum_{i=1}^n | X_i - M |$

V $\ddot{a}$ nnmann's Index C_p(u,v)

V $\ddot{a}$ nnmann (1995) introduced the generalized index C_p(u,v), which reduces to the following capability indices given appropriate choices of u and v: