Encyclopedia And Handbook of Process Capability Indices: A Comprehensive Exposition of Quality Control Measures (Series on Quality, Reliability and Engineering Statistics)

6

)

(7.3)

(7.4)

{ 7J

Consequently, the statistic W (by Central Limit Theorem) is asymptotically normally distributed with mean 0 and variance a2 + b2 , and the natural estimator Spk is approximately (asymptotically) distributed according to N([j,s,crg), where: E(Spk) = Ms = Spk Var(Spk) = a2 = (a2 + b2)/{36n[
and

The asymptotic distribution of the estimate of Spk variance that decreases with n . Thus, the estimator 5 ^ is asymptotically unbiased. As the sample size increases both the sample mean and the sample variance approach to their expected values. The remaining term in (7.1), O p (n _ 1 ) , represents the error of the expansion having a leading term of order n^ 1 in probability. Hence, the approximation is also consistent. In fact, the parameters a and b expressed in (7.3) and (7.4) can also be rewritten as functions of the precision index Cp and the accuracy index Ca in the following manner: d (1 + S) a = V2cr (l + 6) 7

+ (!-<$)

- ^ [ 3 C p ( 2 - Ca)4>{ZCp{2 - Ca)) +

(1 - / P

b = 4>

=

(1 + 6)}

-0 i 7 J { 7 JJ

(3CpCa)

1-6 7 3CpCa{3CpCa)},

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The S„k Index

(The identity: 3CpCa —

1+6 7

is worth noting). Thus, the p.d.f. of the approximate distribution, N^fis,^), can easily be derived to be: (see Pearn, Lin and Wang (2004))

f{x) =

J—4a OO <

2

X <

2

+b

exp

18n(fl3S pfc ))' a2+b2

f {X

_

2 Pk)

OO .

The formula of the first-order approximation of Spk as given in (7.1), is somewhat clumsy, but it produces an adequate approximate distribution for a sample size large enough. Lee et al. (1999) indicate that the approximate and exact distributions are almost indistinguishable as the sample size n reaches 1000 or more. In fact, even for n = 100 the approximation is quite reasonable for practical purposes. However, the accuracy of the approximation is difficult to investigate since the parameters a and b are required to be further estimated from the sample, resulting in a loss of reliability. Thus, the analytical approximation would not be practically useful in practice until this difficulty has been resolved. Moreover, using rather complicated and tedious algebraic manipulations Pearn et al. (2005) have shown that also, the estimator Spk can be expressed in the form of: Spk = Spk + DXZ + D2Y + D3Z2 + D4ZY + D5Y2

+Op'

n~/n i

here Z and Y are distributed according to the joint bivariate normal distribution, and Di , i = l,2,---,5 , are functions of

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6 = (fx — m) / d and 7 = a / d . (These are the basic functions for the specification bound and the mean.) Therefore, the distribution of Spk may alternatively be approximated by the following polynomial combination of the distributions of Z and Y: (Z, Y) -^—^

N((0,

0), S 2 ) , where S 2 =

0 1/2 '

with the bias approximated as: DXZ + D2Y + D3Z2 + D4ZY + D5Y2.

The formulae for the both approximations to the distribution of Spk are quite complicated, and the calculation is somewhat cumbersome. As mentioned above, since the parameters a and b in the approximate formula(s) ought also be estimated in real world applications, a substantial uncertainty may be introduced into the assessments of the performance due to the additional sampling errors resulting from the estimation of a and b . Furthermore, the accuracy of the approximation has not yet been investigated. Consequently, the approximation may not be useful in practice until these issues are successfully resolved. To analyze the actual accuracy of the natural estimator Spk, Pearn and Chuang (2003) investigate the accuracy computationally, using the simulation techniques to obtain the relative bias and the relative MSE for certain common performance requirements, where the relative bias is defined as BiasR(Spk) = [E(Spk) - Spk}/Spk = E(Spk / Spk) - 1 . It measures the average relative (percentage) deviation of Spk from the true value Spk . Moreover the relative MSE is defined as MSER(SSpk)=E[(Spk-Spk)/ Spk f =E[(Spk / Spk) - l ] 2 ; it measures the average of the squared relative deviation of Spk from the true

The S„k Index

141

value Spk . These authors have further considered the statistic [MSER(Spk)f/2 the square root of the relative MSE, which is a more direct measurement of the relative deviation (i.e. the percentage of deviation). Unfortunately either explicit or implicit mathematical formulae for BiasR(Spk) and for MSER(Spk) are analytically intractable. The simulation approach seems to be a best alternative for the accuracy assessment. For commonly used performance requirements which are Spk = 1.00, 1.33, 1.50, 1.67, and 2.00, each combination of the precision measure Cp , the accuracy measure Ca, and the (/J,, a) pair is first assigned generate random normal samples of size n . The sample data is then calculated to obtain the estimator Spk • A total of N =10000 replications are repeated for each sample size of n— 5(5)100, next the average value E(Spk) is computed and is compared with the preset true Spk to calculate the relative bias. The simulation error was also examined, and resulted in not being greater than 5 x 10~3 . These simulation results clearly indicate that the estimator Spk overestimates the true value of Spk in all cases that have been investigated. The amount of the overestimation, in terms of the relative bias, BiasR(Spk), appears to be increasing in Cp at the beginning, and then remained stable after roughly Cp = Spk + 0 . 2 . Particularly, this is valid in all cases for exceeding 15 . After the sample size n > 15 , the fluctuation of BiasR(Spk) remains less than 0.5%, and is no greater than 0.10.2% for n > 60 . This pattern is also apparent for [MSER(Spk)}1/2. Indeed in most cases, the magnitude of the deviation is increasing in Cp at the beginning and then becomes stable approximately after Cp = Spk + 0.1 , particularly, for n > 20 . (See e.g. Pearn and Chuang (2003)) The above described simulation results are useful for practitioners/engineers to measure the process performance for factory applications, especially, if the processes are controlled/monitored on a permanent routine basis.

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7.2.2 Confidence

intervals

for Spk

Based on the first-order approximation as given by Lee et al. (1999-QREI), a 100(1 — a)% confidence interval can be expressed as:

, _ bpk

Za/2

Va 2 + b2

.

6^(3SpkY

bpk

JtfTb2 +Za/2

6Vncf>(3Spk)

where as usual za/2 is the upper a/2 quantile value of the standard normal distribution, and the parameters a and b are estimated from the sampled data. In addition, a 100(1 — a)% lower confidence bound of Spk can also be obtained from the lower (one-sided) confidence limit as: Spk -zax

(a2 + 6 2 ) 1 / 2 /[6Vn0(35 pfc )].

If the calculated lower confidence limit turns to be greater than the predetermined index value, the process is judged to be capable. Otherwise, the process is classified as incapable, and approximate quality improvement activities ought to be initiated. More recently, Pearn and Chen (2003) studied the natural estimator of Spk and investigated computationally its statistical properties, using the bootstrap simulation techniques to obtain the lower confidence bound in order that practitioners/engineers can utilize them to perform quality testing and to determine whether the process meets the preset quality requirements. These authors compared the four following bootstrap methods including (a) the standard bootstrap (SB), (b) the percentile bootstrap (PB), (c) the biased corrected percentile bootstrap (BCPB), and (d) the bias-corrected and accelerated (BCa) bootstrap methods based on the coverage proportion. For each one of these methods, they have calculated the proportion of the simulation runs for which the lower confidence bound is smaller than the corresponding Spk value. This "actual coverage proportion" (viewing the lower bound Spk as a one-sided confidence interval (Spk,oo)), could then be

The Sfk Index

143

compared with the expected value of 1 — a , say 0.95. The simulation results show that the SB method substantially outperforms the PB, BCPB and BCa approaches in the case of normally distributed processes, and is therefore recommended to be used when the assessing process performance Spk based on the yield. Furthermore, for the SB method the coverage fraction is greater than 0.90 in all the cases investigated when the sample size n exceeds 25, and it is also more stable than the other three methods. Therefore, the SB approach ought to be the one recommended for the Spk index, especially for sample sizes greater than 25.

7.3 Hypothesis testing with Spk Based on the asymptotic distribution of Spk derived by Lee et al. (1999-QREI), a hypothesis testing for Spk can be constructed in a straightforward manner. For example, the testing of the null hypothesis Ho : Spk < C , C is a specified value versus an alternative hypothesis Hi '• Spk > C

can be carried out by considering the test statistic: _ 6(Spk -

C)^n(3Spk)

where a and b are the estimates of a and b, and Cp , Ca, /i and a are replaced by Cp , Ca , x and s , respectively. The null hypothesis H0 is reject at a level if the statistic T > za , where as above za , is the upper 100a% point of the standard normal

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distribution. The proposed testing procedure is quite simple to use based on the normal approximation, but a substantial uncertainty (sampling errors) could be introduced into the performance assessments since using the value of d and b calculated from the sample data could be misleading. Calculation of the critical values To assess further the accuracy of testing procedures, Pearn et al. (2003) investigated the critical values using a standard simulation technique to compare with the critical values obtained from the normal approximation as proposed by Lee et al. (1999), under the same performance requirements. The comparison of the critical values CQ calculated by means of the normal approximation and the simulation technique for various selected sample sizes from n = 5 to 150 and various Spk values = 1.00, 1.33, 1.50, 1.67, 2.00, with the risk a = 0.05 . The simulation results indicate that the critical values are more sensitive to the two indices Cp and Ca than to those that are obtained by the normal approximation. For example, given the fixed value Spk = 1.67 with n = 20 , c0 = 2.24 for Cp = 1.67 , and we have c0 = 2.30 for Cp = 1.90 . Thus, we could choose the maximal values of c0 (among these of Cp and Ca ) to obtain conservative bounds on the critical values for reliability purposes. Such an approach ensures that the decisions made based on the critical values have the risk of (wrongly) specifying an incapable process as a capable, to be not greater than the preset type I error a . It also have been noted that the normal approximation substantially under-approximates the critical values, especially, for small sample sizes n < 40 , when the amount of underestimation exceeds 0.10. Therefore, for short run applications (such as accepting a supplier that provides short production runs in the QS-9000 certification), one should definitely avoid using the normal approximation. It was also observed that the underestimation might be as large as 0.07 for n = 50 , 0.03 for n = 110 , and 0.02 for n = 150 .

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145

Therefore, in real-life applications we should strive towards sample sizes larger than 150. Furthermore, sample sizes which are necessary so that the normal approximation converge to 5pfc assuming the values 1.00, 1.33, 1.5, 1.67 and 2.0 within sampling errors less than 0.10, 0.09, 0.08, 0.07, 0.06, 0.05, 0.04, 0.03, 0.02, 0.01, 0.00 respectively (with the accuracy up to the second decimal point, i.e. 5 x 10~3 ). For example, for Sp^ = 1.33 with risk a = 0.05 , a sample size of n > 95909 ensures that the sampling error is no greater than 5 x 10"3, which is negligible. Thus, if Spk > 1.33 , we then may conclude that the actual performance S^ is also > 1.33 . This type of assessment investigation is not practical in applications. However, these computations illustrate the behavior and the rate of convergence using the normal approximation. [1] The comparison of the power analysis (the coverage rate) using a number of approaches (1 st order, 2nd order approximations, standard simulation, bootstrap) has not been carried out as yet. [2] For multiple samples, 1st order approximation by Taylor expansion + an accuracy analysis. [3] For multiple samples, critical values and lower (coverage rate) comparison of several approaches.

bounds

[4] For multiple samples, the MPPAC also can be applied in the same way.

Chapter 8

The CPU/CPL Index

8.1 Process capability and the index CPU/ CPL 8.2 Estimating and testing CPU/CPL: Based on single sample 8.3 Estimating and testing CPU/CPL: Based on multiple samples 8.4 Estimating and testing CPU/ CPL: Bayesian approach with single sample 8.1 Process capability and the index CPU/ CPL We hope that by now our readers are familiar with a number of process capability indices (including Cp , Cpj. , Cpm and Cpmk) that have been commonly used in the manufacturing industry to measure whether a process is capable of reproducing product items within the specified manufacturing tolerance. However the indices described earlier are appropriate measures only for processes with two-sided specifications only (which require specifying both USL and LSL). For the unilateral tolerance situation where only one specification limit is provided, Kane (1986) one of the early pioneers of process capability studies considered the following indices

Cpu

_ USL- n ~ 3a '

n CpL

146

_ fi-LSL ~ 3a

The CPUICPL Index

147

where USL and LSL are as above the upper and lower specification limits, respectively, \i is the process mean, and a is the process standard deviation. The index Cpu measures the capability of a "smaller-the-better" process with an upper specification limit USL, while the index CPL measures the capability of a "larger-the-better" process with a lower specification limit LSL. Process

yield measure

based on Cpu and Cpi

For normally distributed processes with specification limit USL, the process yield is given by

= P(Z <3CPU)=

one-sided

$(3CPU),

where the variable Z follows the standard normal distribution N(0, 1) with the c.d.f. $(z) = (2-K)1'2 P exp(-i 2 /2)dt. Similarly, J —oo

for normally distributed processes with one-sided specification limit LSL, the process yield is

P(X>LSL)=P(^<^=-^) = p(-iz
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(2001)). These recommendations are summarized in Table 8.1 For existing manufacturing processes, the capability should be at least 1.25, and for new manufacturing processes, at least 1.45. For existing manufacturing processes dealing with safety, strength, or critical parameters, the capability must be no less than 1.45, whereas for new manufacturing processes dealing with the same issues, the capability ought to be no less than 1.60. In recent years, due to globalization various companies have adopted criteria for various evaluations of their processes based on capability objectives that are more stringent than the recommended minima. For instance, the popular "six-sigma" program pioneered by Motorola essentially requires that when the process mean is in control, it will not be closer than six standard deviations from the nearest specification limit. In effect, it requires that the process capability ratio will be at least 2.0. Table 8.1. Some minima capability requirements based on C/ for new and special processes. Ci Value 1.25

Production Process Type Existing Processes

1.45

New Processes; or Existing Processes involving Safety, Strength, or Critical Parameters

1.60

New Processes involving Safety, Strength, or Critical Parameters

8.2 Estimating and testing CPU/ CPL: Based on a single sample

8.2.1 Estimation

of CPU and CPL

In practice, sample data ought to be collected in order to calculate the indices CPU and CPL since the process mean fi and standard deviation a are usually unknown (as it was indicated in

The Cpu/ CPL Index

149

the previous chapters). To estimate the indices Cpy and CPL , Chou and Owen (1989) proposed Cpy and Cpi , the natural estimators of CPU and Cpi, defined as: Cpu

_ USL — x ~ 35 '

f, CpL

_x— LSL ~ 3s '

where as above x = Y^. xt / n is the sample mean, and s2 = (n — l ) - 1 ^ . _ A % i — x)2 is the sample variance, obtained from a process that is "demonstrably" stable (under statistical control). Under the normality assumption, the estimator Cpy is distributed as (3Vn) _1 tn_i(6) , where £n_i(<5) is a non-central t random variable with n — 1 degrees of freedom and the noncentrality parameter 8 = 3^/nCpu . The estimator Cpi has the same sampling distribution as Cpy with 6 = 3-fnCpi . However, both the estimators CPU and Cpi are biased. Pearn and Chen (2002) have shown that by adding the correction factor &„_! = [ 2 / ( n - l ) ] 1 / 2 r [ ( n - l ) / 2 ] / T [ ( n - 2 ) / 2 ] to CPU and CPL , one obtains unbiased estimators bn_i Cpy and hn_\ Cpi which have been denoted as Cpy and Cpi. That is, E(Cpu) = CPu , and E(Cpi) = CPL- Since 6n_! < 1, Var(CPU) < Var(CPU) and also Var{Cpi) < Var{Cpi) . The estimators CPU and Cp^ are based only on the complete and sufficient statistics (x,s2), consequently Cpy and Cp£ are the uniformly minimum variance unbiased estimators (UMVUEs) of Cpy and CPL, respectively.

8.2.2 r-th

moment

The r-th moment (about zero) and the variance of Cpy can be obtained as follows: set Z = -Jn(USL — x) / a , this variable is distributed as N(3*fnCPy,l). It is easy to verify that E(CPU) = Cpy under the normality assumption. It follows from the definitions of Z and that of Cpy that

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^_(r[(n-l)/2])r-1r[(n-l-r)/2]

F(r

Var(CPU)

r[(n-l)/2] r[(n-3)/2]

=-

(r[(n-2)/2])

2

1

T

2

PU)

, 1 r[(n-l)/2] r[(n-3)/2] 9n (r[(rz-2)/2])2 The results for the r-th moment, the expected value, and the variance of the dual estimator CPL are analogous.

8.2.3

Distribution

We are changing the variables possessing the non-central t distribution with v = n — 1 degrees of freedom and the noncentrality parameter 8 = 3VnC; . Let X = Cj = 6„_i(3Vn)~ t„(6). The c.d.f. of Cj can be derived directly as: F(x) =

1 2(»-3)/2r[(n-l)/2] 2

poo

x I y Jo y

„ UlL n z 2

e

1

p3^/nxy/bn_l^/n-l

—f== | V2Wo

exp

-f(.-tf

dudy,

Here 6 = 3VnC 7 and 6n_i is the correction factor to CPu . Differentiating the equation of the c.d.f. with respect to y gives the p.d.f. of Cj to as

/(*)

3Vn/(n-l)-2-"/2 bn__l^fT[(n-l)/2]

L

(n-2)/2

x,o „

exp

2/ +

3xyfny [bn-iJn-1

dy,

The CPUICPL Index

151

where, as above, the correction factor is bn_i = 1/2 [2/(n - l)] r[(n - l)/2]/T[(n - 2)/2] and 8 = 3VnC 7 . Figures l(a)-l(b) display the p.d.f. of Ct for various CI =1.00, 1.25, 1.45, 1.60, 2.00 for n = 10 and 30, respectively. Figures 2(a)-2(b) display the c.d.f. of Cj for the same values of C/ and n.

8.2.4 Testing

hypothesis

with CPU and CPL

For processes with one-sided manufacturing specifications (requiring the existence of only one of USL or LSL, but not both), the index Cpy can be used to measure the capability of a smallerthe-better process for upper specification limit USL, whereas the index Cpi to measure the capability of a larger-the-better process with a lower specification limit LSL. To test whether a given process meets the capability requirement, we shall consider the following statistical testing hypothesis with HQ :Cj < C (the process is incapable), versus the alternative Hi :Cj > C (the process is capable). Consequently, one can utilize the test (p(x) = 1 if Cj > C0 , and ip(x) = 0 , otherwise. The test

C0 , with type I error a(C0) = a , the chance of incorrectly judging an incapable process as a capable one. The calculations of p-value (rejection probability), critical value, and power of the test are provided below. Suppose the observed value of the statistic Cj = w , we then can calculate those values as follows (here 6 = 3Vn C): p - value = P{(5 7 > w | Ci < C) ^P\tn_i{8)>^

\Cl
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As above the critical value, C0, is then determined by:

p{c 7 >c 0 \cI = c)

a

= P Ci(«) >

3VnC0

Cj=C

Hence, we have 3VnC0

= tn-i,a(8)

or C0 =

fyI_id,a(tf)/(3Vn).

where, as above, tn__ia{6) is the upper a quantile of a non-central t distribution with n — 1 degrees of freedom satisfying P{tn-i(fi) > ^n-i,a(^)) = a • As indicated above, the sampling distribution of the UMVUE, (5/ , is bn^ (S^/n)"1 £„_i(6) , a noncentral t distribution with n — 1 degrees of freedom and noncentrality parameter 6 = 3VnC/ . Thus, the power of the test can be calculated as *(CI)

= P(CI

>C0\CJ)=P

1^(6)

>

3-JnC0) u

n-l

Under H0 for EC[ (ip(x)) = a , we obtain that the critical value is C0 = bn^i(3-Jn)~1tn_iQ(6), where 6 = 3~JnCj . Therefore, the test ip is a uniformly most powerful (UMP) test of size ECr(
The lot is accepted, if The lot is rejected,

Gt > ^ - t ^ 3vn otherwise.

a(6

= 3VnC 7 ).

The CPUICpL Index

8.2.5 Lower confidence

bound for

153

CPU/CPL

Critical values are used for making decisions in capability testing with designated type I error a which is the risk of misjudging an incapable process (H0 : Cj < C) as a capable one (Hi :Cj > C ). The Rvalues are used for making decisions in capability testing, which presents the actual risk of misjudging an incapable process (HQ :Cj < C) as a capable one (Hx :Cj > C). Thus, if p-value < a we then reject the null hypothesis, and conclude that the process is capable with the actual type I error pvalue (rather than a ) . Both approaches, the critical values and the p-values, do not convey any information regarding the minimal value (the lower confidence bound) of the actual process capability. Determination of the lower confidence bound on the actual process capability is essential for quality assurance. The lower confidence bound not only gives us a clue about the minimal level of the actual performance of the process which is closely related to the nonconforming units, but it is also useful for making decisions in the course of capability testing. Chou et al. (1990) established the lower confidence bounds on Cpu and Cpi based on the natural estimators Cpu and CPL. Let USL = x + kis and LSL — x — k^s , where kx = 3CPU and &2 = 3CPL • A 1007% lower confidence bound Cy for Cpu satisfies Pr (CPU > CJJ) = 7 . It can also be written as: 7 =

p r /^£> c \ =

= Pr

P r /£±^it>^

Z — 3-JnCu > -ki^n) sJa

or, equivalently, lower confidence It can be shown and in_1 (^2) a r e

= Pv(tn_i(6i) > -VnfcO,

Pr(f n _ 1 (6 1 ) < -Juki) = 7 . Similarly, a 1007% bound Ci for Cpi satisfies Pr (Cpi > Ci) = 7 . that Pr(i n _ 1 (5 2 ) < k^Vn) = 7 , where tn_i(8i) non-central t distributed random variables with

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n — 1 degrees of freedom and the non-centrality 61 = 3-JnCu and 62 = 3VnCL , respectively.

parameters

Observe that these two estimators are biased. Pearn and Shu (2003) developed an algorithm (including a Matlab program) for calculating the lower confidence bounds on CPU and Cpi based on the UMVUEs and provide tables for the engineers/practitioners to be used in measuring their processes.

8.3 Estimating and testing CPU/ CPL: Based on multiple samples A common time honored practice of the process capability estimation in manufacturing industry is to implement first a routine-based data collection plans for controlling/monitoring the process stability, and then to analyze the past "in control" data. Kirmani et al. (1991) considered estimation of a and of the precision index Cp for cases where the data are given as multiple samples. Pearn and Yang (2003) proposed an unbiased estimator of Cp for multiple samples, and showed that the unbiased estimator is the UMVUE of Cp , which is asymptotically efficient. Li et al. (1990) have studied the distribution of estimators of Cp and Cpk based on ranges of multiple samples. More recently, under the assumption of normality, Vannman and Hubele (2003) investigated the distributional properties of estimators of a wide class of capability indices, containing the indices Cp , C^ , Cpm and Cpmj; based on multiple samples collected over time for an incontrol process.

8.3.1 Estimations

of CPU and CPL based on multiple

samples

Consider m preliminary multiple samples (subgroups) each of the same size n taken from control chart samples. We shall

The CPU/ CPL Index

155

assume that the sample of the characteristic under consideration, (Xii,Xi2,...,Xin) , follows a normal distribution N(/j,,a2) for i = 1,2,...,TO.The process mean is estimated by the grand average X and the process variance by the pooled sample variance S% from subsamples collected over time for an in-control process. Let x /n and Si 2 n 1 / 2 be the

* = EU v

= E"=i(*« - ^) /( -!)]

i-th sample mean and the sample standard deviation respectively. Then X = Yl™-\ %i I m and S2P = Y17=i ^ Im w i l 1 b e u n b i a s e d estimators of fi and a respectively. Pearn and Lin (2003) suggest an unbiased estimator of CPU and CPL of the following form: f,*

_ bm(n-i)(USL - X) ~ 3^ '

Cpu

^* CpL

~

_ bm(n-i){X - LSL) 35; '

where as above bg = (2 / gfl2T{g / 2)/T[(g - l ) / 2 ] is the correction factor already encountered earlier. Under the normally assumption, [3^/mn /fem(„_i)] x Cpu can be expressed as Zv /y/W Jm{n - 1) , where Zv = Z + 3JrrmCPU ~ N(6u,l) , Z ~ iV(0,l) , we have W = m{n - l)S2 /a2 ~ x ^ ^ , and Zv and W a r e independent. The estimator (3Vmn /& m ( n -i)) x Cpu is distributed as the non-central t variable with m(n — 1) degrees of freedom and the non-centrality parameter 8u — 3-JmnCPU . The estimator (3v?nn / bm(„_i)) x CPL has the same sampling distribution as {3-Jmn /6 m ( n -i)) x CPu but with the non-centrality parameter 8L = 3-Jm,nCPi . Furthermore, since CPU_a,nd CPL based solely on the complete and sufficient statistics (X,Sf,), one concludes that the estimators CPU and CPL are the UMVUEs of CPU and CPi respectively based on multiple samples. Thus, the c.d.f. of X = Cj (which denotes either Cpy or Cpi) can be derived as:

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Encyclopedia and Handbook of Process Capability Indices

F(x) =

2^~2)l2T(g/2) poo

3Jg+mxy/bgJg

-JL

exp

yf2lrJo n Jo

-i(,-*j»

dudy.

We present a derivation in the Appendix (see Pearn and Wu (2003)). Differentiation with respect to x results in the p.d.f. of Cj given as:

m_

3y/(g + m)/g-2- (g+m)/2 bg^WT(g/2)

f °>-1)/2exPl *J 0

y+

3xy/(g + m)y bQJ~g

where as above bg = (2/g) 1/2 T{g/2)/F[(g and 6 = 3~JmnCj .

dy,

- l ) / 2 ] , 5 = m(n - 1)

From the plots of the p.d.f. of Cj for various sample sizes, we observe that (i) for fixed sample sizes m and n , the variance of Cj increases as Cj increases, (ii) for a fixed n and Cj , the variance of Cj decreases as m increases, and also (iii) for fixed m and Cj, the variance of Cj decreases as n increases. For cases with unequal sample sizes, the unbiased estimators of Cpu and Gpi can straightforwardly be generalized to be:

^PU —

T?

3S„

' ^PL —

d&r,

where X = '^2-_,iT'iXi /N is t h e grand mean of t h e overall sample, N = ]>J--i n » is the number of observations in the total sample and S% = X ^ w ( n » — ^)Sf /{N — m) is the pooled sample variance. From the representations

157

The Cpuj CPL Index

[2>^W^i/bN_m] x CpV = and where

Zu/ylW/iN-m),

[ 3 V F ^ T / bN_m] x C*PL = ZL / JW/(N - m) , Zv

and Zj, are as above Zv / yJW /(N — m) and are distributed as tv(6v) ( tv(6L) ), namely according to the non-central t distribution with v = N — m degrees of freedom and the non-centrality parameters 8u = 3VlV Cpu ( 6L = SJNCpi), respectively. We remind our readers that the process ought be stable in order to produce a reliable estimate of the process capability. If the process is being out of control in the early stages of process capability analysis, it would then be unreliable for estimating the process capability.

8.3.2 Testing

CPU and CPL based on multiple

samples

To test whether a given process meets the preset capability requirements and runs according to the desired quality condition, we can examine the following hypothesis: H0 :d C, where C is a preset known constant. A process fails to meet the capability (quality) requirement if Cj < C , and obeys the capability requirement provided Cj > C . The C values commonly used in the capability testing for most industry applications are displayed in Table 8.1 in the beginning of this Chapter. We define the test statistics v?'(x) = 1 ^ @i — ^o a n d
where g = N - m , bg = (2/g)ll2T{g/2)/T[(g - l)/2] , and tg^a{8) denotes the upper a -th percentile of a non-central t

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variable with N — m degrees of freedom and the non-centrality parameter 6 = 3y/~NC[ . The power function may be expressed in the standard manner: (3{Cr,
=PT{C*J

>

C0\Cj}

Pv{tg(6)>3^NC0/bg}.

The size of the test is given by a(C0)=

sup P{Cj,
CreH0

Moreover, Pearn and Wu (2003) have shown that the test statistic
C*x > ^j^tN_m^a{8

= 3VJVC/).

otherwise.

The derivations of the lower confidence bounds, the p-values and the capability calculations for situations with multiple samples can be performed using the same techniques as in the cases with a single sample. However these calculations may be more involved. (See Pearn and Lin (2003), Pearn and Wu (2003) and Pearn et al. (2003) for details).

8.4 Estimating and testing CPU/ CPL: Bayesian approach with single sample 8.4.1 Bayesian approach to the assessment CPL based on single sample

of CPU and

A Bayesian procedure for assessing process capability was proposed in the early stages of process capability studies by Cheng

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159

and Spiring (1989) for the initial primitive basic index Cp under the assumption that the process mean \i is equal to the target value T . However, the restriction that fi = T may not be a practical assumption for numerous industrial applications. (Bayesian approach has not been emphasized in the nineties of the 20-th century the formation and consolidation period of the PCI "technology".). In 1999 Shiau et al. suggested a Bayesian procedure for the general situation without restrictions on the process mean applied to the index Cpm. More recently, Pearn and Wu (2003) utilized a similar Bayesian approach for the capability index Cpk . For the one-sided specification limit, Pearn and Wu (2003) have derive a Bayesian interval estimate for Cpu and Cpi and proposed a Bayesian procedure for testing the process capability (relaxing the restrictions). We shall assume that the observations x = {xi,x2,- •-,£„} are random samples taken from independent and identically distributed (i.i.d.) normal variables with the mean /j, and variance a2 N(^,a2). Under these assumptions, given a pre-specified capability level w > 0, the posterior probability p based on the index Cj that a process is capable can be expressed as: p = Pr{the process is capable| x}=Pr{C/ > w | x}

= /.

o

1 T(a)ya+1 exp

x $ 3Vn

' Cj %

K~\

r V,

dy V (n - l)y— w

(8.1)

where a — (n — l ) / 2 and, as above, $(•) is the c.d.f. of a standard normal variable. Note that the posterior probability p depends on n , w and Cj . When n and w are given, and since Cj can be calculated from the process data, there is a one-to-one correspondence between p and Cj. Therefore, it follows that the minimal value of Cj = C (p) , necessary to ensure that the

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posterior probability p reaches a certain desirable level, can be useful for assessing process capability. In other words, to see if a process is capable (with capability level w and confidence level p), we only need to check whether Cj > C*(p). Pearn and Wu (2003) have provided tables of the minimal values of Cj , C*(p), for various confidence level p = 0.99, 0.95, 0.90, and commonly used capability requirements C / = 1.25, 1.45, 1.60.

8.4.2 A Bayesian approach to assessment CPL based on multiple samples

with CPU and

Posterior Probability for Multiple Control Chart Samples For cases where the data are collected as multiple control samples, we have considered m samples each of size nt and have suggested the unbiased estimators of Cpu and CPL , where Xj is the i-th sample mean, and St is the i-th sample standard deviation, and N = Y^Z.\ ni i s the number of items in the total sample. These estimators are: n* <^PU

_ bN-m(USL - X) a n a x,* _ bN_m(X - LSL) — w > ^PL — w >

where as above bg = (2/ g)1/2T(g /2) /T[(g - l)/2] , g = N-m, n X = Yl™- iXi IN is the overall sample mean, and S2P = ^2™- (n> ~ 1 )^' 2 IW ~ m ) *s * n e P°°l e d sample variance and with Sf = y^m_ (xtj —XifKrii —1). However, for multiple control samples, it is necessary to take into account the variations between them. To assess these variations Pearn and Wu (2003) suggest the ratio of total within sample variation ( SSW ) and total sum of squared variation (SST ) defined as:

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The Cpu/ CPL Index

ssw _ 5 § ( ^ ^ ^ 2

7

ssr

E E (**• - ^)2 i=i j=i

2= 1

: E fo !)4 + E «ite *) i=i i=i

Similarly in these situations, the extension of the posterior probability p for multiple control samples can be expressed as: noo

1

Jo T(a)ya+1 6XP x $

3VF

27

^tf-m '^(N

dy, -m)y — w

(8.2)

where a — ()~]._ n, — 1) / 2 . Compare with (8.1) for a single sample, where the ratio 7 = 1, N — m = n — 1 , a = (n — 1) / 2 . In this case the estimator of Cj, (7/ , is reduced to the estimator Cr discussed before.

Chapter 9

Multi-Process Performance Analysis Charts (MPPAC)

9.1 9.2 9.3 9.4 9.5

Introduction The modified Cpk MPPAC The Cvm MPPAC The Spk MPPAC The Le MPPAC

9.1 Introduction In this short Chapter we shall describe a number of new practical procedures for improving the analysis of process capability based on the indices CP, Cpk, Cpm, Spk and some others. Ever since Shewhart (1944) introduced his pioneering control charts, it has become a common practice for practitioners to use various control charts for monitoring different processes on a routine basis. As an example, when dealing with a variable data, the control chart technique usually employs a chart (such as a X chart) to monitor the process center and a chart (such as an R chart or an S chart) to monitor the process spread. These charts are easy to comprehend, and they effectively communicate the critical process information without necessarily employing words and formulas. Unfortunately, they are applicable only for a single process (one process at a time). T h u s , using t h e m in multiprocess environment could be a clumsy and time-consuming task for supervisors or shop engineers since it may require to analyze each

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Multi-Process Performance Analysis Chart (MPPAC)

163

individual chart in order to evaluate the overall status of shop process control activities. We have already pointed out that process capability index measures the ability of the process to reproduce products that meet certain specifications. When dealing with the capability of a process, there are mainly two characteristics of importance involved, the process location in relation to its target value and the process spread. The closer the process output is to the target value and the smaller is the process spread, the more capable will the process be. However, the fact that process capability indices combine information about closeness to target and the process spread, expressing the capability of a process by a single number, may in some instances be viewed as one of their major drawbacks. When a process is found to be "non-capable", the operator might be interested in knowing whether the non-capability is caused by the fact that the process output is substantially off target or that the process spread is too large, or is a result of a combination of these two factors. In order to circumvent this defect some current researchers in the last 15 years were suggesting that different graphical methods be used to ascertain the improvement initiatives aimed at generating more capable processes (see e.g. Gabel (1990), Boyles (1991), Tang et al. (1997) and Deleryd and Vannman (1999)). A Multi-process Performance Analysis Chart (MPPAC), proposed by Singhal (1990), which evaluates the performance of a multi-process product with symmetric bilateral specifications, provides priorities among multiple processes for capability improvement and reveals whether reducing the variability or the departure of the process mean should be the main task for improvement. While Cpu and Cp\ present the X-axis and Y-axis, respectively in a MPPAC, Cp is the average of Cpu and Cpi , namely, Cp = (Cpu + Cpi) / 2 . Moreover, the Cpk MPPAC provides an efficient route to process improvement by comparing the locations on the chart of the processes before and after an improvement effort has been carried out. Singhal (1991) also provides a MPPAC with several welldefined capability zones by using the process capability indices Cp

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and Cpk for grouping the processes in a multiple process environment, into different performance categories on a single chart (see Figure 9.1). This is very helpful when process performance is measured in terms of capability indices. Different capability zones describe a status of each process which is easy to interpret and assist in grouping the processes into performance categories for the purpose of quality improvement operations. The American giant corporation Motorola Inc. introduced their six sigma (6-cr) program (Mikel (1988)) which is equivalent to a defect rate of 3.4 parts per million (PPM). This program corresponds to a Cp value of 2.0 or more and a Cp^ value of 1.5 or more. All this is done under an implicit normal assumption. A Cp value of 2.0 tells us that the specification width is twice as wide as the distribution of the process (with a defect rate of 0.002 PPM) as long as the process mean is equal to the target value. Under the six sigma program, the process mean is allowed to shift as much as 1.5 sigma and still obtain a Cp^ value of 1.5 or more. (See Chapter 2 for the relation between the PPM defect rate and the values of PCI's)

Figure 9.1. MPPAC with capability and six sigma zones (Reproduced from Singhal (1991))

In practical applications, when a product has several models with required different specifications needs to monitor and control,

Multi-Process Performance Analysis Chart (MPPAC)

165

it may be difficult or very time-consuming to carry out factory control activities. As it was mention above, a MPPAC not only evaluates the performance of a multi-process product with symmetric bilateral specifications but also sets the priorities among multiple processes for capability improvement and indicate whether reducing the variability, or adjusting the departure of the process mean should be the focus of improvement operations. This renders a MPPAC to be an efficient tool for communicating between product designer, manufacturers, quality engineers, and among various management departments.

9.2 The modified Cpk MPPAC Pearn and Chen (1997) proposed a modification to the Cpk MPPAC combining the more-advanced process capability indices, Cpm or Cpmk, to identify the problems causing processes' failings to center around the target value. Under this modification the performance of an entire product, composed of the-nominal-thebest, the-larger-the-better, and the-smaller-the-better characteristics, could be measured. The modified Cpk MPPAC is shown in Figure 9.2, with five capability zones corresponding to the five standard process conditions for Cpk = 1.00, 1.33, 1.67, and 2.00. We mark these process capability zones with bold lines as the critical values of individual process capabilities for the multi-process product on MPPAC chart, and the corresponding fractions of nonconformities (in PPM) are summarized in Table 9.1 (see also Montgomery (2001)) among other sources. In this modified MPPAC, Cpu and Cp\ represent the X-axis and Y-axis, respectively. Recall that Cp is the average of Cpu and Cpi > [ CP = (C pu + Cpi) / 2 ] and Cpk is the minimum value of the X- and Y-axes [ Cpk = min{C pu ,C p ;} ]. Hence, based on Cpk MPPAC, the vertical and horizontal axes of the chart are geared to evaluate the-larger-the-better and the-smaller-the-better characteristics.

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Combining Singhal's MPPAC with asymmetric process capability index Cpa , and considering unilateral characteristics, Chen et al. (2001) introduced a Process Capability Analysis Chart (PCAC) to evaluate the process potential and performance for an entire product composed of the-smaller-the-better, the-larger-thebetter, symmetric and asymmetric specifications. As mentioned above, the process yield of a multi-process product is lower than of any individual process yield. Similarly, when the entire product capability is preset to satisfy the required level, the individual process capability should exceed the preset standard for the entire product. Therefore, the process capabilities are recognized to be capable if all of the individual process capability indices are located within the process capability zone. Conversely, processes must be upgraded when some of the process capability indices are located outside this zone. It is straightforward to distinguish process performance with respect to the locations of process capability indices on a modified Cpk MPPAC. Hence, the modified Cpk MPPAC not just distinguishes process capabilities, but also reveals the degree of quality accuracy for multi-process products. This makes the modified Cpk MPPAC to be an effective and efficient tool for evaluating multi-process products, composed of numerous unilateral and bilateral specifications. C, =• 0.50

C , = 0.75

--7—

/ /

/

/

/

C, = 0.875

—r

/

. /

/

/

, - • "

/ • / /

/

, - • "

•

/ / ,• / / , • • - ' . , -

, • • • • -

/

.*'••

. • • *

" " " ' , - • • - " " " "

I ///sO*

/M*< 0.50

1.00

1-33

167

i m

C„

Figure 9.2. The modified Cpk MPPAC with capability zones for Cpk = 1.00, 1.33, 1.67, and 2.00.

Multi-Process Performance Analysis Chart (MPPAC)

167

Recall (Chapter 5) that the loss function utilized in the G. Taguchi method has the property that the closer the process mean is to the process target the better quality and fewer process losses are encountered. Conversely, the farther the process mean is located from the process target the worse are process capabilities. Keeping the process on-target is crucial for smooth operation. A few subsidiary lines of Ca = 1 — |/x — M\j d can be added on a MPPAC for controlling accurately the process centering. The index Ca measures what is known to be the relative distance of the shift from process mean to the preset target. (The defmition of relative distance is (/J, — M) / d or (M — fi) / d .) Evidently, equal relative distances result in the same values of Ca . Note that the preset target value coincided with the midpoint of the specifications (i.e. T = M). Off-diagonal subsidiary lines are plotted for values of Ca 0.500, 0.750 and 0.875 in Figure 9.2. Ca < 0.875 indicates that the process is not accurate and some actions to shift the process mean closer to the process target are required. Equivalently, Ca > 0.875 indicates that a process possesses quite a good accuracy. In general, Ca should not be too small, since it would imply that the process mean shifts further away from the process target which results in a substantial process loss. Set r = | [i — M | / d , then the values of [i will be M + r x d and M — r x d for each Ca . The slope of the corresponding subsidiary line is (1 + r ) / ( l — r) when the process mean is greater than the process target, and analogously the slope of the corresponding subsidiary line is (1 — r ) / ( l + r) when the process mean is greater than the process target, and analogously the slope of the corresponding subsidiary line is (1 — r) /(l + r) when the process mean is greater than the process target, and analogously the slope of the corresponding subsidiary line is (1 — r ) / ( l + r) when the process mean is smaller than the process target. Table 9.1 displays

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five values of Ca with the corresponding fx, r , and the slopes of the subsidiary lines. Table 9.1. The values of Ca with the corresponding /J,, r , and the slope of lines. Ca 1.000 0.875 0.750 0.500 0.250 0.000

M M M M M M M M

+ + + + -

P M 0.125 0.125 0.250 0.250 0.500 0.500 0.750 0.750 USL LSL

xd xef xd xd xd xd xd xd

r

Slope

0.000 0.125 0.125 0.250 0.250 0.500 0.500 0.750 0.750 1.000 1.000

1.000 1.286 0.778 1.667 0.600 3.000 0.333 7.000 0.143 00

0.000

Concerning the modified Cpk MPPAC, it is useful to observe that: [a]

The parallel and perpendicular lines through the plotted point intersecting the vertical axis (y-axis) and the horizontal axis (x-axis) at the points represented by Cpu = (USL — ^)/(3<x) and Cpi = (/i — LSL)/(3a), respectively.

[b] The 7r/4 subsidiary line (i.e. the line with the slope 1) represents the points where the process mean equal to the target (// = M = T) and the values of the indices Cpu, Cpt and Cpk are equal. [c]

For the points inside the area to the right of the 7r/4 subsidiary line (i.e. with the slope < 1) represents processes with the process mean shifted towards the lower specification limit (i.e. the process mean is lower than the target value), or Cpk = Cpi < Cpu. On the other hand, for the points inside the area to the left of the 7r/4 subsidiary line (i.e. with the slope > 1) represents processes with the process mean shifted

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169

towards the upper specification limit (the process mean is higher than the target value), or Cpk = Cpu < Cv\. [d] The origin point represents a process with Cpk = Cpu = Cpi — 0 which means that the standard deviation of the process is infinite or that the process mean lies at USL or LSL. In quality improvement, reduction of variation from the target is of the same importance as increasing the process yield and reduction of the process spread. The modified Cpk MPPAC combines the two concepts - closeness to the target and narrow spread - in more efficient manner than using a process capability index alone. If the cases when exact values of \x and a are known, the modified C^ MPPAC can easily be applied. In other words, if the corresponding value of {Cpu,Cpi) is inside the capability region, the process is then defined to be capable, and if the value is outside, the process is defined as a non-capable one. In practice, however, we never know the true values of Cpu and Cv\ • Pearn and Wu (2004) developed a procedure to construct the lower confidence bounds on the indices Cpk , Cpu , and Cpi for each characteristic type. These lower confidence bounds can simultaneously be plotted on a single chart to check whether the process output is off target or that the process spread is too wide (or if a combination of these two situations takes places).

9.3 The Cpm MPPAC

Since the index c simultaneously measures process variability and centering (see Chapter 5), a Cpm MPPAC provides a convenient way to identify problems in process capability after statistical control has been established. Based on the definition, set Cpm = h , for various h values, then a set of (n,a) values satisfying the equation:

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a (d/3)

+

f(M-r)l

2

d/3

can be plotted on the contour (indifference curve) of Cpm = h. These contours are semicircles centered at (T, 0) of radius 1/ h . The more capable the process, the smaller is the semicircle. Six contours of the Cpm MPPAC for the six common values for Cpm = 1/3, 1/2, 1, 1.33, 1.67, and 2, are shown in Figure 9.3. Concerning the Cpm MPPAC, it is worth observing that: [a]

The parallel and perpendicular lines through the plotted point intersect the vertical and horizontal axes at the points represented by (a / (d / 3) )2 and ( 0 - T) / (d / 3) f , respectively.

[b] The distance between T and the point at which the perpendicular line through the plotted point intersects the horizontal axis, corresponds to the departure of the process mean from the target. [c]

The distance between 0 and the point at which the parallel line through the plotted point intersects the vertical axis represents the process variance.

[d] For the points inside of the semicircle of the contour (the indifference curve) Cpm = h , the corresponding Cpm values are larger than h . For the points outside of the semicircle of the contour Cpm — h the corresponding Cpm values are smaller than h. [e]

As the point gets closer to the target values the value of the Cpm becomes larger, and the process performance improves.

[f]

For processes with a fixed value of Cpm , the points within the two 7r/4 lines envelop, the process variability is due mainly to the process variance.

Multi-Process Performance Analysis Chart (MPPAC)

171

[g] For processes with a fixed value of Cpm , the points outside the two 7r/4 lines envelop, the process variability is due mainly to the process departure. (o/(d/3))2

9

Target Line

4

1

0.SS5 (U59 0.25

0,359 0.MS

(c/(d/3)f

1

4

Figure 9.3. The contours of a Cpm MPPAC.

The Cpm MPPAC conveys critical information about a multiple processes concerning the process departure and the process variability using a single chart. Current research about MPPAC is restricted to obtaining quality information from single sample of each process ignoring the sampling errors. The information provided from the existing MPPAC chart can therefore be unreliable and misleading resulting in incorrect decisions. Pearn et al. (2004) investigate the accuracy of estimation as a function of the process characteristic parameter £ = (/x — T) / a , given a group of multiple control chart samples. Data regarding the true capability values and the maximum of nonconformities (in PPM) are provided for production quality control. Suitable sample sizes are then recommended to the proposed Cpm MPPAC for multiple processes production quality control. This approach ensures that the critical information obtained from the Cpm MPPAC based on multiple control chart samples, is more reliable.

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9.4 The Spk MPPAC Based on the yield index Spk (see Chapter 7), Boyles (1994) developed the so-called Spk contour plot that is a contour plot of the index Spk as a function of the process parameters (//, a), to be used for monitoring and controlling process performance. In fact, the Spk contour plot is a handy tool for evaluating multiple processes since it allows to obtain the process yield and the process departure ratio by checking the location of the index value appearing on the contour plot. For multiple processes with the same specification limits, the contour plot not only shows the process capability for multiple processes simultaneously, but also provides a quick reference about the parameters that should be targeted for process improvements. We note, however, that the Spk contour plot is applicable only for multiple processes with same specification limits on each single process which may not be appropriate for processes with multiple characteristics when the characteristic specifications are not equal. To enable the applicability of the contour plot for processes with multiple characteristics, Chen et al. (2003) applied the method developed by Deleryd and Vannman (1999) who introduced a process capability plot, called the (6,7) -plot, which is an adjustment of Boyles' (/i, <j)-plot where 6 — (/J, — T)/d , and 7 = a I d . The authors readjusted the definition of Spk in the following manner: bpk —

+ •.

M

3

12

I*"1 [5*

\

a 1 — Qr

,

Cdp

+

.

> .

LSL

^

[1 + C«*r]l ,

C

dp

J.

where Cdr = (// — T) / d , Cdp — a j d and $ is the standard normal c.d.f. Note that Cdr measures the departure ratio and Cdp measures the variation relative to the specification tolerance.

Multi-Process Performance Analysis Chart (MPPAC)

173

If /j, < T (/x > T), we have Cdr < 0 (Cdr > 0 ) , and when/i = T , Cdr = 0 (process is on target). Evidently, if zx = £SX , Cdr = — 1; and for /x = £/5X , C^ = 1. The new expression of Spk is for the case where the specification interval is two-sided with the target value T at M , a most common practical situation. Using Cdr as the x-axis and Cdp as the y-axis, Chen et al. (2003) have plotted the following point set forming the curve of Spk (bold-shape curve in Figure 9.4) on the (Cdr, Cdp) coordinates:

{cdr,cdp)\^-1

1 — Cdr

!« ,

C

dv

+

,

i + cdr

> . ,

C

dp

bpk ,

We note that the process capability plot is invariant to the value of the specification limits. Processes with multiple characteristics having different characteristic specification limits can thus be plotted simultaneously on a single chart. Chen et al. (2003) call this control chart a Multi-Characteristic Process Capability Analysis (MCPCA) chart. In Figure 9.4, the top curve is the capability plot for processes with Spk = 1.00, and the bottom one is the plot for processes with S^ = 1.33. As we have mentioned above (Chapter 7), Spk is a yieldbased index. According to the modern quality improvement theories, reduction of the variation from the target is as important a step increasing the process yield (i.e. meeting the specifications). Therefore, three pairs of control limits I1( I2, and I3 are drawn on the MCPCA chart (in Figure 9.4) to monitor the variation from the target of each one of the characteristics. Corresponding to the three pairs of control limits I1( I2, and I3, the six values of Cdr =(n-T)/d are -1.0, -0.5, -0.25, 0.25, 0.5, and 1.0 from the left to the right, respectively.

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0.35-

0.30.25-

0.20.150.10.05-

0-1

-0.5

-0.25

0

_ 0.25

0.5

1

dr

Figure 9.4. A Sp^ MCPCA chart with several departure control zones.

Table 9.2. Various control regions for a process departure and improvement assessments. Control Region

C^ Value

Process Improvement Assessment

I

±0.25

characteristic departure is tolerable

I

±0.50

characteristic investigation desirable

I

± 1.00

characteristic departure is serious, the overall process should be reviewed.

departure is abnormal, and improvements are

In terms of the six-sigma quality improvement program (as formulated by Motorola, see, e.g., Noguera and Nielsen (1992)) setting d = 6cr , the three pairs of the control limits I t , I2, and I3 correspond to \ y, — T \= 1.5a, 3cr and 6
Multi-Process Performance Analysis Chart (MPPAC)

175

PPM of non-conformities, assuming that the specification limits are 6cr away from the target. Under the assumption of normality and stationarity of the process (see Chapter 2), the three pairs of control limits Il5 I2, and I3 provide various process accuracies (the degree of centering) of the control zones. Various control regions (zones) for process departure and the improvement assessments are summarized in Table 9.2. Practitioners can judge the degree of centering of the characteristic j by checking the location of the corresponding plotted point on a MCPCA chart. For example, the departure measure C^r value is equal to -0.5 for the characteristic A in Figure 9.4. This departure ratio is considered to be significant which calls for an immediate check to determine the abnormal changes of the parameter settings. Using a MCPCA chart, the practitioners can thus effectively control and monitor both the variation and the departure from its target for each single process characteristic.

0.350.30.250.20.150.10.050". -1

, -0.5

-0.25

, 0

. 0.25

. 0.5

. ~dr 1 dr

Figure 9.5. A MCPCA chart with contours of Spk = 1.00, 1.33, 1.50, 1.67, 2.00 (from top to bottom).

In a MCPCA chart (see Figure 9.5), we use standardized measures of process departure and process variation. Consequently, a MCPCA chart can be used for processes with multiple

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characteristics where the individual characteristic specifications are not identical. The MCPCA chart displays all characteristic index values on one chart, and provides the characteristic yield based on the Spk contours. It can present instantly the information about the locations and spreads of all the characteristics under study by their (Cjr, Cdp) - coordinate values. The MCPCA chart also displays the relative magnitudes of the process variation and process departure in terms of the standardized measures (Crfr, C
9.5 The Le MPPAC As we have already discussed in Chapter 7 it were Hsiang and Taguchi (1985) who first introduced the loss function approach to quality improvement which focuses on reduction of variation around the target value. Johnson (1992) introduced the relative expected loss Le for symmetric cases, to provide numerical measures of process performance for industrial applications. Tsui (1997) interpret Le = Lpe + Lot , which results in an uncontaminated separation between the information concerning the process relative inconsistency loss ( Lpe ) and the process relative off-target loss (Lot). The index Le is defined as a ratio of the expected quadratic loss and the square of the half specification width as follows:

=/:

(x - Tf d2

dF(X) = r-\ + dl

ti-T \ d

2

Multi-Process Performance Analysis Chart (MPPAC)

177

where F(-) is the c.d.f. of the measured characteristic. The first term (cr / d)2 is denoted Lpe , and the second [(/i — T) I df is denoted Lot , and Le can thus be rewritten as Le = Lpe + Lot. The mathematical relationships: Le = (3C p m )" 2 , Lot = (1 - Ca)2 , and Lpe = (3C p )~ 2 can easily be established as indicated in Chapter 5. The advantage of using an equivalent index Le over Cpm is that an estimator of the former has superior statistical properties than that of the latter, since the former does not involve a reciprocal transformation of the process mean and variance. (See Chapter 4, Section 2 for a definition of Cpm.) The index Le also provides an uncontaminated separation between information concerning the process precision and the process accuracy. This separation indicates a direction practitioners may follow in modifying the process parameters to improve the process quality. Some commonly encountered values of Le are: 1.00 (a process is incapable), 0.44 (a process is incapable), 0.11 (a process is usually referred to as capable), 0.06 (process is called satisfactory), 0.05 (process is normally called good), 0.04 (process is normally called excellent), and 0.03 (process is designated as super). The corresponding Cpm values are also listed in Table 9.3. Table 9.3. Some commonly used Le values and the equivalent Cpm . Condition

Le

Cpm

Incapable

1.00

0.33

Incapable

0.44

0.50

Capable

0.11

1.00

Satisfactory

0.06

1.33

Good

0.05

1.50

Excellent

0.04

1.67

Super

0.03

2.00

The sub-index Lot measures the relative process departure, which as mentioned above, has been referred to as the relative offtarget loss index. The sub-index Lpe, as it was already pointed out,

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measures process variation relative to the specification tolerance, and has been referred to as the relative inconsistency loss index. Some commonly used values of Lpe are: 0.11, 0.06, 0.05, 0.04 and 0.03. The corresponding quality conditions are indicated in Table 9.4. Evidently, theses 5 values of Lpe are equivalent to the Cp values of 1.00, 1.33, 1.50, 1.67, and 2.00, respectively, thus covering a wide range of the precision requirements needed for numerous real-world applications. Table 9.4. Some commonly used precision requirements. Quality Condition

Precision Requirement

Incapable

Lpi = 1 00

0.11 <

Lpe

Capable

0.06 < Lpe

< 0.11

Satisfactory

0.05 < Lpe

< 0.06

Good

0.04 < Lpe

< 0.05

Excellent

0.03 < Lpe

< 0.04

Super

Lpe

Target

LSL

U = 1.00 Ui = 0AA i-o,= 0.11

J

< 0.03

USL

tof=t).11

Ut = 0.AA ^ = 1.00

Figure 9.6. The Le MPPAC.

Pearn et al. (2003) using the expected loss index Le introduced a novel control chart, called the Le MPPAC. Based on the definition, Le = (// - T) 2 / d2 + a2 / d2 , we set Le = k for various k values, and then determine a set of (/i, a) values satisfying the equation: (/U - T)2 + a2 = kd2 which can be

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179

plotted on the contour (a curve) of Le = k . These contours are semicircles centered at fi = T with the radius yfkd . Analogously to the situation encountered in the case of the Cpm index, the more capable is the process, the smaller is the semicircle. The seven contours on the Le MPPAC for the seven Le values listed in Table 9.3 are shown in Figure 9.6. Note a certain clustering of the semicircles for smaller values of k . A Le MPPAC has the following properties worth of mentioning: (1)

As a point gets closer to the target, the value of Le becomes smaller, and the process performance improves.

(2)

For the points inside the semicircle of the contour Le — k, the corresponding Le values are smaller than k. Analogously, for the points outside the semicircle of contour the Le = k, the corresponding Le are larger than k.

(3)

For processes with fixed values of Le, for the points within the two 7r/4 lines envelop, the variability is due mainly to the process variance.

(4)

For processes with fixed values of Le, for the points outside the two 7r/4 lines envelop, the variability is due mainly to the process departure.

(5)

The perpendicular and parallel lines passing through a plotted point intersect the horizontal and vertical axes at the points that represent its Lot and Lpe, respectively.

(6)

The distance between T and the point, at which the perpendicular line through the plotted point intersects the horizontal axis, denotes the departure of the process mean from the target.

(7)

The distance between T and the point, at which the parallel line through the plotted point intersects the vertical axis, denotes the departure of the process variance.

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In addition, Pearn et al. (2003) obtained the exact upper confidence bound for the Le to provide a measure of the maximal expected loss of a process based on the sample data. The proposed decision - making procedure ensures that the risk of making a wrong decision would not exceed the preset Type I error. The Le MPPAC displays multiple processes with the mean departure and the process variability relative to the specification tolerances on a single chart, which is very appropriate for manufacturing capability control of a group of processes in a multiple process environment. Also the Le MPPAC prioritizes the order in which a further capability improvement effort should be accomplished: either to move the process mean closer to the target value or to reduce the process variation.

Chapter 10

PCIs with Asymmetric Specification Limits

10.1 10.2 10.3 10.4 10.5 10.6

Introduction The Cpk index for asymmetric tolerances The Cpm index for asymmetric tolerances The Cpn index for asymmetric tolerances The Crmk index for asymmetric tolerances The Loss index for asymmetric tolerances

10.1 Introduction

The concert of symmetric tolerance was used tacitly in this beak on numerous occasions. A process is said to have a symmetric tolerance if the target value T is set to be the mid-point of the specification interval [LSL, USL], i.e. T = M = (USL + LSL) / 2 . Most research in quality assurance literature has been focused so far on cases in which the manufacturing tolerance is symmetric. Examples include the earlier paper by Kane (1986), Chan et al. (1988), Choi and Owen (1990), Boyles (1991), Pearn et al. (1992), V a n n m a n (1995), V a n n m a n and Kotz (1995), Spiring (1997), P e a r n et al. (1998), P e a r n et al. (2003a, 2003b) among many others. Granted t h a t cases with symmetric tolerances are quite common in practical situations, cases with asymmetric tolerances ( T vt M) (see Figure 10.1) also often occur in the manufacturing industry. F r o m the customer's point of view, asymmetric tolerances reflect t h a t deviations from the target t h a t are less

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tolerable in one direction than the other (see Boyles (1994), Vannman (1997), and Wu and Tang (1998)). Usually they are not directly related to the shape of the supplier's process distribution. Nevertheless, asymmetric tolerances can also arise in those situations where the tolerances are symmetric to start with, but the process distribution is skewed or follows a non-normal distribution. To deal with this the data are usually transformed to achieve approximate normality. A prominent example of this approach is Chou et al. (1998) who have used the well known, time - honored Johnson (1949) curves to transform the non-normal process data. Unfortunately, there has been so far comparatively few research publications dealing with asymmetric tolerances. (Exceptions are Choi and Owen (1990), Boyles (1994), Vannman (1997), Chen (1998), Pearn et al. (1998, 1999), Chen et al. (1999), and the more recent Jessenberger and Weihs (2000)).

OS

0*

Loss 04

0]

-n

t

35

Figure 10.1 Assymmetric tolerance.

10.2 The Cpk 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

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

USL-T 3a

Cpk -=

T - LSL ' 3a

min[C*pu,C*pi } ,

(10.1) (10.2)

where

r —

USL-T 3a

1-

I V-T\ USL-T

)

> Cpi

—

T

-LSL 3a

\H-T\ T -LSL,

respectively. Observe that the index Cp 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 onesided 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:

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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{D u ,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
PCIs with Asymmetric Specification

185

Limits

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

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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* L

c;k = -^-,

(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

187

Limits

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

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

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)/(USL -T)f,

T < x < USL,

1

<x
otherwise.

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Figure 10.3. An asymmetric loss function corresponding to the asymmetric tolerance (LSL, T, USL) = (10, 40, 50).

Figure 10.4. processes with
values for 10 < fj, < 50 and a = 2 0 / 3 (LSL, T, USL) =

Figure 10.3 displays a typical loss function for processes with an asymmetric tolerance (LSL, T, USL) = (10,40,50) . Figure 10.4 displays the plots of C'^ for processes with 10 < /J, < 50 , CT = 1 0 / 3 (top) and o = 2 0 / 3 (bottom), where (LSL,T,USL) = (10,40,50) is tilted towards USL. It is evident that index C'^ takes into account the asymmetry of the loss function. Thus, given two processes E and F with fiE > T and fiF < T , satisfying (/j,E -T)/Du = (T - fiF) /Di (i.e., processes E and F have equal "departure ratios"), C'^ values given to the processes E and F are the same if aE = aF . For example, consider the processes E and F with \xE = 45 > T and \iF = 25 < T. Clearly, the corresponding departure ratios are 1/2 for both processes: ((45 - 40)/10 = (40 - 25)/30 = 1/2). We thus have Cpk = 0.50 for both processes with aE = aF = 1 0 / 3 , and C'^ = 0.25 for both processes with aE = oF = 2 0 / 3 (see Figure 10.4.). In addition, the index C"k decreases as the mean fi shifts away from target T in either direction. In fact C ^ decreases faster when ytx moves away from T to the closer specification limit than to the further specification limit. This is an advantage since the index would respond faster to the shift towards "the wrong side" of T

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189

as compared with the shifts towards the middle of the specification interval. To estimate the index Cpk , Pearn and Chen (1998) propose the natural estimator: -A* 35

r" — where

A*

=max{d*(x-T)/Du,d*(T-x)/Dl},

(Compare with A , in Equation (10.5)). As usual, x=^2. Xi /n and s = [^._, (XJ — x)2/(n—l)]1'2 are the sample mean and the sample standard deviation, respectively. These can be obtained for a process that is "demonstrably" stable (well in control). In the case when the production tolerance is symmetric A reduces to | x — T | and the estimator Cpk becomes: Cpk = mm{(USL -x)/

3s, (x - LSL) / 3s},

i.e. the natural estimator of Cpk discussed in Kotz et al. (1993). Therefore, we may view the estimator C'v\ as an extension of Cpk . Assuming that the process is normally distributed, the estimator Cpk can be rewritten as

Cnk

W 2 # - i / 2 c[_ (n - i 1) a

,1/2

max

K = (n — l)s 2 / o2 is distributed according to the distribution, and Z = nl'2(x — T) / a is distributed as XnN(6,l) with 6 = nl/2(/j,-T)/a . Furthermore, since x and s are mutually independent (in the normal case), so are the Z and K. To derive the expected value and the variance of Cpk , we first need to evaluate:

where

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-Z A

E

-

I

—

—

1 2

~ Du + A ; (27T) /

and E max*

U

ID.' A J)

+

A

exp

' 2

f <5

/ <5 1

vA.

A <S2^

2

Dl

A2

$(- |6

c2 ^

+

A2

A2;

(27T) 1 / 2

exp

+2

U

[1 - 2 * ( - | 8 |;

2

{^}) (!-»(-!*

max<

The rth moment (about zero) of C'v\ can then be expressed as

E(c;ky = (n x£

1)r/2

max

£(^ r / 2 )E Du ' A J,

r-j

cr

n

1/2

(10.6)

Note that C'v\ is a biased estimator of C ^ . Some numerical values of E(Cpk) and Var{Cpk) are presented in Table 3-5 of Pearn and Chen (1998). A few representative values are given below. The resultant bias is negative for all the cases in the Pearn and Chen (1998) investigation as long as /J, ^ T . For \x = T , the bias is positive for n — 10 but becomes negative for larger values of n . For d j a = 2.0 it is negative for all n > 20 ; for d Io = 3.0 it is negative for n > 30 ; for d /a = 4.0 , it is negative for n > 40 and for d / a = 5.0 , it is negative starting with n > 60 . As n becomes very large, the bias becomes 0. Note that when the tolerance is symmetric, the results are the same as those provided by Kotz et al. (1993). (c.f. Chapter 3)

PCIs with Asymmetric Specification

191

Limits

We shall now denote B = nl,2{d* / a) and Y = 2 [m&x{(d / DU)Z,— (d / Di)Z}] . Using these notations C'^ can be expressed as

C';k = [ V ^ I ( 5 -

<JY)\/{MnK).

Let, as above, <&(•) and (/)(•) be the c.d.f. and the p.d.f. of the standard normal distribution N(0,1), respectively. The c.d.f. and the p.d.f. of the variable Z distributed as N(S, 1) can be expressed as: Fz(z) = $(z — 6) and cj){z — 6) , respectively. Hence, the c.d.f. of Y is given by: FY{y) = $[(£>„ /d*)4y -6}-

$[-(Dt /d*)Jy

- 6} ,

(10.7)

and the p.d.f. is fr(y) = —^-(Du4>[{Du 2d V^ + Dt4[(Dt/d*)Jy

Id*)Jy

- 6} (10.8)

+ 6]).

Therefore, the c.d.f. of the estimatior C'^ can be expressed as follows (Pearn et al. (2003)).

f FK (L(x,y)) fY(y)dy, FA„

(£) =

l-Fy(B2),

x < 0, x = 0,

JFK(L(x,y))fY{y)dy,

(Pearn et al. (2003)), and the p.d.f. of C'^ becomes

x>0

(10.9)

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/ / * ( £ ( * , BH))fy(BH)

k;Sx) =

'dt,

)

X<0,

,

JfK(L(x,

BH))fY{BH) —

(10.10)

-dt, x>0, x

. o

where, L(x, y) = (n - 1){B - Jyf /(9nx2) , i ^ ( - ) and fK(-) are c.d.f. and p.d.f. of K, respectively, FY(-) is the c.d.f. of Y given by (10.7), and /y(-) is the corresponding p.d.f. (10.8). In the cases symmetric manufacturing tolerance (T = M ),DU = Dt = d = d*, B = n 1//2 (d / a ) = D ; consequently the c.d.f. of Cpk given by (10.9) is reduced to (3.1) A decision making rule for testing Cpk Using the index C'v\ , the engineers can access to the process performance and monitor the manufacturing processes on a routine basis. To obtain a decision making rule one can set a testing hypothesis with the null hypothesis Cpk < C for a given constant value C (the process is incapable) and the alternative C'v'k > C (the process is capable). The null hypothesis will be rejected if Cpk > cQ J where the constant ca (called the critical value) is determined so that the significance level of the test is a , i.e., P{Cpk > ca | Cpk = C) = a . The decision making rule will then be: for given values of the risk a and the sample size n , the process will be declared capable if Cpk > ca and incapable otherwise. By using the familiar designation £ = (// — T) / er and b = d I a, the index C'v\ can be rewritten as: C;k =[& + £min{l,r}]/3,for £ < 0 and

Clk =

[b-£ min{l, r}} / 3 for £ > 0,

where r = Dt / Du . Hence, the value of C ^ can be calculated from the values of £, b, and r . Specifically, if the triple ( £ , b, r) = (-1, 3, 3/2), C'lk = [3 + (-l)/max{l, 3/2}]/3 = 7/9 = 0.7778. Ssetting C^ = C , we arrive at b = 3C — £ / max{l, r} for £ < 0

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193

and b = 3C + £/max{l,r} for £ > 0. Recalling that B = n 1 / 2 x (d* /CT) and 6 = d j a we have 5 = n&2. Therefore, for Cp'k = C ,

2

B

n ( 3 ( 7 - £ / m a x { l , r}) 2 , £ < 0 (10.11)

= rc(3C + £

2

min{l, r}) , £ > 0.

Using the definitions of £ and n we use the (central) chisquare and the normal distributions to determine the critical value ca satisfying P(Cpk > ca \ C'p\ = C) = a . The last equation is equivalent to 1 — F^,„ (ca) = a given C'p'k = C . Observe that ca is in general larger than zero, hence we can find ca from Equation (10.9) B2

JFK(L(ca,y))fY(y)dy

= a,

(10.12)

o where B2 is given in (10.11) and L(ca,y) = (n — 1)(B - J~y)2 x (97JC 2 )- 1 in ( 1 0 . 1 0 ) .

As it is the case in the previous cases discussed above the critical values ca for £ = £0 and £ = — £0 are the same, provided T = M . To test if the process meets a capability (quality) requirement, we shall first determine the value of C and the a -risk. Since the process parameters fi and a are unknown, so is the parameter £ = (fj, — T) / a. However, £ can be estimated by calculating £ = (x — T) / s from the sample. If the estimated value Cpk is larger than the critical value ca ( C'p'k > ca ), we conclude that the process indeed meets the capability requirement ( Cpfc > C ). Otherwise, there is no sufficient information to conclude that the process meets the present capability requirement. In such cases, we would tend to believe that C'p\ < C (the process is incapable).

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Following the standard time-honored approach we calculate the p-value, i.e. the probability that Cpk exceeds the observed estimated index given the values of C , £ , r = Dl / Du , and sample size n . This probability is then compared with the significance level a . If the (estimated) C'p' value is c* (given the values of C , £, r , and n) the p-value can be calculated as p - value = P(C'Jk > c \ Cp'k = C) = 1 - %sph( c * ) B2

+ ^4&Jy a a

+ tJn]]dy,

(10.13)

where Du / d = 1 / min{l, r} , Dt / d = max{l, r} (recall that d = mm(Du, De) ), B2 is given in (10.11) and L(c ,y) in (10.10)). The rather cumbersome calculations in (10.13) can be carried out by means of numerical integration. If the j>-value turns out to be smaller than the a -risk, we than conclude that the process meets the capability requirement (C'v\ >C). Otherwise, as above, no sufficient information is available to conclude that the process meets the current capability requirement. In this case, we would tend to declare that Cpk < C (namely that the process is incapable). A simple example could clarify the procedure described above. Consider a normally distributed process with asymmetric specification tolerances LSL = 20, T = 26.5, and USL = 32. Hence d =(USL - LSL)/2 = 6 , D, =T - LSL = 6.5 , Du =USL - T = 5.5, d* = min{D 0 , A } = 5.5 , and r = Dt / Du = 1.18 . To test whether the process meets the capability (quality) requirement, we first assign C = 1.33 , i.e., we define a process with Cpk > 1-33 as capable. With the sample size n = 100 , the sample mean x = 27, and the sample standard deviation s = 1.10, we obtain A* = m&x{d*(x - T)/Du,d*(T - x)ID{\ = 0.5 , |

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195

= (x — T) / s = 0.45 and finally Cpk = 1.515 . By using numerical integration to calculate the r.h.s. of equation (10.13) we arrive at the p-value of 0.055. We thus conclude that the process meets the capability requirement provided the a -risk is set to be larger than 0.055. If the a -risk is set smaller than 0.055, we again do not have sufficient information to assert that the process meets the current capability requirement.

10.3 The Cpm index for asymmetric tolerances Recall (Chapter 4, Section 1) that Chan et al. (1988) proposed the capability index Cpm using in the denominator r 2 = E(X - T)2 instead of a2 = E(X - nf . This index takes into account the proximity of the process mean to the target value and as well as the process variation. For asymmetric cases {USL-T ^T - LSL ), Chan et al. (1988) modified Cpm as follows. Simply shift appropriately one of the two specification limits, so that the new (shifted) specification limits are now become symmetric with respect to the target value T : C*pm = min

USL-T

3r

T - LSL

'

d* 3V
3r (10.14)

T)

where as above d = min{Du,Di} , Du =USL — T , and A =T - LSL . Obviously, if Du =Dt , then T = M = (USL + LSL)I'2 and d* = d =(USL - LSL)/2 , the specification tolerance becomes symmetric and the index C*m defined in (10.14) reduces to the original index Cpm defined in Equation (4.1). Boyles (1994) also notes that this generalization could understate process capability by restricting the process to a proper subset of the actual specification range. For the two processes E

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and F with —fj,E < T , and jiF > T - satisfying the relationship | fJ-F — T | = | T — nE I (an equal absolute departure) - the index values calculated for the processes E and F are the same. Consider now specifically the two processes E and F with the target value T = (3USL + LSL)/A , /j,E = T - 0.5d = M , fiF = T + O.bd = USL , and aE = aF = d / 6 . In this case, we have | MF — T | = | T — nE | = 0.5c? and obtain the same C*m = 0.316. However, process E is eventually substantially better than process since the expected proportions of non-conforming items are approximately 0% and 50% for the processes E and F, respectively. An example for USL = 50 , LSI = 10 , T = (3USL + LSL) / A = 40 , ^ =T - 0.5d = 30 , fiF = T + 0.5d = 50 , and aE = aF = 1 0 / 3 is illustrated in Figure 10.5. Consequently, the index C*pm in appropriately measures process capability in these cases.

Figure 10.5. A example ioiUSL = 50 , LSL = 10 , T = 40 , [iE = 30 , UP = 50 , and aE = o> = 10/3 .

Definition (10.14) shows that Cpm assesses the distance of JJ, from T and the asymmetric specification limits. A natural estimate of Cpm can be defined as

PCIs with Asymmetric Specification

C*pm =

USL-T

m i n

197

Limits

T-USL 3f

3r

where f2 = [^"_ fa - T)2 /{n - 1)] 1/2 . When the process measurements follow a normal distribution, then, for given values of the parameters T , USL , and LSL , the p.d.f. of Y = Cpm is a variant of the inverted non-central x 2 distribution (note f2 appears in the denominator) and is given by:

My) =

9n/2

-exp

(("-1)3pra

y

y

. -/(n-l)C^sfl+,-

7 V(-

+A

E j!r(f+

, y > 0,

J)2 2J '

where A — n(fi — T)2 / a2 . The Cpm index can also be used to assess the process capability of the unilateral specification limit cases by letting LSL = — oo when only an upper specification limit exists and correspondingly letting USL = oo when only a lower specification limit is finite (see Chan et al. (1988), Stoumbos (2002)). Motivated by G. Taguchi's (1986) strong recommendation to apply the loss function to quality assessment procedures, Boyles (1990) generalized the index Cpm for asymmetric tolerances as follows: +

pm

_ USL - LSL

~

(10.14)a

6jE(L^sj

where E(Loss) denotes the expectation of the loss function

Loss(X,T)

k^X -Tf,X

< T,

kziX -Tf,X

> T.

where fc, (i = 1,2) are "standardizing" constants. We have:

(10.15)

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E(Loss) =a2 [(l+e ){h ( l - * ( 0 ) + * & * ( £ ) } - ( * a - W ( 0 ] . e = (/i-T)/(7,

h = fa/ri)ko, h={ri/r2)k>, n=(T

(10.16)

- LSL) /(USL - LSL),

r2 = (USL - T) /{USL - LSL), k0 = max(ri /r2,r2/

n)/[2(r^ + r 2 2 )], and

$(•) and (•) are the c.d.f. and the p.d.f. of a standardized normal variable, respectively. In the symmetric case when r i — r2 = 1/2 , E(Loss) — T2 and hence Cpm reduces to Cpm . Note that k0 is determined in a manner so that Cp = Cpm when \i = T (see Choi and Owen (1990)). As we have mentioned several times the indices Cpm , Cpm and Cpm consider not only the process variation but also the proximity of the process mean to the target value. However, they still do not take into account the location of the process mean in the basic interval [LSL, USL]. The Cpm index has an additional advantage in that it uses an asymmetric loss function to measure differently the capability at /U = T — h and // = T + h for a constant h > 0 in the case when USL — T ^ T — LSL . A drawback is that the calculations of E(Loss) in (10.16) are somewhat complicated, even when utilizing current truly "mighty" computer "capabilities". (Not to be confused with the capabilities in the sense used in this look!) The new generalization Cpm, which assumes a larger value for a more capable process (as the traditional indices Cp , Cpk , Cpm, and Cpm do), have been developed by Chen et al. (1999) to be:

PCIs with Asymmetric Specification

Limits

199

where d*=min{£>u, A } , A* = max{d\n - T)/Du,d*(T - n)/ Dt} , Du =USL-T and Dt =T - LSL (Compare with C'v'k in (10.5)). Obviously, if T = M (a symmetric tolerance), then d = d , A = | n — T | and Cp'm reduces to the original index Cpm . The index C"m is > 0 when the process mean falls within the tolerance limits. However, according to the modern quality improvement theories, reduction of variation from the target is as important as meeting the specifications requirements. The factor A in the definition ensures that the generalization Cpm achieves its maximal value at /j, = T (the process is on-target) regardless of whether the tolerances are symmetric ( T = M ) or not ( T ^ M ). Furthermore, for the processes E and F with a E = &F) MB > T , /JLF < T , satisfying the equal departure ratio relationship (/j,E — T)/ Du = (T — fip)/ Di , the index values assigned to the processes E and F are the same. In fact, the value of Cpm decreases faster, when fi shifts away from T , towards the closer specification limit than that to the farther one. In particular, for processes E and F with <JE = aF, /z# = LSL, and UP = USL, the index values of C'v'm assigned to the processes E and F are the same. In this special case, for normal processes with fiE = LSL and fip = USL , the same index values indicate the same expected proportions of nonconforming items. Furthermore, given Cpm > c , we can obtain a bound on | fi — T \: T - [(1 - R)/3c]Di < ii < T + [(1 - R)/3c]Du

,

(10.18)

where R = \ 1 - r | / ( l + r) and r = Dt / Du . As we have mentioned above, the index C*m cannot differentiate between processes capabilities in a accurate manner because the index values of C*m for two processes with equal variance and equal absolute departure are the same. This measure is therefore not very reasonable, especially for processes with asymmetric tolerances. On the other hand, the index values of Cpm assigned to processes with equal variance and equal departure ratio are the same. In this case, both processes have an

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equal average loss. Therefore, it can be concluded that Cpm is superior to C*m based on the criteria of process targeting which is related to process loss. Let Xi,x2,- •-, xn be a random sample taken from a normal distribution N(fi, a2). To estimate the new generalization C" , we propose the natural estimator given by <%»=

,f

>-,

(10-19)

where x = ^ . _ xi / n , A = max{d(x — T)/ Du, d(T — x) / Dt} and s\ — ] P _i (xi ~ %)2 In • This estimator should be obtained from a process that is "demonstrably" stable (in-control). If the production tolerance is symmetric, A is simplified to | x — T \ and the estimator C'pm becomes

which is the natural estimator of Cpm discussed in Boyles (1991). The natural estimator C'p'm (10.19) can be rewritten as:

^-iznr-

(ia20)

where R = n1/2d* j a, K = nsl/a2, and Y = [max.{{d/Du)Z,-(d/Dl)Z}f with Z = nll2{x - T)/ a. Under the normality assumption, the statistic K is distributed as Xn-i > % as N(6,l), 6 = nll2(n — T)/ a and the p.d.f. of Y is expressed in terms of /y(-) as:

+ >0 ^)=2^(^(-^/*) ' 2Vy [ 4 " V ^ / - / • ^^ ( ^ / * ) ] ^

(1 21)

°-

where d\ = d / Dl and (4 — d / Du regardless of the specific distributions. Hence, the p.d.f. of Cpm can be expressed as:

PCIs with Asymmetric Specification II 3

fd" 0*0

c.pm •1 1 /if = 27z 4 Jo Jo7T VI X

dy

[

Cpm (1

—

201

Limits

*)

2

9x

k' „pm m V £

+ -rfz

3xdi

3x^2 J

dt, x > 0

(10.22)

(See Chen ei a/. (1999) for details.) Since the statistic Z2 follows a non-central chi-square distribution with one degree of freedom and the non-centrality parameter 62 , Chen (1998) defines the distribution of Y as a weighted non-central chi-square distribution with one degree of freedom and the non-centrality parameter ( 62 under the assumption of normality). This author also presents the p.d.f. of Y , in an alternative form (Equation (10.21)):

fr(y)

e~x'2

(V2gy r /i + j

2V7T

, i=l

y>0.

(10.23)

a

i

(Recall the connection between the central and non-central x 2 distributions.) Here A = <52 and Yj is distributed as Xi+j • Hence, the p.d.f. of Cpm can also be explicitly expressed as: 2^-n12

fc- (*) = d-U+V

rjnx~(n+l)

3T((n-l)/2) r\1_y){n-3)/2yU-D/2 J0

-A/2

uscy

2SKJ^J\\SX}

U

e x p f " ^2 (1 - y + d~2y) 18x

a; > 0. (see Appendix A of Chen et al. (1999))

•dy

(10.24)

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If the production tolerance is symmetric (i.e. dx = d± = 1) the p.d.f. of Y reduces to the ordinary non-central chi-square distribution with one degree of freedom and the non-centrality parameter A = 82, and the p.d.f. of C"m becomes nl—n/2/~
/
-exp

AC 2 ) 2 k36x

jlT

3"n x„n+l oo

C2 18z2

A 2

[2+j

n

•, x > 0.

(10.25)

3=0

which coincides with the p.d.f. of Cpm (as given by Vannman and Kotz (1995)). Observe that the estimator Cpm is biased. The magnitude of the bias is Bias(Cpm) = E(Cpm) — C'p'm . The mean square error of Cp'm can be expressed as MSE{Cv\n) = Var(C'pm) + [Bias(Cpm)f . The bias and the mean square error of C'pm where calculated by Chen et al. (1999) for various values of £ = (fi — T) / a , d j a , di = d j Di, dq = d j Du , and the sample size n. The results show that as | £ | increases, the MSE decreases. As the sample size n increases, both the bias and the MSE decrease. Furthermore, when the value of d J a increases, both the bias and the mean square error increase for fixed d\, dq, £ and n .

10.4 The Cpn index for asymmetric tolerances Choi and Owen (1990) proposed a process capability index Cpn that takes into account all the three factors, the process variation, the proximity of the process mean to the target value and the location of the mean in the interval [LSL, USL] . The index is given by Cpn = mm{CPNU,

CPNL] ,

PCIs with Asymmetric Specification

where CPNU =

USL

~ M , and CPNL =

n

M

3T

203

Limits

~ LSL or

Also, we define Cpn = 0 for CPNU < 0 for CPNL < 0 . It can easily be verified that Cpn

=

Cpm ( 1 — K) ,

where k = 2 | T - \x \/{LSL +USL) as defined in Chapter 2. Similarly to Cpk and Cpfc , Cpn = 0 when k > 1. Choi and Owen (1990) also express Cpn in terms of the loss function as follows

C

"» = lS^TLS\ '

(10 26)

-

where E(Loss) denotes the expectation of the loss function Loss(X, T) = w(^)(X - T) 2 ,

(10.27)

with w(n) — 1/(1 - kf for A; < 1 and w(fi) — oo for A; > 1. The definition of C pn as given in (10.26) is the same as that of Cpm defined in (10.14a) except for the difference between the loss functions as defined in (10.15) and (10.27). Observe that the weight function W(/JL) reflects the centering of the process mean via the quantity \ M — fi \ while the indices Cpm , Cpm and Cpm do not account for | M — [i \ in their definitions. Choi and Owen (1990) provided an estimator of Cpn in the usual manner as: _ . \USL-x x-LSL] Cpn = mm | — , — J,

(10.28)

where, as above, f is the estimator of r obtained by replacing /j, and a2 by their UMVUE estimators x = ^ ajj / n and 2 12 s = [^._ (xi — x) /(n — I)] ' , respectively. The estimator C ^ i s

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evidently consistent. For a given positive constant probability that Cpn > w can be expressed as:

w , the

P(Cpn > V>)

P

Z + Vn(Z\ 3yl^K + (Z + yR)2

> w, • , ^-^lK

-Z + -Jndv + {Z + 4<)

>w

(10.29)

where as above dj = (/z — LSL) / a , d? = (USL — fi) / a and £ = (/J, — T) I a . Also the random variables Z and K have the standard normal and the chi-square distributions with n — 1 degrees of freedom, respectively, and are independent. After some algebraic manipulations, for w > 1 / 3 , the probability in (10.29) can be expressed as P(CP„ >w) = P{0
D,A(K)
B(K)},

(10.30)

where D=

A(K) =

-Jn l-9w2

{n-l)

K +
(dlj - rfi + 2£)2

36wJ

•{(dw^-di)

+ V(9w2£ - dx)2 - (1 - 9«/2)[d? - 9^ 2 {ir/(n - 1) + e 2 }]},

and fl(tf)

Vn •{(Q^-dx) l-9w2

-V(9tw2C " ^i) 2 - (1 - ^2)[di

- 9w2{K/(n

- 1) + e 2 }]}.

(see Choi and Owen (1990) for details). Since the joint density function of the independent variablese K and Z is g(k,z) =

2 1 l .(n-3)/2 e -(ft+z )/2 ) V2^r[*j*]2(n- l)/2

random

PCIs with Asymmetric Specification

Limits

205

where 0 < k < oo , and —oo < z < oo , the probability in (10.30) can be represented as P(Cpn >w) = J o j =

m

g(k,z)dzdk

T[^-^fo\^m)-*{A{k))}k^l\-*l*dk.

(10.31)

There seems to be no closed form for the integral in (10.31). For a given a , however, the values of w can be obtained by means of numerical integration satisfying P(Cpn > w) = a . 10.5 The Cpmk index for asymmetric tolerances Recall that the index Cpmk is more sensitive to departure of the process mean from the target value T than the other three indices Cp , Cpk , and Cpm . Along these lines Pearn and Chen (1998) considered a generalization of Cpk for processes with asymmetric tolerances. The generalization takes into account the asymmetry of the tolerance, which would reflect the process capability more accurately than the original Cpk . Applying the same idea Pearn et al. (1999) proposed a capability index Cpmk , which is a generalization of Cpmk , to deal with processes possessing asymmetric tolerances. The generalization Cpmk is defined as Cpmk —

; 0 -') 3V
(10.32)

where d = {USL - LSL)/2,A = max{d(/x - T)/Du, d(T - /i)/D t } , Du =USL-T , Dt = T - LSL , d* = mm{Du,Dt} , and A* = Taax{d*(n-T)/Du,d*(T-fi)/Di} . Evidently, if T = M (a symmetric case), A = A* =\ ^ — T \ and Cpmk reduces to the

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original index Cpmk. We note that Cpmk > 0 for a process with mean fi falling within the tolerance limits, analogously to the indices Cpmk , Spmk , and the yield-based index Cpk . However, as we have emphasized above, according to the modern quality improvement theories, reduction of variation from the target is as important for an index as meeting the manufacturing specifications. In fact, being on-target would be a highly desired condition for a process. The factors A and A ensure that Cpmk obtains its maximal value at [i = T (process is on-target) regardless of whether the tolerances are symmetric (T = M ) or not (T ^ M). Furthermore, for processes E and F with aE = aF, HE > T , fiF < T , satisfying the relationship (/j,E — T) / Du = (T — fxp)IDi (i.e., having the same departure ratio), the index values for the processes E and F are identical. Actually, the value of Cpmk decreases faster when n moves away from T to the closer specification limit, but decreases slower when fi moves away from T to the farther specification limit. Since under the normal assumption the Yield > 2Q(3Cpk) — 1 and C'p'mk < Cpmk < Cpk , given the value of Cpmk , one can calculate a lower bound of the process yield as 2Q(3Cp'mk) — 1 (Boyles (1991)). For example, given a process with capability Cpmk — 1 ) the process yield is guaranteed to be no less than 2<1>(3) — 1 = 99.73% . On the other hand, an upper bound on the departure ratio can be calculated as A/dT , and is A/d = (T - fi)/Dt for /x < T. To estimate the index Cpmk , Pearn et al. (1999) considered a natural estimator Cpmk defined as follows. First replace the process mean \i and the process variance a2 by their conventional estimators x and s2. , which are obtained from a stable process. Then the estimator is given by

PCIs with Asymmetric Specification

C';mk =

207

Limits

d* -A* » ^ A ^ 2,

(10.33)

zjst + i

where d = min(Du, D{) , A = max{d(x — T)/Du,d(T — x)/D{\ , J4 = max{d (x — T)/Du, d (T — x)/D[} , and x and s 2 as above. If the manufacturing tolerance is symmetric, d = d , A = A =\x—T\ , and the estimator Cpmk reduces to Cpmk = (d-\x -m |)/{3[s 2 + (x - T) 2 ] 1 / 2 }, which is the natural estimator of Cvmk considered by Pearn et al. (1992) for symmetric case. Define now D* = n1/2d* /a , D = nl/2d/a , K = ns2n /a2 , Z = nll2{x-T)/a , Y = [max{(d / DU)Z, - (d/ D^Z})2 , 1 2 2 8 = n / (/i — T) j a , and A = 8 . Using these definitions, the estimator Cvmk can be rewritten as: &,

Cpmk -

_D*-(d*/d)JY ^K + Y •

(10-34)

It remains to recall that under the normality assumption, K is distributed as Xn-i ( a chi-squared variable with n — 1 degrees of freedom), and Z obeys the normal distribution N(6,l) with mean 6 and variance 1. Let as above <&(•) and (/>(•) be the c.d.f. and the p.d.f. of the standard normal distribution JV(0,1) , respectively. Then, the c.d.f. and the p.d.f. of Z can be expressed as: Fz(z) = $(z — 6) and fz(z) = 4>{z - 8) respectively. Consequently, the c.d.f. of Y is given by: FY(y) = P(Y

®{-d{l4v

^Jy)

- S).

where dx = d / Dx and d^ = d / Du (Don't confuse with dif i = 1,2 , defined in 10.4). The p.d.f. of Y is then given by

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fy{y) =

f-FY{y) 2Vy

(^""VWr1 w -s) + df Wi'JV

T h e c.d.f. of C'^ FA„

+ s)).

(10

can now be expressed as:

=

(X)

o,

x < --

(D* - u< JS(x)if •K /

Ox1

•

-S(x)t

fY(S(x)t)S(x)dt,

< x < 0, (10

1

1

l-^(d^ D-6)+^(-dr D-6),

I-]FK

(D* - u' JS(x)if

i = 0,

-S(x)t

fy(S(x)t)S(x)dt,

(where if1 = — , d — min(Du,

x > 0,

Dj))

(See Appendix to Pearn et al. (1999)) and the p.d.f. of Cpmfc becomes:

PCIs with Asymmetric Specification

Limits

209

/ <Spink? " » =

JIK

b{X)t

9^

/ctr^2S(x)(D*-u<JS(x)i) fv (S(x)t) i — -9x6

'— dt,

< x < 0,

(10.37)

JfK

(D* -u<JS(xjif - S(x)t 9a? 2S(x)(D*

fy (S(x)t)

,{

9 g

-u'JS(xji) 3 **>

x > 0,

where D* = n1/2d* / a , u' = d* / d , S{x) = [D* /{u' + 3a;)f , F^-(-) is the c.d.f. of K , fx{-) is the p.d.f. of K , and /y(-) is the p.d.f. of Y given by (10.35). Pearn et al. (1999) derive the r-th moment of Cpmk without utilizing the distribution of Cp'mk • Note that the estimator Cpmk is biased. The magnitude of the bias is Bias(C';mk) = E{C';mk) - d;mk , and the MSE is MSE{C';mk) = VariC^ + iBiasiC'^f ^ where Var(C'p'mk) = E{C';mkf [E(Cpmk)f is the variance of Cpmk . The results indicate that the bias of Cpmk is positive when H ^ T . That is, C'p'mk is generally overestimated by Cpmk . On the other hand, when \i = T , we have A = A — 0 and Cpmk — d /(3cr), the bias of Cpmk tends to be negative for certain values (see Pearn et al. (1999)). Thus, C'p'mk may be smaller than Cvmk and the bias be negative when n = T . This is partially due to the fact that both A and A* are calculated to be positive even for \i — T while A = A = 0 . Clearly, the presence of A and A* decreases the value of the calculated C'p'mk . As the sample size n increases, the mean square error of Cpmk decreases. Consequently appropriate sample sizes for capability estimation are essential.

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The smaller the sample size is, the higher value of C'v'mk will be required to ascertain the true process capability. We have mentioned above that Vannman (1994) investigated a general class of capability indices for processes with asymmetric tolerances. Her generalizations have been defined as

3yja2 + v(ju - Tf where u, v > 0 . Vannman (1994) showed that among numerous (u,v) values, the pairs (u, v) = (1,3) and (it, v) = (0,4) generate two indices which are most sensitive to process departures from the target value. For u > 1, the indices Cpa(u,v) decrease when mean fi shifts away from the target T in either direction. In fact, Cpa(u)w) decrease faster when ^ shifts away from T to the closer specification limit than that to the farther one. This is quite an advantage since the index would respond faster to the shift towards "the wrong side" of T rather than towards the middle of the specification interval.

10.6 The loss index for asymmetric tolerances Johnson (1992) developed the relative expected loss Le (discussed in Chapter 5 Section 1) for symmetric case, which is defined as the ratio of the expected quadratic loss and the square of the half specification width: = /.

(x - Tf dF(x) 2 d

( M - T ) 2 + a 2 _ / M - T \ 2 , lu\ + 7 , d2 \ d j \dl

(10.38)

PCIs with Asymmetric Specification

Limits

211

where F(-) is the c.d.f. of the measured characteristic. Under asymmetric tolerances conditions would be risky and it is quite possible that the results obtained may be misleading. Consider the following classical example with the asymmetric tolerance, (LSL, T, USL) where T = (3 USL + LSL)/A and a = d/3. For the processes A and B with \i^ = T — d / 2 = M (the midpoint of the specification interval) and \IQ = T + d / 2 = USL both processes have the index value Le = 13/36 and an equal degree of clustering around the target, namely, | fi — 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. Evidently, Le measures process capability inconsistently in this case, and is inappropriate for the situations with asymmetric tolerances. To overcome this consistency, Pearn et al. (2003) proposed a modification of the expected loss index, which is referred to as L'J , to deal with processes in both symmetric and asymmetric tolerances cases. In developing the generalization, the term | fi — T | in Le have been replaced by A. Adding the terms (d J Du) and {—d/D{) to (fi — T) according to whether /i is greater or less than T, the asymmetry of the loss function is taken into account. Regardless of whether the tolerances are symmetric or not, these modifications ensure that the new index attains the minimal value at /i = T . Additionally, the half specification width d is substituted by d where d is given below. Actually Pearn et al's (2003) generalization of the expected loss index can be expressed as L" =

(A)

2

/

x2

+ [Y) >

( 10 - 39 )

where A = max{(/x - T)d/ Du, (T - fi)d / £>,} , Du = USL - T , Di = T — LSL and d = vnm{Du, D{\ . The quantity d will appear in numerous expressions related to the estimation of Le" . Note that Le is sensitive to target value T and attains larger

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value when T is away from the mid-point between the upper and the lower specification limits (M = (USL + LSL)/2). Denote (A/dJ by L':t , (a/d*)2 by L'Je, obtaining l£ = C + L^ . The terms (d/ Du) and (—d/D{) ensure that if the processes A and B with fiA > T and \xB < T , respectively, satisfy (fiA — T) / Du = (T — fiB)/Di, then the index values obtained for A and B are the same. Evidently for symmetric tolerances ( T = M ), A = \ n - T \ , Du=Dl=d , and d* = d = (USL - LSL)/2 . Accordingly, in such a case the generalization L" reduces to the original index Le. To estimate a loss index L" Pearn et al. (2003) proposes a natural estimator of the form:

L'I = A_ d*

' a

+ |4

\2

(10.40)

where A = max{(X — T)d/Du, (T — X)d/Di} , the mean // is estimated by the sample mean X = ^ J . Xt j n and the variance a2 by Si = YTi=SXi -X? /n, which is the maximum likelihood estimator (MLE). In the case where the production tolerance is symmetric, A becomes \ X — T \ . Therefore, the estimator L" reduces to Le = (n~ld~2)'^2l. (JQ — T)2 , the natural estimator of Le proposed by Johnson (1992) (Chapter 5 Section 3). Consequently, we may view the estimator L" as a direct extension of Le.

The rth moment about zero of L" has also been derived by Pearn et al. (2003) using the properties of a non-central Xv distribution:

E(L':Y r X

E i=0

a nd

e-V2 2VTF

2rr E(^) j=0

r([(l + j ) / 2 ] + i) T([(n + j)/2] +

n+ j

\(di-iy +

+ r

{-iy{df-iy\

(10.41)

PCIs with Asymmetric Specification

Limits

213

where />• = W / O ' O , <4 = d/Du , dh = d/Dx , yu = (y/d2u) , Vi — (y / df) , X = 62 and 6 = V7T(/x — T) / a . In particular, the expected value and the variance of Le are:

E(W) =

({n-l)a nd

2

2 \

+

a n
i+i

U-A/2 2V?f

(I + J K + H W

(10-42)

3=0

and

Var(W) -A/2

(7 2J*4

2V¥

n^d

2 O"

A/2

^

2V?f

nd

(2(n-l)a4)

+

£(W

2/4

l + j (i + i)(3 + j ) [ ^ + (-iy^ 4 ]

°°

E(W

l + j

(i +

j)[dl+(-iydf}

3=0

(10.43)

nd

The estimator Le" is biased as are many of the estimators discussed in this Chapter. The bias of I/e" may be computed in a standard manner to the Bias(L") = E(L") — L" , and the MSE, which is more relevant to an analysis of process quality, is given by MSE{L'^ ) = Var(L? ) + [Bias(W )f . The results in Table 2 of Pearn et al. (2003) indicate that as | a | increases, so do the bias and the mean squared error. Furthermore, as the sample size increases, the bias and the MSE decrease. In those cases when the production tolerance is symmetric, which implies du = du — 1, L" is an unbiased estimator of L" , ( Bias(L") — 0 ). The unbiased estimator depends only on the complete, sufficient statistic (X,S2) for (/i, a2) , by the Lehmann-Scheffe Theorem (Hogg and Craig (1989)) L" is an uniformly minimum variance unbiased estimator (UMVUE) of L".

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Let Xi,X2,...,Xn be a random sample of size n from a normally distributed process N(fi,a2) . Then the p.d.f. and the c.d.f. of Le can be expressed as:

fL^)=J^^fK(Bx(l-t)) o x (di1(f>(di1^Bxi

+ 6) + d^1(l)(d-1yfBxi-6))dt,

(10.44)

FL„(x)= f FK(Bx - y)-L= x {d^dj1 Jy + 6) V2/ o + c ^ V K " 1 ^ - S))dy , for x > 0,

(10.45)

Bx

and

respectively where fK(-) and F^Q denote the p.d.f. and the c.d.f. of K, distributed as %n-i.

A decision

making rule for testing

L"

Based on the sampling distribution of the L" . (Equations (10.44), (10.45)), we shall develop a decision making rule, which can be used for testing capability of a process. Using the index L" , the engineers can access the process performance and monitor and control the manufacturing processes on a routine basis. Thus, we can consider a testing hypothesis with the null hypothesis and the alternative hypothesis to be: H0 : L" > C (incapable), versus Hl : L" < C (capable). The null hypothesis H0 will be rejected if Le" < ca , where constant ca , called the critical value, is determined so that significance level of the test is a , i.e., P(L" < ca \ L" = C) = The decision making rule to be used is then stated as follows:

the the a . for

PCIs with Asymmetric Specification Limits

215

given a-risk (the probability of wrongly rejecting the null hypothesis when it is true) and sample size n, the process will considered to be capable if L" < ca and incapable otherwise. Note that, by defining the characteristic parameters a = (/J, — T) J a and b — d / a , the indices L„t and Lpe can be rewritten as L"t — (dia/b)2 for a < 0, Lpe = (dua/b)2 for a > 0, and Lpe = (1/6) 2 , where du = d/Du, and dt = d/Dt. Hence, the value of L'J = L"t + Lpe can be calculated given values of the four tuple (a, b, du, df). For example, if (a, b, du , d{) = (1, 3, 2/3, 2) 4 " = (2x 1/3)2 + (1/3) 2 = 5/9. If 4 " = C, using 4 " = L^ + L%e, we obtain: (deaf 1 . ^-V 2 - + -o2 for 6 6

^ M , „ (d„a)2 1 a < 0 and C = v V + -5£r ir

In this case b2 = [(dea)2 +1] / C for a < 0 and 62 = [(dua)2 + 1 ] / C for a > 0. Moreover, we have B = nd*2 J a2 = nb2 . Hence, if L" = C, the term B becomes:

and

B = n[(dea)2 +1]/C

for a < 0

B = n[(dua)2 +1]/C

for a > 0.

(10.46)

Furthermore, we have the equality 6 = n 1 ' 2 (/i — T)/a = n1'2a (see (10.41)). With the result of (10.45), we can use the central chi-square and the normal distributions to find the critical value ca satisfying P(L" < ca | L" — C) = a , i.e., F^,(ca) = a given Lg = C, or Bca

J FK{BCU - y)^-={dil(t>{dil4y + -Jna) + d~l(l>{d~ljy - -Jna))dy = a,

(10.47)

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where <j> is the standard normal p.d.f. and B is given in (10.46). Evidently determination of ca may require tedious calculations. Note that the distribution characteristic parameter a = (jU — T) J a in the expression (10.47) is usually unknown, and has to be estimated in real-world applications, substituting [i and a by the sample mean X and the maximum likelihood estimator (the sample variance) S% = ^ ™ ( J Q — X ) 2 / n . To reveal explicitly the relationship between a and ca, we shall examine the behavior of the critical value ca against the parameter for a = -3(0.05)3. Note that a = (/i - T) / a = -3(0.05)3 covers a wide range of real world situation involving process capability analysis. The results of these calculations indicate that the critical value reaches its minimum at a = 0.5 (corresponding to a small deviation between /J, and T or large value of a) in all the cases with accuracy up to 10~3 . Making

decisions

by means

of a critical

value

Whenever the calculated value of L" is smaller than the critical value ca ( L" < ca ), we shall conclude that the process meets the capability requirement (L" C (i.e. the process is incapable). Making

decisions

by means

of a

p-value

An alternative decision rule can be stated as follows. After calculating the estimated statistics via the sample information, we determine the corresponding p-value, i.e., the probability that Le does not exceed the observed estimated index given the values of C, du, dt, a, and the sample size n, and then compare it with the significance level a . If the estimated index value is 1® , (given the values of C, du , dt, a, and the sample size n), the p-value can then be calculated using the:

PCIs with Asymmetric Specification

Limits

217

p-value = J FK(Bl0 - y)^-=(^V(^ _ 1 >/y + >/*"») + (d^id^Jy

- Vna))dy ,

(10.48)

where B is given in (10.46). Compare with (10.47) the numerical calculations can easily be carried out using the computer software, to integrate the function based on the chi-square distribution and the normal distribution. If the p-value is smaller than the a -risk, we shall conclude that the process meets the capability requirement ( L" C (i.e. the process is incapable).

Chapter 11

Supplier Selection Based on PCIs

11.1 11.2 11.3 11.4

Introduction Tseng and Wu's MLR selection rule based on Cv Chou's approximate selection rule based on Cpu and Cp[ Huang and Lee's approximate selection rule based on Cpm

11.1 Introduction

In an initial stage of production setting, the decision maker often faces the problem of selecting the best manufacturing supplier from several available candidates. There are many factors, such as quality, cost, service and so on, which ought to be considered when selecting the best supplier. Several selection rules have been proposed for selecting the means or variances in the analysis of variance (ANOVA) (see Gibbons, Olkin and Sobel (1977), G u p t a and Panchapakesan (1979), G u p t a and Huang (1981) more details). T h e vast majority all of the selection rules is based on the ordered sample variances. A common persistent drawback of these selection rules is t h a t the information available sample d a t a cannot be efficiently used. As it was mentioned in the earlier Chapters of this book, process capability analysis is applied to determine whether the process capability of a supplier conforms to customer's requirements, by evaluating the process capability index (PCI) to a controlled process. 218

Supplier Selection Based on PCIs

219

In the first 10 chapters of this book we have studied the most common PCIs. In the nutshell, PCIs provide common quantitative measures of the manufacturing capability and production quality, to be used by both producer and supplier as a guiding reference when signing a contract. Purchasing personnel may use the PCI to decide whether to accept or reject products provided by suppliers. As we have indicated on numerous (perhaps too numerous!) occasions in this book in practice the process mean and the process variance are usually unknown. In order to calculate the index value sample data ought to be collected, and a great degree of uncertainty could be introduced into the capability assessments if the sampling errors are ignored. A multitude of methods have been developed to evaluate whether a single supplier's process conforms to customer's requirements. However, few studies have been addressed to the problem of selecting between two or several suppliers' PCIs. There are two common methods that are available to determine the better suppliers' PCI. First 100% inspection is performed to calculate separately the PCI for each supplier, the suppliers can then be compared according to their respective true PCI values; this approach is very expensive and time-consuming and is therefore rarely used in practice. The second method involves sampling implementation, and statistical testing is then used to assess suppliers' process capabilities. 11.2 Tseng and Wu's MLR selection rule based on Cp In a situation when the manufacturing process is in control we are going to assume that the quality characteristic X is normally distributed, USL and LSL are usually fixed and are determined in advance. Tseng and Wu (1991) consider the problem of selecting the best manufacturing process from k available processes based on the "precision" capability index Cp (see Chapter 1) and have proposed a modified likelihood ratio (MLR)

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selection rule. Suppose that there are k available manufacturing processes, 7^ , 7r2 , ..., ^ , and that the product quality characteristic data of each process 7Tj follows a normal distribution N(fa, of) , i = l,---,k . Without loss of generality, it may be assumed that each process has the same process mean (i.e. fa = • • • — fa = fj, ). Kane (1986) proposed the statistic, Cp(i) = (USL — LSL)/(6ai) representing the process capability ratio (PCR) index 7Tj , where, as above, USL and LSL are the upper and the lower specification limits of the process, respectively. Let (Xn, Xi2,- • •, Xin.), i = 1,2,- ••, k , be a random sample of size r^ , taken from the population of the process 7r,. The estimated PCR (to be denoted by Cp(i)) of the process TT{ is given by A ,.*

6p(j)

=

USL-LSL 6s,

where Si

—

Based on the estimated PCR values, the MLR selection rule S = (6i,62,- • ;6k) f° r selecting the best manufacturing process is as follows: For each i = 1,2,- --,k the rule St states: select the process 7r, if and only if

(n^V/lCM2

<

di,

Z(nt-l)/[Cp(j)? i^i

where d^ is a constant to be determined below. This MLR selection rule is then completely specified provided the sample sizes (ni,n2-l- • -, nk) and the critical values (d^d^,- • •, dk) are known. In order to determine these values Tseng and Wu (1991) propose the following two criteria.

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1. Probability of correct selection: The selection of rule <5j is called a correct selection (CS) if 7Tj is the best process and it can be correctly selected by 6t. Usually, it is required that the probability of CS exceeds a minimal value P , referred to as the P -condition. 2. Error probability: The selection of rule <% is called an error selection if there is no significant difference in Cp index values among these k processes and 7Tj is selected arbitrarily. Usually, it is required that the probability of the error selection be less than a maximum value a , referred to as the a -condition. It is reasonable to propose the ratio r^ = [Cp(i)]/[Cp(j)] as a measure of separation between the processes 7Tj and TTJ . For a fixed i , we consider T = (TH,- •;Tii_i,Tii+i,- • -,Tik) as the separation vector of process 7Tj from the remaining processes Ttj , j vt %. Evidently, process 7Tj would be called the best process if and only if r^ > 1 for all j ^ i. If f^ represents the Ith preference region and Q0 represents the indifference region, then they can be suitably defined as follows: fij = { r | Tij > A, for all j ^ i}, A > 1, and ^o = { T I Tij = 1, for all j ^ i} . Now for i = 1,2,- • •, k the P -condition and the a -condition can be stated as follows: P r ^ C S I ^ ] > P*

for all r G ^ t ,

Pr T [CS| 6t] < a*

for all r e fi0 ,

and

where Pr,.[CS | <5j] represents the probability of correct selection of rule <5j under r .

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More details related to theoretical derivations of the MLR selection rule are given in Tseng and Wu (1991). Some Tables of the sample size and of the critical values for selecting the best manufacturing are also computed by controlling the probability of a correct selection (CS) and the error probability based on the proposed MLR selection rule. Furthermore, when the product quality characteristic data for each process follows a non-normal symmetric distribution, a simulation study was carried out to examine the robustness of the selection rule. The results indicate that the proposed MLR selection rule is insensitive to a number of non-normal symmetric process distributions (such as logistic and uniform).

11.3 Chou's approximate selection rule based on Cpu and Cpl As mentioned above, we cannot compare two suppliers directly since the 100% inspection is not practical to use. We are compelled to sample some products made by these two suppliers, and then use a statistical analysis to determine which one has the superior process capability. Chou (1994) developed an approximate method for selecting a better supplier based on onesided capability indices Cpu and Cp\ when the sample sizes are the same, which deals with comparing two one-sided processes and selects a better one with higher process yield. Let X 1 1 ,X 1 2 ,...,X l n and X2i,X22,...,X2m be the measurements of two samples independently drawn from two suppliers 7Tj following the normal distributions N^i^af) , for i = 1,2 . Based on the hypothesis testing for comparing the two Cpu values, H0 : Cpui > Cpu2 versus Ht : C p u l < Cpu2 • Chou (1994) has carried out a variable transformation so that Wj = USL — X, will follow normal distributions N(USL — fa,erf), for i = 1,2 . The sample mean and the sample standard deviation, Wi and 5, , are calculated for suppliers i , i = 1,2 . Thus, the test is equivalent to testing H0 : Wi J ox > w2 J o2 versus Hi : w^ / (Ji < w2 / a2 • Furthermore

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under the assumption that n = m , Miller and Karson (1977) showed that the likelihood ratio test statistic is given by A =

2YY {(W? + 2Y1 f/ (W22 + 2Yi)1'2 2 2

- WXW7

where Y2 = (n — 1)S? / n is the maximum likelihood estimator of the variance a2 , for i — 1,2 . Using the likelihood ratio test, the null hypothesis H0 is rejected if

Wl/S1

<W2/S2

and A < c

for some constant c satisfying 0 < c < 1. The above is equivalent to Cpui < CPu2 and A < c , where A =

[(aCPui + 2)

{o-CpU2 + 2)

— aCpuiCpu2

and a = 9 n / ( n — 1). Since yl is a function of (7pnl and C ^ > it's cumbersome to derive the exact distribution of A which cannot to the best of our knowledge be simplified to a closed form. Observing however that — 2\nA is the log-likelihood ratio statistic approximating a chi-square distribution with one degree of freedom under the stipulations of large-sample theory and that HQ is true we can provide a test with the critical value c that can be expressed as

exp

Xi 2 (l-2a)1

0 < c < 1.

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Here —xi(l — 2a) is the (1 - 2a) -th quantile of a chi-square distribution with one degree of freedom. Consequently, the rule rejects H0 provided C pul < Cpu2 and the condition A < c = exp{-xi(l - 2 a ) / 2 } is satisfied. The selection method proposed by Chou (1994) utilizes some approximating results but provides no indication on how one could further proceed with selecting the better supplier by testing process capability index Cpu or Cpi . Pearn et al. (2004) investigated the selection power analysis of the method via simulation. The accuracy analysis provides useful information regarding the sample size required for designated selection power. To simplify the presentation, we shall use Cpu only. When only the lower specification LSL is given, we calculate Cp[i , i — 1,2 for the zth supplier and a selection procedure also can be constructed in the same way. That is, if the minimum requirement of Cpu values for two candidate processes, and the minimal difference of Cpus between these two suppliers, 8 = Cpu2 — Cpui, are determined in advance how many sample sizes are required such that the suppliers can be differentiated with the designated selection power. The selection power corresponds to calculating the probability of rejecting the null hypothesis H0 : Cpui > Cpu2 , when actually Cpu\ < Cpu2 is true, by using simulation technique with 20,000 replications. To render this method practical for inplant applications, a two-phase selection procedure is developed by Pearn et al. (2004) to select the better supplier and examine further the magnitude of the difference between the two suppliers.

11.4 Huang and Lee's approximate selection rule based on Cpm We have already mentioned on several occasions that, according to modern quality improvement theory, reduction of the process loss is as important as increasing the process yield. The use of loss functions in quality assurance settings has been invigorated with the introduction of the G. Taguchi's philosophy. The index Cpm incorporates the variation of the production items

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225

with respect to the target value and specification limits preset in a factory. Based on the Cpm index a some what mathematically complicated approximation method is developed by Huang and Lee (1995) for selecting a subset of processes containing the best supplier from a given set of processes. Under these circumstances, a search for the larger Cpm which are used to provide unitless measure of the process performance is equivalent to a search for the smaller 7 2 . We emphasize again that the term 2 = a2 + (/j, - Tf = E[(X - T)f is the major part of the 7 denominator of Cpm , which incorporates two variation components: (i) variation with respect the process mean and (ii) deviation of the process mean from the target, (c.f. Chapter 4, Section 1) The method essentially compares the average loss of a group of candidate processes, and selects a subset of these processes with a small process loss 7 2 , which, with certain level of confidence, contains the best process. Since the specification limits are usually fixed and determined in advance, searching for the largest Cpm is equivalent to searching for the smallest j 2 . The selection rule of Huang and Lee (1995) is that one retains the population i in the selected subset if and only if, j 2 < c x m i n ^ ^ j , . , j 2 where the value of c is determined by a function of parameters, which can be in turn determined by calculations from the obtained samples. Let 7Tj be the population with mean /ij and variance a2 , i = 1,2,'••, k , and Xiy,Xi2,...,Xin. be the independent random samples from -ni. If the populations are ranked in terms of 7 2 , our interest would be to select the best process with the smallest value of 7 2 . A correct selection is coded as CS, and the ordered 7 2 in the form of 7r2, < 7,22| < • •• < 7^, are assumed. Now denote by 7 ^ the population associated with 7(21 , i = 1,2,- ••,k. In this case, the best population is ir^. We wish to define a procedure with a selection rule R such that the probability of a correct selection be no less than a pre-assigned number p * where 0.5 < p* < 1 , i.e. Pr(CS | R) > p * . This is

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requirement is referred to as the p *-condition. The selection rule R based on the unbiased and consistent estimators •yf of 7? , i = 1,2,- •-, k , defined as follows:

E(^-r)2

(rii -l)Sf

+n,(xi

-Tf

h

where

Sf =

n,; -

-. /__j \Xij 1

X

i)

)

x

i

2—4 xij •

— Ui

j=l

j=l

Comparing the estimators Cpm (c.f. Chapter 4 Section 2) among k populations is equivalent to comparing the values of jf, i = 1,2,-••, k . Hence, using the result of Pearn, et al. (1992), concerning the distribution of Cpm :

if ~— xl,(\), \

(IM-T)

m


where, as above, Xn W) i s the non-central chi-squared distribution with n^ degrees of freedom and the non-centrality parameter A,. Huang and Lee (1995) consider the problem of selecting the populations with the smallest 7 2 . The selection rule R is here as follows: Consider 7Tj as the better supplier if and only if 7I < c • min1<J<j.i j^i 7j , where c > 1 . In order that the p * - condition, be satisfied, the appropriate c ought to be q = exp- -2 A

T

+

1_

_1

I Til]

l[k})

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Supplier Selection Based on PCIs

Choose now the value of c which is larger than 1 and choose the value as small as possible, so that c = min{c 1 ,c 2 }, if q > 1 and <% > 1, c= q

if q > 1 and c^ < 1;

c = q,

and c = Oi if c? > 1 and q < 1;

where the factors are

A =

-dz + yj4 ~ 4*443

24 fc-i

A, -

-£fe - Vdf - 4di^3 2di

Jfc-1

tij = a x y ] 1 + O/t

+a

i=i

fc-i

£-" <:-i

ak

ab T-^ yjdi +

£i = l a>i +a U7= iE * = »xEJi+? fc-l

d, =

4a

ak^/aj

(k-l

.

\

\2

.

In[2p 2* _ W2a •

1 V ffl,

a = 0.5 — a

E

\ 2= 1

a

k »

6 = -0.513277 Mi

a = -0.085514.

«i +2Aj

,and \ = Hi

Here ^ is an estimator of vt, i = 1,2,- • •,fc, and the ordered «j are denoted by v^ < i\2} < " < %] • The method, however, provides no indication on how one could further proceed with selecting the best population among the chosen subset of populations. Pearn et al. (2004) have investigated this method for the cases with two candidate processes. In practice, if a new supplier II wants to compete for

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the orders by claiming that its capability is better than that of the existing supplier I, he (she) must furnish convincing information justifying his (her) claim with a prescribed level of confidence. Thus, the sample size required for a designated selection power must be determined in order to collect actual data from the factories. If the minimum requirement of Cpm values for the two candidate processes, Cpm0 , say and the minimal difference $ = Cpm2 ~ Cpmi a r e determined, the sample size required need to be such that the suppliers could be differentiated with designated selection power. Thus, based on the proposed selection procedures, if 72 < c ' 7i a n d 7i > c • 7I we shall conclude that the Cpm of 7r2 is better than that of 7rx. Otherwise, we would assume that the existing supplier I is better than the new one II since we don't have sufficient information which would allow as to reject the null hypothesis. The accuracy of the selection method is investigated using simulation technique with simulated 10,000 numbers. For users' convenience in applying this procedure in practice, the sample size required for various designated selection powers = 0.90, 0.95, 0.975, 0.99 with various capability requirements Cpm = 1.00, 1.33, 1.50, 1.67 and the differences 6 = 0.05(0.05)1.00 under the p * -condition = 0.95 are tabulated in Pearn et al. (2004). The selection power which amounts to calculate the probability of rejecting the null hypothesis HQ : Cpm\ > Cpm2 > while actually Cpmi < Cpm2 is true, using a simulation technique. (See above). It should be noted that the required sample size is a function of Cpm , the difference 8 between the two suppliers capability indices and the designated selection power. From the tables of Pearn et al. (2004), it can be seen that the larger the value of the difference 6 , the smaller the sample size will be required for a fixed selection power. For fixed 6 and Cpm the required sample size increases as the designated selection power increases. This phenomenon can easily be explained, since the smaller is the difference and the larger is the designated selection power, the larger sample size will be required to account for a smaller uncertainty in the estimation.

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229

The following two-phase selection procedure was developed by Pearn et al. (2004) to select a superior supplier and further to examine the magnitude of the difference between the two suppliers. Phase I - Supplier Selection In most applications, the supplier selection decisions would solely be based on the hypothesis testing, comparing the two Cpm values: H0 : Cpml > Cpm2 versus Hx : C p m l < Cpm2 . If the test rejects the null hypothesis H0 : C p m l > Cpm2 , there is then sufficient information to conclude that the (new) supplier II is superior to the (original) supplier I, and the decision "to replace" would be suggested. Phase II - Magnitude Outperformed

Measurement

We have just seen that In the Phase I of supplier selection problem, the decision would be solely based on the hypothesis testing comparing the two Cpm values (without a further investigation of the magnitude of the difference between the two suppliers). In other applications, supplier selection decisions would be based on the hypothesis testing comparing the two Cpm values: HQ • Cpmi + h> Cpm2 versus Hx : CpmX + h < Cpm2 , where h > 0 is a specified constant. If the test rejects the null hypothesis H0 : CpmX + h > Cpm2 , there is sufficient information to conclude that supplier II is significantly better than supplier I by a magnitude of h , and the replacement would then be made. In this case, however, one would have to compare the test statistic 7i , i = 1,2 , with the selection value c corresponding to the preset capability requirement for a given sample and designated selection power to ensure that the magnitude of the difference between the two suppliers exceeds h . Note that CpmX must be greater than the preset capability requirement, and = Cpm2 Cpmi + h , where h = max{ h! \ test rejects Cpmi + h' > Cpm2}. The basic problem of checking whether or not the two suppliers meeting the preset capability requirement could

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be accomplisher by finding the lower confidence bounds on their process capabilities. We recommend consulting Huang and Lee (1995) and Pearn et al. (2004) for further clarifications and details.

Chapter 12

Acceptance Sampling Plans Based on PCIs

12.1 12.2 12.3 12.4 12.5

Introduction Acceptance sampling Acceptance sampling Acceptance sampling Acceptance sampling

plans plans plans plans

based based based based

on on on on

Cpk Cpm Cpmk C pu and Cp[

12.1 Introduction A reader having diligently studied the previous 11 chapters may find some of the material in Chapter 12 to be repetitious. T h e reason is t h a t we strive to have this chapter to be self contained. Beginners a n d / o r practitioners may be inclined to the start reading the book with this chapter. Acceptance sampling plans have been for the last 50 years one of the most practical tools in classical quality control applications, which deal with a quality contracting of product orders between t h e factories and their customers. Such a plan provides the vendor and the buyer with a general criterion for lot sentencing while meeting their preset requirements for the quality of product orders. A well-designed sampling plan can substantially reduce the difference between the required (expected) and the actual supplied product quality. Acceptance sampling plan, however, cannot avoid the risk of accepting unwanted poor product lots, nor can it avoid the risk of rejecting good product lots unless 100% inspection is implemented. Acceptance sampling plan is a statement regarding the required

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sample size for product inspection and the associated acceptance or rejection criterion for sentencing each individual lot. The criteria used for measuring the performance in an acceptance sampling plan, are usually based the operating characteristic (OC) curve which quantifies the risks for vendors and buyers. The OC curve plots the probability of accepting the lot against actual lot fraction defective, which provides the discriminatory power of the sampling plan. That is, the OC curve shows the probability of accepting a lot submitted with a certain fraction of defectives (nonconformities) which results in the producer and the buyer having a common base for judging whether the sampling plan is appropriate. For product quality protection and company's profit, both the vendor and the buyer would focus on certain points on the OC curve to reflect their benchmarking risks. The vendor (supplier) usually would focus on a specific level of product quality, traditionally referred to as AQL (average quality level), which would yield a high probability for accepting the lot. The AQL also represents the lowest level of quality for the vendor's process that the consumer would consider acceptable in the form of a process average. Therefore, a preferred sampling procedure would be one which gives a high probability of acceptance at the AQL that is usually specified in the contract. The consumer would also focus at another point at the other end of the OC curve, called LTPD (lot tolerance percent defective). The LTPD is the lowest level of quality that the consumer is willing to accept for an individual lot. Note that the LTPD is a level of quality specified by the consumer, corresponding to the specified low probability of accepting a lot with the defect level as high as the LTPD. For completeness we shall now briefly survey the fundamental concepts of Type I and Type II errors. Acceptance sampling plan consists of a sample size ( n ) and an acceptance criterion (C 0 ). Sampling involves risks that the sample will not adequately reflect the totality of quality conditions of the product. Type I error (a)

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233

is the probability, (for a given (n,C0) sampling plan), of rejecting the product that has the defect level equal to the AQL. The producer suffers when this occurs because here a product with acceptable quality is rejected. Thus, a is called the producer's risk with values commonly ranging from 0.01 to 0.10. In most cases, the values of Type I error a is set somewhere between 0.010.10. The Type II error (/?) is the probability for a given (n,C0) sampling plan accepting the product with defect level equal to LTPD. Here the consumer suffers when this occurs, since a product of unacceptable quality is accepted. Thus, j3 is called the consumer's risk with the values typically ranging also from 0.01 to 0.10. There are a number of papers in the statistical literature providing historical justification for using these values (which are often chosen to be 0.05). While theoretically any two points on the OC curve could be used to construct a sampling plan, it is customary in the industry to use the AQL and LTPD for this purpose. When the levels of lot quality specified are p± = AQL and p2 = LTPD, the corresponding points on the OC curve are usually referred to as the producer's risk point and the consumer's risk point, respectively. Consequently, a well-designed sampling plan must provide the probability of at least 1 — a of accepting a lot if the lot fraction of defectives is at the contracted value AQL. Analogously, the sampling plan must also provide the probability of acceptance no more than (3 if the lot fraction of defectives is at the LTPD level which is an undesired level designated by the buyer. Thus, the acceptance sampling plan must have its OC curve passing through those two designated points (AQL, 1 — a) and (LTPD, {3). There are a number of different ways to classify acceptance-sampling plans. Two major approaches are by attributes or by variables. Assuming that the product quantity N is large, the distribution of the number of defectives d in a random sample of size n of inspected items are Binomially distributed with parameters n and p , where p is the fraction of defectives in the lot. The sample size

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n and the acceptance number c of an attribute sampling plan are the solution to the two nonlinear simultaneous equations:

(These equations must be solved simultaneously, for c and n which appear in these equations in the most unexpected positions!) There is no simple direct method for solving them. A graphical procedure presented in Montgomery (2001) can be utilized. Two lines are drawn on a nomograph, one connecting pi and 1 — a , and the other connecting p2 and 0 . The intersection of those two lines gives the region of the nomograph in which the desired sampling plan is located. In addition to the graphical procedure described above, tabular procedures are also available for the same purpose. When a quality characteristic is measurable on a continuous scale and is known to have a distribution of a specified type, it may be possible to use as a substitute for an attributes sampling using instead the based on sample measurements such as the mean and the standard deviation of the sample. These variables sampling plans possess an advantage that the same OC curve can be obtained with a smaller sample than it is required by an attributes plan. The precise measurements required by a variables plan would probably cost more than the simple classification of items associated with by an attributes plan, but the reduction in sample size may offset this constant expense and result in savings. Such savings are especially substantial if the inspection is destructive and moreover the items are expensive (see e.g. Schilling (1982) and Montgomery (2001)).

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We shall now present a few comments of historical nature. Duncan (1986) provides a good description of these available techniques. His book served for many years as a comprehensive reliable compendium. Guenther (1969) has developed a systematic search procedure, which can be used along with Binomial, Hypergeometric, and Poisson tables to obtain the desired acceptance sampling plans. Stephens (1978) provides a closed form solution for single sample acceptance sampling plans using the normal approximation to the Binomial distribution. Hailey (1980) has written a computer program to obtain single sampling plans (with a minimal sample size) based on either the Poisson or Binomial distributions. Hald (1981) gave a systematic exposition of the existing statistical theory for the sampling inspection and provided careful tables for sampling plans which are still quite useful after some 25 years. The basic concept and models of statistically based sampling plans were described in Jennett and Welch as early as (1939). Since 15 years later, Lieberman and Resnikoff (1955) obtained tables and operating characteristic (OC) curves for various AQLs for the popular MIL-STD-414. Owen (1967) considered sampling plans based on the normal distribution, and developed sampling plans for various values of the probabilities of the Type I error when the standard deviation is unknown. Das and Mitra (1964) investigated the effect of nonnormality on the performance of the sampling plans. Bender (1975) considered sampling plans for assuring the percent age of defective in the case of the product quality characteristics obeying a normal distribution with unknown standard deviation, and presented a procedure using a iterative computer program calculating the somewhat cumber some non-central ^-distribution. Govindaraju and Soundararajan (1986) developed sampling plans that match the OC curves of another popular MIL-STD-105D plan. More recently, Suresh and Ramanathan (1997) developed a sampling plan based on a more general symmetric family of distributions.

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As the rapid advancement of manufacturing technology manifests itself the customers are starting to demand that the fraction of nonconforming products to be quite low (in parts per million, PPM). In such situations, the required number of inspection items must be quite large. Here the process capability enters into our discussions. Briefly a process capability index is a function of process parameter and manufacturing specifications. It measures the ability of a process to reproduce product units that meet these specifications. Consequently, process capability indices can be used as a quality benchmark for product acceptance. Consider a variables sampling plan to control a lot or a process fraction defective (or nonconforming) items. Since the quality characteristic is a variable, there will exist either an upper specification limit (USL) or a lower specification limit (LSL), or both, that define the acceptable values of this parameter. As indicated earlier, selection of a meaningful critical value for a capability test requires specification of an acceptable quality level (AQL) and a lot tolerance percent defective (LTPD). The AQL is simply a standard against which one judges the lots. It is envisioned that the vendor's process will operate at a fallout level that is considerably better than the AQL. In practice, both the vendor and the buyer will lay down their requirements in a contract: the former demands that not too many "good" lots shall be rejected by the sampling inspection, while the latter requests that not too many "bad" lots will be accepted. A sampling plan will be chosen to reconcile and meet these somewhat opposing requirements. Let (AQL, 1 — a ) and (LTPD, (3 ) be the two points on the OC curve under consideration. To determine whether a given process is capable, we could first consider the following testing hypothesis procedure. H0 : p = AQL (process is capable) Hi : p = LTPD (process is not capable)

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Hence, as we have already alluded above the required inspection sample size n and the critical acceptance value C 0 for the sampling plans are simply the solution to the following two nonlinear simultaneous equations. P {Accepting the lot| proportion defective p = AQL} > 1 — a and P {Accepting the lot| proportion defective p = LTPD} < (3 .

12.2 Acceptance sampling plans based on the Cpk index If the Cpf, index (see Chapter 3) is used as a quality benchmark for acceptance of a product lot. Then the null hypothesis about proportion defectives H0 : p = AQL is equivalent to testing process capability index with •^o : Cpfc > CAQL > where C^QL is the level of acceptable quality for Cpk index corresponding to the lot or the process fraction of nonconformities AQL. For instance, if the proportion of defectives p = AQL of vendor's product is less than 66 NCPPM (nonconforming percentile per million), then the probability of consumer accept the lots will be larger than 100(1 — a)% . On the other hand, if the proportion of defectives of vendor's product, p = L T P D , is more than 2700 NCPPM, then the probability that consumer would accept the lot would not exceed 100/3% . Thus, from the relationship between the index value and the fractions of defectives, we could obtain the equivalent values CAQL = 1-33 and the C L P T D = 1 . 0 0 based on the capability index Cpk . For processes with the target value set at the mid-point of the specification limits (i.e. T = M), the index may be rewritten as: Cpk = (d I a— \ £ | ) / 3 , where £ = (/J, — M) / a (see Chapter 3). As noted earlier, the sampling distribution of Cpk is expressed in terms of a mixture of the chi-square and a normal distributions. Given Cpk = C , the ratio b = d j a can be expressed as b = 3C+ | £ |. Thus, the probability of accepting a product can be expressed as:

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•XA(Cpk) = P{Cpk

> C0 |

Cpk)

-L

2

9nco x (^(t + §Vn) + ^ ( « - § V n ) ) d t .

(12.1)

(c/. Chapter 3 Equation (3.8)). Recall that G(-) is the c.d.f. of chisquared distribution and (/>(•) is the standard normal density. The required inspection sample size n and the critical value Co of Cpk for the sampling plan are obtained as the solution to the following two nonlinear simultaneous equations: >6i -Jn

Jo

G '{n-l){b\Jn

-if

9nco

x {
and

P>

n

rG Jo

(12.2)

'(n - l)(&2Vn - t)2 "i 9nco

x ((t + ^Vn) + c/)(t -

c\Jn))dt,

(12.3)

where ^ = 3CAQL + I ^ I a n d h = 3CLTPD + I £ I > noting that CAQL > CLTPD • Observe that the required sample size n is the smallest possible value of n satisfying equations (12.2) and (12.3), and we shall use [n] as the sample size, where mi denotes the least integer greater than or equal to n . However, since the process parameters fj, and a are unknown, so is the distribution characteristic parameter, £ = (// — M) /a , which has to be estimated in real-world applications. As it has been already mentioned above such approach introduces additional sampling errors from estimating £ in finding the critical acceptance values and the required sample sizes. To eliminate the need for this additional estimation procedure of £, Pearn and Wu

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(2004) carried out extensive calculations and have obtained the critical acceptance values CQ and the sample size n for £ = 0(0.05)3.00, for various values of CAQL a n d CLTPD • Observe that the parameter values £ = 0(0.05)3.00, cover a wide range of applications with process capability Cpk > 0 and provides the same values whenever £ is replaced by — £ . The results of the calculations show that the critical value Cg is: (i) increasing in £, (ii) reaches its maximum at £ = 1.00 in all cases, and (hi) remains the same for £ > 1.00 for all CAQL (with the accuracy up to 10~ 6 ). The critical acceptance value c0 reaches its maximum at £ = 0.50 and remains unchanged for £ > 0.50 as the sample size n exceeds 30 (and for n > 100 , the maximum is at £ = 0.35 with the accuracy up to 10~ 4 ). Hence, one could solve equations (12.2) and (12.3) with £ = 1.00 to obtain the critical acceptance value and the sample size n without having to estimate the parameter £. This approach ensures that the decisions which are carried out based on these critical acceptance values may be more reliable. To solve the two nonlinear simultaneous equations (12.2) and (12.3) given above, let /•»<wn

Si(n>Cb)= / *J 0

'(n-l)(6iVn-i)2' G 9nco

x (cj)(t + ^Vn) + <j>(t - Z,Jn))dt - (1 - a) (12.4) and 5 2 (n,c 0 )= /

G

'(n-l)(6j2Vra - £ ) 2 ' 9rcco

x ((t>(t + ^^n) + (f)(t-^^n))dt-p

•

(12-5)

Figures 12.1-12.2 display the surface and contour plots of the equations (12.4) and (12.5) jointly under CAQL = 1-33 and

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=1-00 with the a-risk = 0.05 and the /?-risk = 0.10, respectively. From Figure 12.2, one observes that the interaction of ^ ( n , c0) and S2(n,c0) contour curves at level 0 is (n,^) = (61.74, 1.1477), which is evidently also the solution to the nonlinear simultaneous equations (12.2) and (12.3). Namely, in this case, the minimum required sample size rni equals to 62 and the critical acceptance value c0 is 1.1477 for a sampling plan based on the capability index Cpk . We observe in conclusion that in practice to bring the value of the PCI Cpfc to acceptable level is to design equipment and develop a process in which the process variable is robust to external disturbances or alternatively to use in situ sensors and alternative feedback control. See El-Awady et al. (1996) for a detailed and lucid discussion. CLTPD

50

Figure 12.1. Surface plot of S\ andS^ .

100

110

n Figure 12.2. Contour plot of S\ and£2 •

12.3 Acceptance sampling plans based on Cpm As described above, due to the increasing importance in applications of clustering around the target, rather than adherence to specification limits, and a deeper understanding of loss functions there appears to be a necessity for an alternative to PCIs Cp and Cpk . The Cpm index discussed in Chapter 4 can be used as a quality benchmark for acceptance of a productions lot. Recall

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the definition of this index as given in Chapter 4, and remark that the index could be rewritten as Cpm = d /[3a x (1 + £ 2 ) x ' 2 ], where, as usual, £ = (/J, — T) / a . Hence, given Cpm = C , the ratio b = d J a can be expressed as b = 3C(1 + £2)*' 2 . The probability of accepting the lot is then becomes ^A{Cpm) — P(Cpm

> C 0 I Cpm — C)

6Vn/(3C 0 ) 0

G

(b2n 9Cn2

[<j>(t + £Vn) + (j)(t - £Vn)]d£-

(12.6)

The required inspection sample size n and the critical acceptance value C0 of Cpm (the natural estimator of Cpm , see Equation (4.5)b) for a given sampling plan will be the solution to the following two nonlinear simultaneous equations (12.7) and (12.8): &,Vn/(3C„)

1-a

<

bin

i:

t2

9Cn2

x [cj)(t + £Vn) + <j)(t -

and

&2Vn/(3C0) J 0

' ljn

G

£~Jn))dt

(12.7)

2

9Cn2

x [<j)(t + £Vn) + (f>(t -

$,Vn)]dt,

(12.8)

where b, = 3C AQL (1 + £ 2 ) 1 / 2 and 62 = 3C LTPD (1 + £ 2 ) 1 / 2 • (Observe that CAQL > CLTPD •) I n order to solve the above two nonlinear simultaneous equations (as above), we set

Si(n,CQ)=

f

G

bfn _

t2

9Cn2

x [
(12.9)

242

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s2M)

f^G[^-S

= x

[4>{t + £Vn) + <j>{t - £Vn)]dt

- (3 • (12-10)

Figures 12.3-12.4 display the surface and the contour plots of equations (12.9) and (12.10) jointly for C A Q L = 1.33 and CLPTD =1-00 with both the a and the /3 -risks = 0.05. From Figure 12.4, one deduces observe that the interaction of the Si(n,CQ) and the S2(n,C0) contour curves at the level 0 is (n,C0)= (68, 1.1668), which serves as the solution to nonlinear simultaneous equations (12.7) and (12.8). Namely, in this case, the minimal required sample size n is 68 and critical acceptance value C0 become 1.1668 for the sampling plan which is based on the "new" capability index Cpm.

50

Figure 12.3. Surface plot of Si a n d ^ .

„

100

1J0

Figure 12.4. Contour plot of Si and 52 •

The sample size n and the critical acceptance value C0 based on Cpm for sampling plans with various CAQL , CLPTD > a -risks and /?-risk are provided in Pearn and Wu (2004). One observes that the larger are the risks (a and/or /?) which producer or customer could tolerate, the smaller will be the required sample size n. This phenomenon has an intuitive explanation, since if we

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243

insist that the chance of wrongly classifying a deficient process as good one (or specifying good lots as inadequate ones) be smaller then more sample information is needed to asses the lots. Furthermore, for fixed a-risk, CAQL and C LPTD > the corresponding critical acceptance values become smaller as the /?-risk increases. On the other hand, for fixed (3 risk, CAQL a n ( i C'LPTD > the corresponding critical acceptance values become larger as the a-risk increases. This later fact can also be explained by the same reasoning as above. Also the required sample size becomes smaller as the difference between the CAQL a n d CLPTD increases since such a decision will be more appropriate.

12.4 Acceptance sampling plans based on Cpmk We have already emphasized above that although Cp^ contains the most of the information concerning the yield, it, unfortunately provides the least insight about the location of the parameter fj, . Conversely, the index Cpmk (Chapter 6) provides the most information about the location of // but the least amount concerning the yield. Although Cp^ index is still serves a the more popular and widely used index on the floor, the index Cpmk is now increasing is applicability as to be a refined measure and a more sensitive index for processes with two-sided specification limits. For the processes with target values set at the mid-point of the specification limits, this index may be rewritten as Cpmk=(d/a-\t\)/[3(l

+

e)1/2},

where, as usual, £ = (/j, — T) / a (see the definition of Cpmjt index in Chapter 6, Equation (6.1)). Furthermore, given Cpmi, = C , the ratio b = d / a can be expressed as b = 3C(1 + £ 2 ) 1 ' 2 + | £ |. The probability of accepting a lot is consequently can be expressed as

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KAyCpmk) — P{Cpmk > C0 | Cpmk • 6Vn/(l+3x) J 00

G

[(bJn-tf 9x2

t2)

x [4>{t + (Vn) + (f>(t -

£Jn)]dt.

(12.11)

where, as above, Cpmk is the "natural" estimator of Cpmk . It thus follows that the required inspection sample size n and the critical acceptance value C0 of Cpmk for acceptance sampling plans are expressed as the solution to the following two nonlinear simultaneous equations (12.12) and (12.13): biJn/(l+3x)

1

o

G

{{b^n-tf 9z2

2)

x [(f>(t + £Vn) + cp(t - £-Jn)]dt 62Vn/(l+3x)

and

G

J0

{{bJn-tf

(12.12)

t2)

2

9a;

[4>{t + £Vn) + <j>(t -

£-Jn)]dt,

(12.13)

where h = 3<7AQL(1 + £ 2 ) 1 / 2 + | £ I and fe = 3C LTPD (1 + ^ 2 ) 1 / 2 + | £ | (noting that CAQL > CLTPD)- To solve these two nonlinear simultaneous equations, we set as above &!Vn/(l+3x) 0

((bVn-Q2 G 9a;2

f2)

x [(j)(t + £Vn) + cj)(t - £Sn)]dt - (I - a)

h^fn/(l+3x) t w n ni+ix)

/ 0

G

{{b^n-tf

(12.14)

^

2

9a;

x [0(t + £V7i) + 0(i - i^n)\dt

- p.

(12.15)

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Figures 12.5-12.6 display the surface and contour plots of the equations (12.14) and (12.15) jointly for CAQL = 1-33 and CLPTD =1-00 with the a-risk = 0.10 and the /?-risk = 0.05, respectively. From Figure 12.6 one concludes that the interaction of S^n^o) and S2(n,C0) contour curves at level 0 is (n,C0) = (82, 1.1870), which is therefore the solution of the nonlinear simultaneous equations (12.12) and (12.13). Namely, in this case, the minimal required sample size n is 82 and the critical acceptance value C0 equals 1.1870 for the sampling plan based on the capability index Cpm),.

Figure 12.5. Surface plot of 5! andS 2 •

Figure 12.6. Contour plot of 5V and 5;

Pearn and Wu (2004) investigate the behavior of the critical acceptance values and the necessary sample sizes for various parameters by solving simultaneously equations (12.12) and (12.13). The results indicate that the larger is the risks that producer or customer could tolerate, the smaller is the required sample size n. As in the situation with the Cp\. index this allows an intuitive explanation, since if we expect that the chance of incorrectly assenting that a defective process satisfactory or classifying good lots as inadequate ones is smaller, the more sample information will be require to decide about the quality of

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the lots under inspection. Furthermore, for fixed a , j3 risks and CLPTD I the required sample sizes decrease as CAQL increases.

12.5 Acceptance sampling plans based on Cpuand Cpi For processes with one-sided specification limits, the indices C/ (Cpu o r CPL) c a n D e applied in similar manner (see Chapter 8 Section 1). Since the sampling distribution of Cj is given as a multiple of a non-central t distribution, specifically as 6n_1(3Vn)~1 tn_i(6) , where £„_i(<5) is a non-central t distribution with n — 1 degrees of freedom and the non-centrality parameter 6 = 3VnC/ , the probability of accepting a lot can thus be expressed as: *A(CI)

= P{Cj >C0\d=C)=P

t^S)

>

3VnC0

(12.16)

"n-l

Therefore, the required inspection sample size n and the critical acceptance value C0 of Cj (a natural estimator of Cj) for the sampling plan under consideration are the solution to the following two nonlinear simultaneous equations (12.17) and (12.18): P and

P

3VnC 0 ' K-i , 3VnC 0 )

K-\

>\-CL

(12.17)


(12.18)

where ^ = 3Vn CAQL and 62 = 3Vn C LTPD , with CAQL > CLPTD • The required sample size n is the smallest possible value of n satisfying equations (12.17) and (12.18), and we use the rnl being least integer greater than or equal to n. To illustrate the manner in which the above two nonlinear simultaneous equations, are solved, we set:

Acceptance Sampling Plans Based on PCIs

2—F[n^l]J

247

0 3VnCox

V2IF 1

exp - ^ - < y

0

POO

„ Z±.

2Xpptl ^ 0

1

,-

2

ducfc - a

(12.19)

3VnCn:r "—Q"

V27T J 0

xexp - - ( u - < 5 2 ) 2 dvdx

-{1-15).

(12.20)

Figures 12.7 and 12.8 display the surface and contour plots of equations (12.19) and (12.20) jointly with the a-risk = 0.10 and the /?-risk = 0.10 for C A Q L = 1.50 and C LPTD = 1.00, respectively. From the Figure 12.8, one concludes that the interaction of 5i(n,C 0 ) and 5 2 (n,Co) contour curves at level 0 is (n,C0) = (24.49, 1.2200), which is also the solution to nonlinear simultaneous equations (12.17) and (12.18). Namely, in this case, the minimal required sample size is [n] = 25 , and the critical acceptance value C0 equals 1.2200 for the sampling plan based on the one-sided capability index Cj . 1

1

!

,.

1.3-

C„

"-^, —--i»

. -

-

-

•

-"=4^-__m—1—

Figure 12.7. Surface plot of Si andSj,-

—

•

-

"

M^-

Figure 12.8. Contour plot of Si andS 2

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An acceptance sampling plan in principle consists of a sample size (n) and an acceptance criterion (CQ). Since the sampling cannot guarantee that every defective item in a lot will be inspected, the sampling involves risks of not adequately reflecting the quality conditions of a lot. Risks are becoming to be more significant as rapid advancements in the manufacturing technology occur and a stringent customers demand is enforced. In particular, when the fraction of defectives is very low, more rigorously measured in parts per million (PPM), the required number of inspection items could be exceedingly large in order to reflect adequately the actual lot quality. To summarize process capability indices are function of process parameter and manufacturing specifications. It measures of the ability of a process to (re)produce product units that we meet the specifications. Consequently, utilizing these acceptance sampling plans based on PCIs, practitioners all able to determine the number of inspection units required assess the critical acceptance value, and to make more sound decisions.

Chapter 13

Process Capability Measures in Presence of Gauge Measurement Errors

13.1. 13.2. 13.3. 13.4. 13.5. 13.6.

Introduction Estimating and testing Cp in presence of gauge measurement errors Estimating and testing Cpk in presence of gauge measurement errors Estimating and testing Cpm in presence of gauge measurement errors Estimating and testing Cpmk in presence of gauge measurement errors Estimating and testing Cpuand Cpl in presence of gauge measurement errors

13.1 Introduction

T h e inevitable variations in process measurements comes from two sources: the manufacturing process and the gauge. Gauge capability reflects the gauge's precision, or lack of variation, but is not the same as calibration, which assures the gauge's accuracy. As it was emphasized on the numerous occasions, process capability measures the ability of a manufacturing process to meet preassigned specifications. Nowadays, many customers use process capability to judge supplier's ability to deliver quality products. Suppliers ought to be aware of how gauges affect various process capability estimates. The gauge capability includes two parts: repeatability and reproducibility. Repeatability, (actually the lack of it), is the gauge's experimental or random error. This means t h a t , when

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measuring the same specimen several times, the gauge will never return exactly the same measurement. Reproducibility (actually the lack of it) is the (often annoying) inability of several inspectors or gauges to arrive at the same measurement from a given specimen, i.e. the variability due to different operators using the gauge (or different time periods, or different environments, or in general, different conditions). To summarize we have: 2 "Measurement error

2 "Gauge

2 ""repeatability

2 > ^reproducibility •

Estimates for ^repeatability a n d "reproducibility come from a gauge study, or an GR&R (gauge repeatability and reproducibility) study. Hradesky (1988), Barrentine (1991), Levinson (1995) and Montgomery (1997, 2001), among others, provide procedures for gauge studies. Gauge capability is a gauge's ability to repeat and reproduce measurements. Its measurement is the % of tolerance consumed by (gauge) capability (PTCC). Montgomery (1991) refers to it as the precision-to-tolerance (or P / T ) ratio. It is the ratio of the gauge's variation to the specification width; its smaller numbers are of course preferable. Denoting the gauge's standard deviation by °Gauge ,

w e

n a v e :

PTCC

=

usT^fsLxlmy-

Some authors and practitioners use the coefficient 5.15 instead of 6 (see e.g. Barrentine (1991) and Levinson (1995, 1996)). This formula uses 6 a as the natural tolerance width for the gauge based and motivated by the normal distribution assumptions. A substantial majority of appeared in the literatures do measurement errors. However significant effect on process inaccurate measurement system

capability research works that not take into account gauge the gauge capability has a capability measurement. An can thwart all the benefits of

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improvement endeavors resulting in poor quality. Analyzing process capability without considering gauge capability may often lead to unreliable decisions. It could result in a serious loss to producers if gauge capability is not being considered in process capability estimation and testing. On the other hand, improving the gauge measurements and properly trained operators can reduce the measurement errors. Since measurement errors unfortunately can not be avoided, using appropriate confidence coefficients and power becomes an essential task. However, the reality is that no measurement is free from error or uncertainty even if it is carried out with the aid of highly sophisticated and precise measuring instruments. Montgomery and Runger (1993) point out that the quality of the data related to the process characteristics relies substantially on the gauge. Any variation in the measurement process has a direct impact on the ability to execute sound judgment about the manufacturing process. Conclusions about capability of a process based only on the single numerical value of the index are not reliable. Analyzing the effects of measurement errors on PCIs, Mittag (1993) and Levinson (1995) developed very definitive techniques for quantifying the percentage error in process capability indices estimation in the presence of measurement errors. Gauge repeatability and reproducibility (GR&R) studies focus on quantifying the measurement errors. Common approaches to GR&R studies, such as the Range method (Montgomery and Runger (1993)) and the ANOVA method (Mandel (1972), Montgomery and Runger (1993)) assume that the distribution of the measurement errors is normal with a mean error of zero. Let the measurement errors be described by a random variable M ~ N(0, a2M) , Montgomery and Runger (1993) present the gauge capability A by means of the formula:

X

= UsBlSL*m%-

<'«>

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For the measurement system to be deemed acceptable, the variability in the measurements due to the measurement system ought to be less than a predetermined percentage of the engineering tolerance. The Automotive Industry Action Group recommends the following guidelines for gauge acceptance presented in Table 13.1 right below. Table 13.1. Guidelines for gauge capabilities Gauge Capability A <10% 10%< A <30%

30%< A

Result Gauge system is O.K. May be acceptable based on the importance of applications, cost of the gauge, cost of repairs, and so on. Gauge system needs improvement; make every effort to identify the problems and have them corrected.

In the Sections 13.2-13.6, we shall survey estimation and testing procedures for various PCIs in presence of gauge measurement errors. 13.2 Estimating and testing Cp in presence of gauge measurement errors Denote by X~N(fi,a2) the relevant quality characteristic of a manufacturing process and consider this process, capability in the measurement error system. Because of measurement errors, the observed random variable Y ~ N(fiy = /i, oY =
Process Capability Measures in Presence of Gauge Measurement Errors

C¥ = • ° \

2-

253

(13.2)

Since the variation of the observed data is larger than the variation of the original one, the denominator of the index Cp = (USL — LSL)/(6a) becomes larger, and the variable Y will underestimate true capability of the process. Pearn and Liao (2004a) listed a number process capabilities with A = 0.05(0.05)0.50 for various true process capability indices Cp = 0.50, 1.00, 1.33, 1.50, 1.67, 2.00, and 2.50. It is obvious that the gauge becomes more crucial as the true capability improves (Levinson (1995)). If A = 0.50 (50%), CYP = 0.49 with the true process capability is Cp = 0.50, while Cp — 1.56 when the true process capability being Cp = 2.50. Utilizing a "perfect" measuring instrument will aid substantially in the case of processes with higher capability. Suppose that { Xt, i — 1, 2, • • -, n } denote the random sample of size n from the quality characteristics X . To estimate the precision index Cp , the natural estimator CP = (USL — LSL) / 6s is used, where S = [ZliWCXj — X)/(n — 1)]1//2 is the conventional estimator of a , which should be obtained from a stable process. Adding to eliminate bias the well-known correction factor bn^i = [ 2 / ( n - l ) ] 1 / 2 r [ ( n - l ) / 2 ] / r [ ( n - 2 ) / 2 ] to CP , and denoting CP = bn_i Cp (see Chapter 1), it was shown by Pearn et al. (1998) that Cp is the UMVUE (uniformly minimum variance unbiased estimator) of CP. However, the sample observations are not { Xi, i = 1, 2,- ••, n } but we actually { Y{ , i = 1, 2, • ••, n } . We thus obtain an estimator of CP to be -Y ^P - A - H

USL - LSL r^

)•

Note that Cp estimates Cp , using Sy = 1/2 [E"=i(li - F ) / ( n - l ) ] . Utilizing the arguments of _Chou and Owen (1989) and Pearn et al. (1998), the p.d.f. of Cp can be expressed as

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f(y) = 2

( V ( n - l ) / 2 C P / V l + A 2 C|)"- 1 (yTn T[(n-l)/2]

X exp

-(n-l)CP(2y2y 2 1 + X2nlCp

The expected value and the variance of the estimator Cp are easily obtained using E(Xn-i) a n d Var(Xn-i) to be: CP

E(CYP) =

yfl + ^C]

, Var{CYP) = ( ^ - U' n2- l - 1 ) n —3

cl

z 1 + X2r<2 Cp

For A > 0, it is obviously that Cp is a biased estimator of CP = (USL - LSL) 16a , and the bias is (1/Jl +\2Cp -l)CP which decreases as A increases. Since n is a finite positive integer, [(n — l ) / ( n — 3)](&n_x)2 — 1 is positive (recall that bn_i > 1 ), we have Var(Cp) < Var{CP). The reader should differentiate between CP , CP , (7P , C^ and (7^ . The mean square errors of the two estimators CP = bn^Cp and Cp are MSE(CP) = MSE{Cl)

n-l n

bn-i

i c:p' 6'?n - l

l 1 + \2n2 C

Vl + A2C|

G„

To determine whether a process under consideration meets the present capability requirement and runs according to the desired quality condition, statistical testing hypothesis can be employed by setting H0: CP < c versus Ex:CP>c. A process fails to meet the capability requirements if CP < c , and meets then for CP > c. The critical value c0 can be determined with a given a -risk (the chance of incorrectly judging an incapable process as capable). In the presence of measurement errors, the true capability of the process will unfortunately be underestimated if one calculates

Process Capability Measures in Presence of Gauge Measurement Errors

255

capability index using the estimator Cp instead of Cp. Thus, the a -risk using Cp to estimate CP is less than the a -risk using CP to estimate the same parameter, and the power using Cp to estimate Cp is likewise less than the power using CP as an estimator of Cp . Pearn and Liao (2004a) indicate that both the a-risk and the power of the test decrease when the gauge measurement error increases. If the producers do not take into account the effects of the gauge capability on process capability estimation and testing, it may cause substantial and serious loss. In that case producers are unable affirm anymore that their processes will meet the capability requirements even if they are indeed sufficiently capable. The producers may incur substantial costs as (large) quantities of acceptable product units will be incorrectly rejected. Improving the gauge measurement accuracy and training the operators to proper by execute the production procedures at each step are essential for reducing the measurement errors. Even so the stubborn measurement errors may be unavoidable in many manufacturing processes. Pearn and Liao (2004a) adjusted the confidence intervals and the critical values in order to ensure the intervals possess the desired confidence coefficients and will improve the power of the test with an appropriate «-risk. For the desired confidence coefficient 1 — a , the adjusted confidence interval of CP , with the confidence interval bounds L* and U*, turns out to be: P{L*

= P L* < = P


r*2

*2

C„

V(n - l)bU{KYY'

- {\CYvf

(n-l^i^r 1 -^) 2 (cYp"i2 (n - ! $ _ ! ( # {cYY\1 v)

Y\2 {\CD


< 1

256

Encyclopedia

= P

and Handbook

r*1

(n -

of Process

Y l)bU{KY\~l )

+ P 1 + £T2A2 < U*2

Indices

< 1 + L** Z2X\ 2

Y\2

K)

Capability

(n-l)bU(KY) (cYPf

Y\-l

,2

= P

(«- m-x2 Y

r*2

(C f(l


+ L* X2)

(n - 1K2_! (C ) (l + U*2\2)

< U *2

=

Y 2

1-a.

(13.3)

Expression (13.3) can be verified by carrying out appropriate substitutions. Here KY = (n — l)Sy / Oy is distributed as x\-\ • Further simplifications show that the adjusted 100(1 — a)% confidence interval bounds can be written as L* =

y Xn-i,i-a

/ 2^;

^n-l^U-iXClfxl^

a/2

V^-l,a/20f

U*

J(n - \)bU - (\CYfxUa/2

where X^_n_ a /2 ^s the 1 — <*/2 percentile of the chi-squre distribution with n-1 degrees of freedom. With the adjusted confidence interval bounds, the desired confidence coefficient can be achieved. Moreover, to improve the power of the test, Pearn and Liao (2004a) obtain adjusted critical values (denoted by CQ ) to be Y a = P{C >c*0\cp = c) = p

= p K-i-Jn ~

1C

2

\_i>/n

VF

,S- > >IKY I C = n

{ co*Vl + X C:

-ICY

> co* I Cp = c

Process Capability Measures in Presence of Gauge Measurement Errors

2

2

c >K* = P ( t f2 _ i ( n - l )2„2\ U ( l + A2c2) —

257

^ &g-i(n - l ) c 2 '

v = P Xn — 1 — *2/-i , \2 2\ co (1 + A V ) ,

Note that the last probability statement involves both c and CQ . To ensure that the a;-risk is within the preset magnitude, we set a* = a , and from the last probability statement we have 6n-i>/n - \c Co

=

V(l + A 2 c 2 )x2-i,i-a '

and the power of the test (denoted by IT* ) can be calculated as K*{CP) = P{CYP >4\CP)

= P

= P

{bn_xyJn

VF

'CPN/(I + A2C2)X2-I,I^ 2

Xn-1 <

> co I Cp

_pgr

2

Cyjl + X C

Cp

-lCp

p

2„2 \ 1 + X'c

1 + A2C2

Xn—1,1—a

The reader is advised the compare the expressions for the adjusted critical values and the power. 13.3 Estimating and testing Cpk in presence of gauge measurement errors The well-known process capability index Cpfc (discussed in some detail in Chapter 3) has been widely used in the manufacturing industry for measuring process quality reproduction capability. Most research works related to Cpk are carried out under the assumption of no gauge measurement errors. Unfortunately, such an assumption does not accommodate adequately real world situations even with the modern highly sophisticated measuring instruments and devices. Let

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X ~ N(n,cr2) represent the relevant quality characteristic of a manufacturing process and Cpk measures the "true" process capability. However in practice, we deal with the observed variable Y (the measurement error) rather than with the "true" variable X . Assume that X and M (the measurement error) are stochastically independent. In this case we have Y ~ N(HY = M) °Y — v2 + 0M) , and the empirical process capability index Cpk will be obtained after substituting aY for a . The relationship between the true process capability Cpk = min j (USL — /x) / 3
pk

_ d— I Y — m \ 3Sy

obtained by replacing the process mean /i and the process standard deviation a by their conventional estimators Y = E"=i Yi/n and SY = [E?=i(*i ~ Y) /(n - 1)] 1 / 2 , from a "bona fide" stable process. Accordingly, applying the technique used in Kotz and Johnson (1993) and Pearn and Lin (2003), the c.d.f. of CYpk is obtained as:

Process Capability Measures in Presence of Gauge Measurement Errors

•ACl-Sn F*Y(X) C

P*V

= 1 '

\ JO

G

259

(n-l)(3CjVn-i) 2 ^ 9nx2

x $[* + 3(CY - C^kyjfi] + *[t - 3(<# - C p K ,)Vn]^ , where CYp = Cp / Jl + \2C2p , is CYvk = Cpk / yjl + X2C2p , $(•) is the c.d.f. of standard normal distribution and, as above, A = 6aM /(USL - LSL) x 100% . To determine if a given process meets the preset capability requirement, we could consider the statistical testing with the null hypothesis H0 : CPK < c (the process is not capable) and alternative hypothesis H0 : Cp% > c (the process is capable), where c is the required critical value. If the calculated process capability is greater than the corresponding critical value, we reject the null hypothesis and conclude that the process is capable. Discussions in Pearn and Liao (2004b) indicated that the true process capability would be severely underestimated if Cpk is used. The probability that C\\ is greater than c0 would be less than when using Cpk . Thus, the a-risk using Cpk is less than the a -risk of using Cpk when estimating Cpk . The power of the test based on CYk is also less than that based on Cpk . That is, the a risk and the power of the test decrease with the measurement error (as in the case of the Cp index (Section 13.1)). Since the lower confidence bound is severely underestimated and the power becomes small, the producers cannot firmly state that their processes meet the capability requirement even if their processes are indeed sufficiently capable. Adequate and even superior product units could be incorrectly rejected in this case. To remedy the situation, Pearn and Liao (2004b) consider the adjustment of the confidence bounds and of the critical values to provide a superior capability assessment. Suppose that the desired confidence coefficient is 9, the adjusted confidence interval of Cpk with the lower confidence bound L*, can be shown to be:

260

Encyclopedia and Handbook of Process Capability Indices

e =

p(cpk>L*) G 0

(n - l)(6*Vn - i),2: 1 9n(Cj,) 2

x [$(* + £ y Vn) +$(t - iYJn)\

dt,

where b* = 3L* /Jl + \2C2p + | £Y \ , and £ y = 3(
18L* + yj324L*2 - 4(9 - X2)(9L*2 - 1) 2(9 - A2)

= C.pi

To improve the power of the test, revised critical values CQ can be determined from a* = P(CYpk > 4 I Cpk = c) nzc* vn

~ Jo

((n - 1) {3CY Vn - tf )

{

9n(c0*)2

JX

X $[* + 3(CY - CYk)Vn) + $[< - 3(C y - C^)V7T]^ , where a* is the a -risk corresponding to the test using Cpk and CQ , CYp = Cp/Jl + \2C\ , and CYpk = c/^l + \2C2p and c is the capability requirement. To eliminate the need for further estimation of the characteristic parameter Cp , we set Cp = CYk + 1/3 as suggested by Pearn and Lin (2003) and find a reliable adjusted critical values CQ , where Cp can be obtained by using the equation Cp / yjl + \2C2p = c/^l + X2C2p + 1/3 to be:

Process Capability Measures in Presence of Gauge Measurement, Errors

Cp

_ 18c + V324c2 - 4(9 - A2)(9c2 - 1) _ ~ 2(9^) -

G

261

^

To ensure that the a -risk is within the preset magnitude, we let a* = a and solve the above equation to obtain CQ . The revised power (denoted by IT* ) can be calculated as 7r

*(Cpfc) = P\Cpk > c0 | Cpk,Cp = Cp2) 3CT>/n

= /. o

'(n-i)(3C^Vn-*)2' [$(£ + Vn) + $ ( i - V n ) ] d t . G 9n(c0*)2

2 where C^ = C^fc + 1/3, and C ^ = Cpk / Jl + X2C p2

When estimating the capability, the estimator C ^ in the case of contaminated data, severely underestimates the true capability in the presence of measurement errors. Consequently, if a statistical test is used to determine whether the process meets the capability requirement, the power of the test would decrease substantially. Consequently, lower confidence bounds and critical values ought to be adjusted to improve the accuracy of capability assessment. Example 13.1 The LM 2576 series of regulators, as depicted in Figures 13.1(a)-(b), are monolithic integrated circuits, which provide active functions for a step-down (buck) switching regulator, capable of driving 3A load with excellent line and load regulation. Those devices are available in fixed output voltages of 3.3V, 5V, 12V, 15V and an adjustable output version. Requiring a minimum number of external components, those regulators are simple to use and they include internal frequency compensation and a fixed-frequency oscillator. The LM 2576 series offers a highefficiency replacement for popular three-terminal linear regulators. It substantially reduces the size of the heat sink, and in some cases

262

Encyclopedia and Handbook of Process Capability Indices

no heat sink is required. A standard series of inductors optimized for use with the LM 2576 are available from several different manufacturers. This feature greatly simplifies the design of switchmode power supplies. Other features include a guaranteed ±4% tolerance on output voltage within specified input voltages and output load conditions, and ±10% tolerance on the oscillator frequency. External shutdown is included, featuring a (typical) 50 /J, A standby current. The output switch includes cycle-by-cycle current limiting, as well as a thermal shutdown for full protection under fault conditions.

Top View

0

0

TT™i5-cs/CFF "1 :c::::u-fadbaek 1 1 "13-OfciufliJ " E l 2-Output 31-Vh,

Figure 13.1 (a). LM2576 series stepdown voltage regulator product (top

Side View ^ = 3 PIUS 1 , 3 , 4 3

Figure 13.1 (b). LM2576 series stepdown voltage regulator product (side

Following this (positive) description of the LM 2576 series of regulators, Consider a supplier manufacturing step-down voltage regulator products in Taiwan, the producing LM 2576-3.3 type with specifications of output voltage: T = 3.3V, USL = 3.366V, LSL = 3.234V for conditions of VIN = 12V (input voltage), ILOAD — 0.5A (load current), and Tj = 25°C (temperature). A total of 70 observations are collected and displayed in Table 13.2. histogram and a normal probability plot show that the collected data follows the normal distribution. The Shapiro-Wilk test is applied to further justify this assumption. (The reader is urged to carried out the normal probability plot and the Shapiro-Wilk test). Now to determine whether the process is "excellent" (Cpfc > 1.50) with the unavoidable measurement errors A = 0.25, we first determine that c = 1.50 and a = 0.05. Then, based on the

Process Capability Measures in Presence of Gauge Measurement Errors 263

sample data of 70 observations, we obtain the sample mean Y = 3.299, the sample standard deviation SY = 0.013, and the point estimator Cjfc = 1.632. From Table 4(c), we obtain the critical value Cfl = 1.595 based on a , A and n . Since Cpk > CQ , we conclude that the process is "excellent". Moreover, by inputting Cpk , A , n , and the desired confidence coefficient 0 = 0.95 into the appropriate computer program we obtain the 95% lower confidence bound on the true process capability to be 1.542. We can see that if we ignore the measurement errors and evaluate the critical value without any correction, the critical value will be calculated to be as CQ — 1.758. In this case we would reject that the process is "excellent" since Cpk is not the estimator greater than the uncorrected critical value 1.758. Table 13.2 observations for output voltage (unit: V) 3.292

3.289

3.293

3.323

3.279

3.304

3.306

3.288

3.287

3.319

3.299

3.269

3.300

3.294

3.308

3.285

3.292

3.278

3.285

3.282

3.297

3.278

3.311

3.295

3.319

3.303

3.305

3.326

3.315

3.298

3.321

3.315

3.284

3.319

3.302

3.314

3.308

3.303

3.294

3.312

3.297

3.305

3.306

3.295

3.286

3.293

3.288

3.314

3.318

3.295

3.309

3.296

3.296

3.305

3.293

3.298

3.305

3.289

3.288

3.315

3.308

3.279

3.292

3.293

3.265

3.283

3.307

3.314

3.303

3.305

Estimating and testing measurement errors

Cpm

13.4

in

presence

of

gauge

We have already mentioned that most of research works (both theoretical and applied) related to Cpm have been carried out under the assumption that no gauge measurement errors are present. See Marcucci and Beazley (1988), Boyles (1991), Pearn et al. (1992), Zimmer and Hubele (1997), Zimmer et al. (2001), and Pearn and Shu (2003) among other publication. Such an assumption, however, does not accommodate satisfactorily realworld situations by even using the advanced top-line measuring

264

Encyclopedia and Handbook of Process Capability Indices

instruments. Quite likely conclusions concerning the process capability that are based on empirical indices are not fully reliable. The relationship between the true process capability Cpm and the empirical process capability C^pmm IisD

£>L =

^1 + ?

(i3 4) l j

cpm vr+^cj+F' where £ = ( / i - T ) / c r .

Since the variation of the observed data is larger than the variation of the original one, the denominator of the index Cpm = (USL - LSL)/^a2 + (/i - T) 2 ] increases, and the "true" capability of the process is therefore understated if the calculations of the process capability index are based on the empirical data Y . In practice, sample data must be available to estimate the empirical process capability Cpm . For a stably normally distributed process, the empirical data (namely the observed measurements contaminated by errors) Yt for i = 1,2,...,n is collected and then the maximum likelihood estimator (MLE) of Cpm is defined as ~Y ^pm

USL - LSL

Gjsl+iY-Tf

d

3^+(f-T)

2

where Y = ] T } - - i ^ / n > "^« — Xy--i(^» ~ ^ ) l n ' anc * ^ u s u a , l d = (USL - LSL)/2 is used. We note that F and S 2 are the maximum likelihood estimators (MLEs) of \i and a\ respectively. The term S\ + (Y - T)2 = ^ " = 1 ( K i - T) 2 In in the denominator of Cpm is the uniformly minimum variance unbiased estimator (UMVUE) of cr2y +{n-Tf = E\(Y-T)2) appearing in the denominator of Cpm.

Process Capability Measures in Presence of Gauge Measurement Errors

265

Evidently, if the a^ — 0 , the estimator of process capability (C m) results in the basic estimator Cpm introduced by originally Boyles (1991). Observe that Cpm is distributed as Y

C

d 3ov"^A/<«

Pm

where OQ — a2 + a\

which can be alternatively expressed as CY ~

d I n 3oy 1

CY

P

I n

~ V<«

V*iU

= c

^ v 1 + ^I \

n

(see (13.5)),

X

n,SY

where %2 ^ denotes the non-central chi-square distribution with n degrees of freedom and the non-centrality parameter 6Y = n^y and £y = (M ~~ T)/<JY • The c.d.f. of CYvm can also be expressed in terms of a mixture of a chi-square and a normal distributions: F^y

-x

(x)

&rVn/(3z) ?K

(6yVn) 9x2

t2

x [(* -£y>/ra)]di, for £ > 0 , where by = d/ay = 3 C j , FK(-) is the c.d.f. of the ordinary central chi-square variable Xn-i a n d
(Y

=

M

A*

aY

2 (C Y \ w

pm

1 > C„ —

c„

vr+A^

(see (13.2)),

266

Encyclopedia and Handbook of Process Capability Indices

rY

and

-

(c.f. (13.5)).

^ pm

We again remind our readers to differentiate between Cpm, f wpm

j

a l l u

Cpm,

pG ^ pm •

Pearn and Shu (2004) note that using the described procedure, the lower confidence bound is underestimated and the a -risk and the power of the test decrease with the measurement errors. Suppose that the desired confidence coefficient is 7 then the adjusted confidence interval of Cpm with the adjusted lower confidence bound LA can be obtained by solving the equation

1-2

f

MJ[ZC\
•K

Jo

where bA = 3LA/yjl + X2Cp and the value of Cp can be obtained by solving Cp = Cpm . To improve the power of the test, then we shall revise the critical values CQ to satisfy c0 < c0 . Thus, the probability P(Cpm > CQ J will be greater than P(C^m > c0) . Both the a -risk and the test power increase when CQ is used as a new critical value in the testing. Suppose that the a -risk using the revised critical value CQ is aA ; the revised critical values CQ can then be determined by P(Cpm > c0 \Cpm < c) = aA . Consequently 6 G Vn/(3c^) J 0

?K

(bG^nf [9(^) 2

-r

(t)dt =

aA,

where bG = 3c(1 + A 2 Cp) -1 / 2 and Cp is obtained as stated above; thus Cp = c . To ensure that the a-risk is within the preset magnitudes, we let aA = a and solve the above equation to

Process Capability Measures in Presence of Gauge Measurement Errors

267

obtain CQ . The power of the test (denoted by irA ) can be calculated as the follows: KA(Cpm)

= P{CGpm

2 ^Vn/(3c£)

= f JO

>ct\Cpm>c) . ?K

(by^nf

iw

_t2

\

(t)dt,

"If

where by =

VT+AV

Example 13.2 The product investigated is the pH sensor combining process-hardened pH electrodes, double junction reference electrode, temperature compensation element, and a solution ground in a durable, reliable, high performance design. It is ideal for process control applications in the most aggressive streams found in the traditional processing industries, including chemicals, paper, metals and mining, utilities, food, pharmaceutical, etc. For wide range measurements where high accuracy is required at either or both extreme ends of the pH scales, a spherical glass with a minimal sodium error is considered good quality. For applications involving abrasive processes, a rugged glass with a thicker membrane is recommended. The rugged glass is most accurate in the range of 1 to 12 pH. The high temperature construction allows the sensor to be used for the process pH measurements at the temperatures up to 120° C (250°F). An integral lOOplatinum RTD (resistance temperature detector), compatible with many common pH transmitters and monitors, is a standard feature of this device. The reference electrode utilizes a double junction design to inhibit silver ions from contacting the process solution, thereby preventing junction fouling from silver compound precipitates. To help facilitate a noise-free signal, the sensor incorporates a solution ground post of titanium metal. 3/4 NPT threads are provided at both sensor ends to allow connection to insertion or submersion type mountings.

268

Encyclopedia and Handbook of Process Capability Indices

The measuring accuracy of the pH sensor is an important factor have a significant effect on the pH sensor quality. A type of the pH sensors has the specification limits, T = 0.0 pH, USL = 0.05 pH, and LSL = -0.05 pH. A total of 70 observations were collected and displayed in Table 13.3. A histogram and a normal probability plot show that the collected data follows the normal distribution. The Shapiro-Wilk test is applied to further justify this assumption. To determine whether the process is "excellent" ( Cpm > 1.33 ) with the unavoidable measurement errors A =0.30, we first determine that c = 1.33 and a = 0.05 . Then, based on the sample data of 70 observations, we obtain the sample mean G = 0.0200, the sample standard deviation Sn =0.0109, and the point estimator C$m = 1.4629 . From the Table 5(b), we obtain the critical value CQ =1.436 based on a , A and n . Since Cpm > c0 we conclude that the process is "excellent". However if we ignore the measurement errors and evaluate the critical value without any correction, the critical value will be c0 =1.547. In this case we would reject that the process is "excellent" since Cpm is not greater than the uncorrected critical value 1.547. Input now T = O.OpH, USL = 0.05pH, and LSL = -0.05pH, the 70 observations, A =0.30 (provided by the gauge manufacturing factory), and the desired confidence coefficient 7 = 0.95 into the Matlab computer program (available upon request) the 95% lower confidence bound on the true process capability is to be obtained 1.415. We thus can be assured that the production yield is 99.9978%, and the number of the nonconformities is less than 21.78 PPM (Parts Per Million). Table 13.3 observations for the measuring accuracy (unit: pH) 0.0236 0.0257 0.0304 0.0289 0.0057 0.0188 0.0204

0.0433 0.0281 0.0198 0.0227 -0.0018 -0.0046 0.0153

0.032 0.0246 0.0172 0.0186 0.0385 0.0322 0.0154

0.0157 0.0198 0.0234 0.0066 -0.0086 0.0136 0.0413

0.0129 0.0335 0.0311 0.0265 0.0178 0.0187 0.0089

0.0085 0.0253 0.0278 0.0399 0.0130 0.0249 0.0344

0.0406 0.0115 0.0035 0.0168 0.0240 0.0157 0.0273

0.0047 0.0306 0.0289 0.0240 0.0317 0.0250 0.0236

0.0047 0.0097 0.0044 0.0040 0.0200 0.0167 0.0158

0.0254 0.0194 0.0188 0.0235 0.0137 0.0083 0.0185

Process Capability Measures in Presence of Gauge Measurement Errors 269

13.5

Estimating and testing measurement errors

Cpmk in presence

of gauge

For the third-generation process capability index Cvm^ , we assume that X ~ N ([i, a2 ) represents the quality characteristic of the manufacturing process under consideration. We know that in practice, the observed variable Y (in the presence of gauge measurement errors) is actually measured instead of the true variable X. Assuming that X and M are stochastically independent, we then have Y ~ N(/x,o\ = a2 + <j2M) , and the empirical process capability index Cpmf. is obtained by substituting aG for a. The relationship between the actual process capability Cpmk and the empirical process capability Cpmk can be expressed as

cPmk ~~ v i + \ 2 c \ + e ' where, as above, £ = (/J, — T)h . To estimate the empirical process capability CGmk sample data must be available. For a bona-fide stable normal process, the empirical data (the observed data contaminated by errors) (7j, for i = 1, 2,.., n , is collected. The maximum likelihood estimator (MLE) of Cpmk is defined as USL-Y 2

^S

Y -LSL 2

j 2

+(Y- Tf ' 3yJS + (Y - T) J '

a n d § = are where Y = E"=i^/n " ELi(^ " Vf/n respectively the MLEs of [x and a\ • Note that the statistic S2n + (Y - Tf = E " = 1 ( i i -T)2 /n in the denominator of CYpmk is the uniformly minimum variance unbiased estimator (UMVUE)

270

Encyclopedia

and Handbook

of Process

Capability

Indices

of uY + (fi - Tf = E[(Y - T)2] . For processes with symmetric manufacturing tolerance (T — M), the estimator CYmk reduces to

rY

-

^ pmk

d-\Y-M\ 3^S2n+{Y-Tf,

where as above d = (USL — LSL )/2 . Evidently, if aM = 0 then the empirical process capability CYmk The c.d.f. of CYmk can be becomes the basic index C,pmk expressed in terms of a mixture of the chi-square and the normal distributions as

pmk

bY-fnj{l+'ix)

W-i-f

{by-Jn-t) ?K

2

9x2

x [(t + £yVn) + (t -

£YVri)]dt

for x > 0 where by = dloY = 3 C j , and, as above, FK(-) is the c.d.f. of the (central) chi-square distribution Xn—i and
fi-M

CY p -

+

(
+

3

{Cpmk ) ~ Q {Cp )

{Cpmk j

(CY ? -\

where CY =

p n

2

n

pmk J

, and C\mk =

Vl + A ^

Q

pmk

Vl + A2C72 + £2

Process Capability Measures in Presence of Gauge Measurement Errors 271

Assume that the desired confidence coefficient is 7 . Then the adjusted confidence interval for CYmk with the adjusted lower confidence bound LA can be expressed as bAJn/(l+3C%mk)

{bAVn -tf •K

/.

x [0(* + O.5Vn) + ^ ( t - O . 5 V n ) ] d i = 7 ,

(13.6)

where bA = 3.75LA/yjl. 25 + X2C2P + 0.5 and Cp can be obtained by solving the equation CYp = JT?XCYpmk + 1 / 6 .

Thus

C„

1.25L 1 2 2 + T7Vl.25 + X C p 6

2 2

Vl + X C p

Given A , L (the 95% lower confidence bound using Cpmk ), bisection and interpolation methods can be used to find the Cp values. In order to improve the test power, we shall revise the critical value CQ to satisfy eg < c0 . Thus, the probability P\C^mk > co) w m De greater than P[CYmk > CQJ. Both the a risk and the test power increase when CQ is used as the new critical value in testing. Let the a -risk using the revised critical value CQ be aA , the revised critical values CQ determined from P (Cpmk > c^ Cpmk < c J = aA, can be expressed in explicit form as: byJn/{l+3c^)

L

(byJn-t)

_t2

^K

X [4>(t + 0.5VTT) + (p(t ~0.5Jn)}dt

= aA,

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Encyclopedia and Handbook of Process Capability Indices

where analogously to (13.6) by = obtained

by

solving

3 75c , + 0.5 and C„ is 2 2 Vl.25 + X C p

equation

Cp = y/l.25Cpmk + 1/6

Therefore , °p = , °5c +1. 2 2 Vl + X C p Vl.25 + X2C2P 6 To ensure that the a-risk is within the preset magnitude we let aA = a and solve the equation to obtain c^ . The power of the test (denoted by nA) is calculated as *A {Cpmk ) = P{C^

> <* | Cpmk > C) , (pG^/n-t)

r 6 K Vn/(l+3c 0 ^)

i.e.

TtA{Cpmk) = j

1

_t2

^

bK x [<j>{t + 0.5Vn) + 4>(t-

0.5-Jn)}dt,

O • I D O yjrfik

where bG =

. + 0.5 . Vl.25 + X2C2p

The results imply that employing the above estimator for sample data contaminated by measurement errors causes severe a underestimation of the actual capability, resulting in imperceptibly smaller test power. To measure the actual process capability, one must use modified confidence bounds and critical values that are suitable to practitioners for their factory applications. 13.6 Estimating and testing Cpu and Cpi in presence of gauge measurement errors Bordignon and Scagliarini (2002) performed a statistical analysis related to estimating Cp and Cpk . The empirical process capability indices Cpu and C$ are obtained by substituting oY for a. (Cpu or Cpi were discussed in Chapter 8) The degree of

Process Capability Measures in Presence of Gauge Measurement Errors

273

error contamination r has been defined to be (Mittag (1997)).

r = ^L a Thus the relationship between the empirical process capability index Cj and the actual process capability index Cj: Cj _

1

where as above CYU or CYt is denoted here as Cj and Cj denotes Cpu or Cpi. Since the variation of the observed empirical data is greater than the variation of the original data (with no measurement errors), the denominator of the index C/ increases, and we would have underestimated the true capability of the process when calculating the process capability based on the empirical data from Y . Since the process parameters (i and a are unknown, we cannot evaluate the actual process capability. However, given (independent) sample data {Yi,i = l,...,n} taken from the process, the estimators of Cpu and Cpi become ^y ^pu

T UoLi — Y f^y , = "n-1 To ' ^ p i = 0n_i

i

— L/oLi — .

Following the arguments in Chou and Owen (1989) and in Pearn and Chen (2002), we can show that the estimator Cj (Cpu or C^i) is distributed as d-tn^i(6Y), where d = 6„_ 1 (3Vn) _1 (don't confuse it with d =(USL — LSL)/2\) and tn_1(6Y) is a non-central t distribution with n-1 degrees of freedom and the non-centrality parameter 6Y = 3VnC7 / >/l + r 2 , with T = aM / a . Let the desired confidence coefficient be 6, then the adjusted confidence interval on Cpu with the confidence interval bound C*u , can be shown (using the methodology employed in this book on numerous occasions) to be

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e = p{cpu >c:) = p

(USL-

, {Y + k[SY - HY > = P 3<7y

= P

= P

nY

vr+^j > -fc/Vn

Sy I Oy Z - 3VnC:/^/^T7?

_ 3(7pu

Sy / Gy

= P *n-l($/ =

Vi + r 2 > c u * |

y

n-l

-3VnC,

VT+ r

r) > *T

where Z~N(0,1) *F = - ( 3 C £ lK_^Jn

and k? = 3 Cju / bn^ ,

Analogously, the adjusted confidence interval on Cpi with the confidence interval bound C*, can be obtained as

9 = P tn-l({>L =

3VnCf

}

< ^

VI + T<

where tf = (3C;,/^_i)Vn . Note that in this case, the probability that the interval with the bound C\ contains the actual Cpu value is greater than the corresponding probability for the interval with the bounds Cu or Cu , while the probability that the interval with the bounds Cu or Cu contains the actual CPU value is the constant 0.95. In order to improve the power of the test, the revised critical q$ ought to satisfy a* = P(C]

>cl\CI=c)

= P{?>JnCj > 3VncS | C1 = c)

Process Capability Measures in Presence of Gauge Measurement Errors

{K-\

K-\

= p[tn^(6Y = P

275

= SVnCj)

> ^-ct

tn_^ =^-1J=)>^2

Vi + r '

| Cj = c) A

K-\

CO

To ensure that the a-risk is within the preset magnitude, we let a* = a. Then CQ can be obtained as Co

=

hn l - t ^ {*Y 3Vnn a

= 3Vn

C

4i + ^-

and the power it* becomes 7r*(C/) = P(Cj > 4 | Ci) = P(3^nCf = P T I °n-l

W

> 3Vnco* | C 7 )

> -7 Co | W °n-l

= P "n-l

= P tn-\(6

= 3Vn

C/

VI +

f) > * n - l , a ( ^ = 3Vn TZ

VTT^J

Example 13.3 TFT-LCDs (thin-film-transistor liquid crystal display) consist of a lower glass plate on which the T F T is formed, an upper glass plate on which the color filter is formed, and the injected liquid crystal between both glass plates (see Figure 13.2 (a)). The T F T plays a critical role in transmitting and controlling electric signals, which determines the amount of voltage applied to the liquid crystal. The liquid crystal controls light permeability using different molecular structures that vary in accordance with the voltage. In this way, the desired color and image is displayed as it passes through the color filter (see figure 13.2 (b)). The TFT-

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LCD consumes less energy compared to a CRT (cathode-ray tube), is slimmer and weighs less. The TFT-LCD becomes the most widely used display solution, because of its high reliability, viewing quality and performance, compact size and environmentfriendly features. Because of the heat resistance, non-conductance, and simple processing steps. Non-alkali thin film glass is the major material of manufacturing TFT-LCD. While manufacturing nonalkali thin film glass, flatness is one of the critical quality characteristics. If the flatness of glass is not in control, the TFTLCD products may result in a certain chromatic aberration.

AA » » » » *

M f f ColorRtterArroy

Liquid Ovsto!

r—:

Back light Plasma' PelaVlIlPtJj '

IFTArfOY

, .

5?Z.««vCT\ Cotern,t8r

Back plate with discharge eiectrode pattern

LC transparent electrode

Bock Light

Figure 13.2(a). Structure of a TFT- Figure 13.2(b). Structure of a TFT-LCD. LCD.

The measuring accuracy of a pH sensor is an important factor which has significant effect on the pH sensor quality. A type of the pH sensors has the specification limits T = 0.0 pH, USL = 0.05 pH, and LSL = -0.05 pH. A total of 70 observations were collected and displayed in table 6. A histogram and a normal probability plot indicate that the collected data follows the normal distribution. The Shapiro-Wilk test is applied to further justify the stated assumptions. To determine whether the process is "excellent" ( Cpm > 1.33 ) with unavoidable measurement errors A =0.30, we shall first determine that c = 1.33 and a = 0.05 . Then, based on the sample data of 70 observations, we arrive at

Process Capability Measures in Presence of Gauge Measurement Errors

277

the sample mean G = 0.0200, the sample standard deviation Sn =0.0109, and the point estimator Cpm = 1.4629 . From table 13.4, we calculate the critical value c$ =1.436 based on a, A and n. Since Cpm > CQ , we thus conclude that the process is "excellent". Also one can verify that if we ignore the measurement errors and evaluate the critical value without any correction, the critical value becomes c0 =1.547. In this case we would reject the hypothesis that the process is "excellent" since C^m is not greater than the uncorrected critical value 1.547. Moreover, inputing T = O.OpH, USL = 0.05pH, and LSL = -0.05pH, 70 observations, A =0.30 (provided by the gauge manufacturing factory), and the desired confidence coefficient 7 = 0.95 into the Matlab computer program (available upon request), the 95% lower confidence bound of the true process capability is calculated to be 1.415. We thus can assure that the production yield is 99.9978%, and the number of the nonconformities is less than 21.78 PPM (Parts Per Million). Table 13.4 observations for the measuring accuracy (unit: pH) 0.0236

0.0433 0.032 0.0157 0.0129 0.0085 0.0406 0.0047 0.0047 0.0254

0.0257

0.0281 0.0246 0.0198 0.0335 0.0253 0.0115 0.0306 0.0097 0.0194

0.0304

0.0198 0.0172 0.0234 0.0311 0.0278 0.0035 0.0289 0.0044 0.0188

0.0289

0.0227 0.0186 0.0066 0.0265 0.0399 0.0168 0.0240 0.0040 0.0235

0.0057 -0.0018 0.0385 -0.0086 0.0178 0.0130 0.0240 0.0317 0.0200 0.0137 0.0188 -0.0046 0.0322 0.0136 0.0187 0.0249 0.0157 0.0250 0.0167 0.0083 0.0204

0.0153 0.0154 0.0413 0.0089 0.0344 0.0273 0.0236 0.0158 0.0185

To summarize our discussion: when estimating capability the confidence bounds are substantially underestimated in the presence of measurement errors. When we use statistical testing to determine if the process meets the capability requirements, we find that the power of the test decreases with the measurement errors. Since the measurement errors are, unfortunately, unavoidable in many branches of manufacturing industry, to obtain a more

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accurate confidence bound and to improve the power with an appropriate a-risk, adjusted confidence bounds and the critical values should definitely be applied.

Chapter 14

Process Capability Assessment with Tool Wear

14.1 Introduction 14.2 A review of various approaches 14.1

Introduction

Manufacturing systems are nowadays geared up to meet the challenges of a quality-based competition. Process capability studies and analyses have become critical issues in process control; indeed a number of guidelines are available for process capability assessment. Moreover, certain conditions such as normally distributed output, statistical independence of observed values and the existence of only random variation resulted from chance causes are stipulated for this assessment. All these conditions may not be fully satisfied in a practical set-up and departures are quite likely to occur. Tool wear is, evidently, constitutes a dominant and inseparable component of variability in many "machining" processes, and hence it represents a systematic assignable cause. Process capability assessment in such cases may become a bit tricky since the standard procedure may not provide accurate results. Process capability studies play an important role in the process control since they assist to decide whether a manufacturing process is suitable and the applications meet the necessary quality standards. The intensified quality-based competition has shifted the focus from the product control to the process control in manufacturing 279

280

Process Capability Assessment with Tool Wear

industries. The assessment of process capability, which now appears to be quite simple, involves significant dimensions in a practical set-up. This is due to the fact that some of the conditions necessary to establish process capability are not fully satisfiable. These conditions stipulate that the process has to be under statistical control, that the process output has to follow the normal distribution, be free of assignable causes, and the observed values of the quality characteristics be statistically independent. Since these conditions are not fully satisfied in many manufacturing situations, the process capability analysis is becoming to be a critical issue. It just does not make sense to proceed with a full-scale production without first conducting process capability studies: This situation was of course realized by many researchers and resulted in numerous publications in the literature. A process capability analysis is therefore valid only when the process under the investigation is free of any special or assignable causes (i.e., is in-control). Not much has been done in the area of process capability for the situations where the presence of an assignable cause is known and tolerated. A process is said to have a "tool wear problem" when a variation due to a systematic cause is present. There are, in fact, two areas of interest when studying a process: process stability and process capability. It is important to have clear guideless about control before developing the plan for a tool wear process. Specifically the intent of the plan to detect changes in the process or is the goal just to monitor the tool? An action to be taken in an out-of-control situation should be determined by the intent of the plan. Statistical process studies and the ongoing control are quite complicated when dealing with machine processes possessing a tool wear. Indeed such a wear is a fact, and it is essential for processes that exhibit tool wear to be controlled, to maintain high part quality and to maximize the tool life. In its simplest and most common form, tool wear data tend to have an upward or a downward slope over time. To determine this trend, a best-fit line to the data ought to be generated. For standard control charts, the grand average and the control limits are

Process Capability Assessment with Tool Wear

281

usually horizontal. In contrast, when the tool wear is involved, the control limits will be parallel to the tool wear slope. Once control has been assessed, the capability of the process can be then determined. 14.2

A review of various approaches

First of all it is necessary to ensure that the process at hand is under statistical control in order to assess its capability. Porter and Oakland (1991) pointed out that the two specific conditions which make the process capability assessment to be difficult are: 1) ensuring stability of the mean and of the standard deviation; and 2) an absence of any special causes. It is also expected that observations be statistically independent, which is not always the case. Indeed, processes with uncontrollable but acceptable trend are common in practice. The causes such as tool wear are responsible for inducing autocorrelations and are not therefore physically removable from the process. The issue of a correlation among the samples and its effect on the control chart limits has been examined by numerous authors in the last 25 years. Vasilopoulos and Stamboulis (1978) have shown that the limits for an X chart can vary by a wide margin especially if the correlation effect is not taken into account. These authers have suggested factors that should be used with an S chart or an X chart, provided the data can be fitted within either a first or second order autoregressive model. However, the effect of correlation in estimating standard deviation, a, has not been considered. Burr (1979) has analyzed the problem of estimating a from the aspect of constructing control limits. Initial setting of a tool has also been studied to prolong the utility of the tool before its replacement or resetting. Accordingly in these situations a is estimated as R / d^, where R refers to the range of values over a short run and ^ is a constant based on the sample size. The control limits are then calculated in the usual manner. Montgomery (1985) suggested fitting the first order autoregressive model to the correlated data,

282

Process Capability Assessment with Tool Wear

which could be of the form: xt = £>+cj>xt_l +et,

(14.1)

where xt is the value of the variable at time t, £ and
here cf> refers to the correlation coefficient. The effect of correlated subsamples in statistical process control was also observed by Neuhardt (1987), who commented that the correlated measurements within subgroups tend to increase Type I error rate in the case of X charts. Accordingly, the actual variance is given by, a2 (1 — r + kr), where r= the pairwise correlation among the subgroups, and A; is the common subgroup size. Time series modeling trend data had been also suggested by Alwan and Roberts (1988), who recommend using residuals in monitoring the process. As the residuals do not have any engineering significance, it is necessary to plot the individual values. An important aspect in deciding on the control limits for the individual observations chart is proper estimation of a. The commonly accepted procedure is to from ranges by taking differences of successive values. Then the average of these moving ranges of size two is applied to estimate a using the standard formula:

Process Capability Assessment with Tool Wear

.

283

MR

where MR is the average of the moving ranges and c^is a factor based on the subgroup size used extensively in control charts analysis. Ryan (1989) object to the above procedure on the grounds that a moving range of two will smooth out any trend presented in the data, which is inadvisable if the trend is a permanent part of the data. Furthermore, it is demonstrated that when successive values do not differ much and are strongly correlated, the moving average approach substantially underestimates a. The alternative suggestion is to estimate a as: s o = —, c4 where s is the sample standard deviation for at least 50 (calculated from an unbiased correction factor involving Gamma function) observations that are independent and c4 is another factor based on sample size. It is also stated that s / c4 gives a better estimate of a than that MR / d^ even in the situations when the observations are not independent as long as at least 50 observations represent the data generated by the process. Furthermore, Ryan (1989) has also suggested that an appropriate method to estimate a would be to fit a time series model and then perform the estimation. For instance, the variance a\ of an autoregressive process of the first order, is given by _2 _

°z

where ^ = is the correlation coefficient between successive values, and a\ = 1 is the variance of residuals from the fitted model. Although several authors have looked at the issue of correlation from the point of control charts, the process capability aspects did not received due attention. There are some situations when

284

Process Capability Assessment with Tool Wear

assignable causes are systematic, such as tool wear, so that their effects can be decomposed before the capability is estimated. This approach is also referred to as a constant or consistent process drift, and other examples which include accumulation of contaminants and temperature change drift must be quantified and removed before the remaining variability can be analyzed for statistical control (Kotz and Lovelace (1998)). Some approaches attempt to remove the variability associated with the systematic cause. For example, Yang and Hancock (1990) recommended that in computing the basic Cp index, an unbiased estimator of a can be obtained as a/(\ — pfl2, where p is defined as the average correlation factor and is given by, P = er^/[fc(A;-l)], where {r^} , i, j = 1 to k is the correlation matrix, with the entries r- • —

1

for all i — j

r\j

I fij \< 1, for i * j

V i,j = 1 to k ,

Some other authors make a general assumption of linear degradation in the tool. For instance, Quesenberry (1988) suggests that tool wear can be modeled over an interval of tool life by a regression model and assumes that the tool wear rate is known or a good estimate of it is available, and that the process mean can be adjusted after each batch without an error. However, the task of model-building does not appear to be neither very easy, nor directly applicable to realistic conditions. The procedure suggested by Long and De Coste (1988) first removes the linearity by regressing on the means of the subgroups and then determines the process capability. Long and De Coste (1988) discuss the techniques for obtaining the best-fit line through the data, calculating the control limits, comparing the slopes to determine different tools, and calculating the capabilllty of the process. These techniques are based on the

Process Capability Assessment with Tool Wear

285

assumption that tools are "consistent" within their tool groups. As the data are recorded over several tools, the subgroup averages are plotted over time. A best-fit line is then determined using standard linear regression analysis. The standard equations for calculating the best-fit line are:

y = b0 +b1x, E(s - x)(y - y)

where ^ = — ^ ^ —

2 —-

Yl(x — x)

and b0 = y — }\x .

The best-fit regression line for every tool is estimated by the points that are the averages of each subgroups, comprised of five consecutive parts. Once the best-fit lines have been determined for each of the tools under investigaton, the slopes of the lines are tested for a statistically significant difference. One possible method for determining significant difference is the "£" test. First, the largest and the smallest slopes are tested for significant difference. The procedure of testing hypothesis is here: HQ : bi — b} = 0 versus Hj : ^ — 65 ^ 0, and the test statistic

JMSE/J2(x - xf ' where b\ and bi are the slopes of best-fit lines 1 and 2, respectively, and MSE is the residual mean square error for line 1. The calculated test statistic is compared with a critical " t" value. If the test statistic is larger than this critical " t " value, the null hypothesis is rejected. This shows that a statistical difference exists between the slopes. If there is no significant difference between the largest and smallest slopes, then evidently "statistically", no difference exists between the remaining intermediate slopes. However, if a statistical difference does exist

286

Process Capability Assessment with Tool Wear

between the largest and smallest slopes, one then repeats the procedure with the next largest or smallest slopes. In summary, we repeat this procedure until no statistical difference is evident between the slopes. When two or more slopes are found to be statistically equivalent, the data can then be considered to have come from the same population and can be combined and studied together. In particular, the average slope is calculated and the equations ought to be adjusted so that the y-intercept, b0, becomes zero. After the equations have been adjusted (to have all zero intercepts) the slopes are being averaged. Those slopes that turn out to be statistically different indicate that their data cannot be considered to have come from the same population and ought to be studied separately. The next step involves calculation of the control limit around the average best-fit line. Quite a few methods exist for determining the so-called "three-sigma" control limits. The method suggested by Long and De Coste (1988) incorporates the average range and the subgroup size to estimate the three sigma limits. The procedure for calculating the control limits is: Upper Control Limit = Y + (A2)(R)(^Jk / m) Lower Control Limit = Y - (A2)(R)(->Jk/ m), where Y is average best-fit line for i = 1 to the maximum number of subgroups A% = a constant based on a subgroup size R = an average of subgroup ranges k = subgroup size used to calculate the average slope line and m = subgroup size used on the control chart. While calculating the control limits, we compute the average range over all the tools that were found to be statistically equivalent. Separate control limits must be calculated for tools that are found to

Process Capability Assessment with Tool Wear

287

be statistically different, and control limits are plotted around the average best-fit line. Calculations for the range chart follow the standard procedure. The spread of these control limits serves as the guideline for the ongoing control. For a tool wear processes, the ongoing control plan incorporates the control limit spread calculated from historical data along with the starting and the ending points of the control tool. An estimate of the starting point must be provided for each new tool. The starting point is approximated by sampling three successive subgroups. From these subgroups, the mean is calculated and the best-fit line is estimated using the slope of the average best-fit line. Once the estimated best-fit line is plotted, the control limits, calculated from the sample data, are drawn around the line. Next one has to develop an appropriate sampling procedure taking all sources of variation into account. The minimum or maximum starting and ending values should be developed to ensure that the material produced is within engineering specification limits at a certain confidence level. The starting and ending values are calculated by adding or subtracting a confidence factor from the specification limits. The confidence factor is determined by multiplying the standard deviation of the best-fit line by a constant to ensure a specific level of confidence. A constant of three or four is usually used. If the process being studied has a positive slope, then the sample starting point must be above the minimum starting value. The last sample should not exceed the maximum ending value. For a negative slope, a maximum starting value is of interest as is the minimum ending value. It should be noted that the specification limits are used only as a reference in determining the tool life and are not drawn on the actual control chart. Once the charting commences the stability of the process or simply the tool wear behavior can be checked. A reaction to an out-of-control situation depends upon the purpose of the chart. If the chart is solely for tool monitoring, the samples are taken as

288

Process Capability Assessment with Tool Wear

scheduled and the points are recorded until the actual number of tool cycles reaches the maximum ending point. At this point the tool is replaced. However, the process knowledge is also of importance, the control limit spread, based on the expected data, is centered around the best-fit line of the current tool. This allows for detection of variation other than tool wear. This procedure was suggested by Long and De Coste (1988) and it requires that the cycle of the tool be identical over the entire lifetime. This is usually not the case in situations where a tool is retooled several times or even when an old tool is discarded at the end of each life cycle. The tool must therefore be considered as a potential source of variability as well. When systematic assignable causes are present and tolerated, the overall variation of the process ( a 2 ) is composed of the variation due to random causes ( a 2 ) and the variation due to assignable causes (cr2,)- That is, a2 = a2 + a\ . The traditional PCI measures fails to acknowledge that portions of the overall variation, (in the presence of tool wear), will be due to assignable causes. Hence any estimates of the process capability will confound the true capability with these two sources. In order to get a true measure of process capability, any variation due to an assignable cause must be removed from the measure of process capability. However, the above approaches tacitly assume a static process capability over a cycle. By allowing the process capability to be dynamic within a cycle, as well as from a cycle to cycle, one could circumvent some of the problems encountered. Spiring (1989, 1991) viewed this as a dynamic process which is in a constant change as a process, tool, age, etc. In this dynamic model, the capability of the process may vary, possibly in a predictable manner. Spiring has devised a modification of Cpm index for this dynamic process under the influence of systematic assignable causes. In this scenario the goal is to maintain some minimum level of capability at all times. As a result, the capability will be cyclical in nature, its period being defined by the frequency of the process/tolling adjustments. Even

Process Capability Assessment with Tool Wear

289

when the assignable cause variation is not systematic, as is in the case with tool or die wear, one ought to be able to deal with random fluctuations of the process mean over time. Often in practice, deviations from the target value are due to easily determined assignable causes, such as shift-to-shift changes, differences in the raw material batches, environmental factors, etc. The proposed measure of process capability for dynamic process proposed by Spiring (1991) is

mm{uSL-T,T Cpm =

- LSL}

3 ^ + (* - Tf

'

where USL , LSL and T are the usual standard entities used in the assessing of Cpm and other PCIs but where /xt represents the mean and o>(-the variation (due to random causes only) of the process at time period t. As we have already remarked (perhaps too often!) the actual value of m and oft a r e rarely known, hence in order to get some assessment of process capability these values ought to be estimated. Analogously to the population values, the estimators will have to incorporate various existing sources of variation. Monitoring process's capability will thus require obtaining the value of Cpm or of a suitable estimate at various times t over each cycle during the lifetime of the tool. A sampling scheme that is used to monitor the capability of a process ought to reflect an array of factors associated with the process. Several factors, such as the frequency and the magnitude of other assignable causes, can be dealt with in a manner similar to the approach when dealing with the control chart sampling schemes. See, e.g. Montgomery (1985) for a discussion of "rational subgroups". However other (new) factors, such as the length of the sampling "window", will be influenced by the factors that are normally not encountered in control chart sampling schemes. The proposed sampling scheme is similar to the schemes used in monitoring a process for control charting procedures. The general

290

Process Capability Assessment with Tool Wear

format is to gather k subgroups of size n from each cycle (e.g., the period from t0 to tx) over the lifetime of the tool. The value of k will be unique to each process and, in fact, may change from cycle to cycle within the process. On the other hand, sample size of less than five (i.e., n < 5) are cautioned against, while larger samples (e.g., n > 25) may also pose problems. The optimal sample size for assessing process capability in the presence of systematic assignable causes will vary for each process under consideration. The manner in which the samples are taken is also of importance. In most situations consecutive sampling is suggested (i.e., the samples are collected as they come off the line). However, if desired, a "systematic 1 in R sampling scheme" can be used (i.e., within a period every second unit off the line is collected). However as R increases, problems may be encountered. Assuming that the effect of the tool deterioration is linear over the sampling window, estimates of Cpm are available which would not involve contribution of the assignable causes. Such an estimate is:

mm{uSL-T, Pm

~

T - LSL]

3ylMSEt+^(Xt-Tf

This measure of process capability considers only the proximity to the target value T and the variation associated with random causes as the linear effect of the tool wear is effectively removed by using n

MSEt = ^ * ra-2 of sequentially selected points (i.e., t^,^,^,---) rather than the common estimator s2. The MSEt is the mean square error associated with the regression equation xa. = aa + /3ta. + ea. and where ta. is the sequence number of the sampling unit and

291

Process Capability Assessment with Tool Wear

ea. ~ N(0,1). The coefficient 0 denotes the linear change in the tool wear given a unit change in time/production. Using the ordinary least square (OLS) estimates of a and f3 and assuming the sampling scheme to be sequential, the computational formula for Cvm becomes

h

' pm *-* rym.

—

. \USL-T min | 3

12

E<

'

ix 2

Y,( 0

T-LSL] i=l 3 J (n - 1) 12

2

i-i

^E(^) »=i

n(n - l)(n - 1) + - (n - 1) 2

2n(2n + 1) _ X (n - I)2~ k

+

n(xta - Tf n-l

where n denotes the subgroup sample size, T the target value and xt representes the i th value of the quality characteristic in the sampling period ta , while 3c(<j = Z"=i xta. n • The above method was proposed by Spiring (1991) who have suggested (as it was already mentioned above) that the problem can be tackled by viewing the process capability as being dynamic rather than static. This entails calculating a new index and monitoring it as the process advances. When the index reaches a preset minimum value, processing is terminated and resetting/replacement is carried out. The method has unfortunately some drawbacks: •

In addition to the regular charts, a new chart showing various values of the process capability will have to be drawn. This necessitates deciding on the sample size, the number of samples, the associated confidence level and the frequency of sampling which may involve a substantial number of calculation.

•

Great effort may be required for a graphical analysis and interpretation.

292

•

Process Capability Assessment with Tool Wear

The usefulness of the procedure described ought to be ascertained in a typical practical set-up.

Thus it would seem that there is no generally agreed upon procedure for assessment of process capability in the presence of assignable causes that affect the process in the case of a tool wear situations. Wallgren (1996) has also studied the properties and implications of the index Cpm when the consecutive measurements represent observations of dependent variables stemming from a Markov process in discrete time. This occurs, for example, when consecutive measurements from a process are serially corrected. The author had also developed an augmentation of the index Cpm , denoted Cpmr based on the first order autoregressive model (AR(1)). Unfortunately this report to the best of our knowledge has been not published as yet.

Chapter 15

Process Capability Assessment for Non-normal Processes

15.1 Introduction 15.2 A brief review of various approaches 15.1 Introduction We shall very briefly sketch the basic assumptions and properties of process capability indices. Process capability indices have been widely investigated as a means of summarizing process performance relative to a set of specification limits. These indices are effective tools for both process capability analysis and quality assurance. The proper use of process capability indices, which are statistical measures of process capability, is based on several assumptions. One of the most essential is that the process monitored is supposed to be stable and the output is approximately normally distributed. When the distribution of a process characteristic is non- normal, PCIs calculated using conventional methods could often lead to erroneous and misleading interpretation of the process's capability. A number of quality control experts have provided useful and insightful information regarding the errors in interpretation of the values of the PCIs that occur due to the misapplication of the indices to non-normal data (see, e.g., Sarkar and Pal (1997), Chou et al. (1998) among others). English and Taylor (1993) have examined the effect of the non-normality assumption on PCIs and

293

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have concluded that Cpk is more sensitive to departures from normality than Cp . Somerville and Montgomery (1996) studied the errors that can occur in calculating Cp or Cpj. for a nonnormal distribution and making inferences about the PPM nonconforming under normality assumption. They have reached a conclusion for the four non-normal distributions, most common in engineering and reliability applications: the t, the gamma, the lognormal, and the Weibull distributions, that although the magnitude of an error can vary substantially depending on the true (unknown) distribution parameters, the errors are of consequence in nearly all cases. Hence, if the capability indices based on the normal assumption concerning the data are used to deal with non-normal observations, the values of the capability indices may, in a majority of situations, be incorrect and quite likely misrepresent the actual product quality. Actually, a leading investigator of quality control problemsnon-normal processes occurs frequently in practice. Gunter (1989) - a leading investigator of quality control problems - has mentioned the special case of the hole diameter of a drilled hole. Due to the nature of the process a minimum hole size is available in the vicinity of which the majority of hole diameters will cluster, however some very large holes occasionally occur due to such factors as vibrations, dull tools, or crooked setups among others. Pyzdek (1995) also informs us that distributions of certain chemical processes such as zinc plating thickness of a hot-dip galvanizing process are very quite often skewed. Choi (1996) presents an example of a skewed distribution in the "active area" shaping stage of the wafer's production processes. Bittanti et al. (1998) address the problem of defining and computing reliable estimates for process capability indices (in particular for Cpk) for non-normal processes; specifically, a curve-fitting approach to the estimation problem is employed and the problem of providing confidence intervals for the estimates of the "non-normal" PCI's is considered. Applications examples of non-normal process

Process Capability Assessment for Non-normal Processes

295

capability indices in various manufacturing industries are provided in Pearn and Chen (1995, 1997). For these skewed processes, the proportion of nonconforming items for fixed values of standard PCIs tends to increase as skewness increases, i.e. the standard PCIs simply ignore the skewness of the underlying population. For example, if the underlying distribution is Weibull with the shape parameter (3 = 2.0 or 1.0 (i.e. the skewness is 0.63 or 2.00, respectively) and LSL = —3.0 and USL = 3.0 , then the expected proportions of non-conforming items below and above the lower and upper specification limits are 0.56% and 1.83%, respectively, for the same value of |U = 0 and a = 1 . Hence Cp = Cpk = 1.0 , whereas the expected non-conforming proportion for a normal population is 0.27% being. Therefore, a method of adjusting the values of a PCI in accordance with the expected proportion of non-conforming items by considering the skewness of the underlying population would be very desirable (see, e.g., Gunter (1989), Pyzdek (1992), English and Taylor (1993), Somerviile and Montgomery (1996), and Kotz and Lovelace (1998)). In the recent years, several approaches to the problems of PCIs for the non-normal populations have been suggested. One suggestion is to use data transformation techniques such as the Box-Cox power transformation, Johnson's transformations and quantile transform techniques (see e.g. Somerviile and Montgomery (1996), Kotz and Lovelace (1998), Tang and Than (1999)). In general, however, practitioners may feel uncomfortable working with transformed data and may have some difficulty in reversing the results of the calculations back to the original scale. A second method is to replace the unknown distribution by an empirical distribution or by a known three or four-parameter distribution. Examples include Clements (1989), Pearn and Kotz (1994), Franklin and Wasserman (1991, 1992), Shore (1998) and Polansky (1998). Castagliola (1996) introduced a non-normal PCI calculation method by estimating the proportion of non-conforming items using the Burr's distribution. This method is complicated and

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requires proivitably large samples, so that most of the quality engineers inclined towards simple methodology may hesitate to use this approach. A third way is to modify the standard definition of PCIs in order to increase their robustness. This approach makes the PCI as insensitive as possible to the shape of the distribution (see, e.g., Pearn et al. (1992) and Rodriguez (1992)). Practice convey to us that the performance of this approach, is however, often unsatisfactory. A fourth method is to use heuristic arguments to develop new indices. Bai and Choi (1995) proposed weighted variance (WV) PCIs based on Choobineh and Branting (1986) ideas, and Wu et al. (1999) suggest new WV PCIs by modifying the WV method of Choobineh and Branting (1986). Chang et al. (2002) developed a novel heuristic method of constructing simple PCIs for skewed populations based on a weighted standard deviation (WSD) method. This method adjusts the values of PCIs in accordance with the degree of skewness in the underlying population by employing different factors when computing the deviations above and below of the process mean. When the underlying population is symmetric, these indices are reduced to the standard PCIs. Tang and Than (1999) review several methods and provide a comprehensive evaluation and comparison of their ability to handle non-normal situation. For more details about process capability indices, the two books, (Kotz and Johnson (1993) and Kotz and Lovelace (1998)), each have devoted a chapters for dissecting such problems. The recent review paper of Kotz and Johnson (2002) also discusses the problems and consequences of non-normality. As a consequence, capability indices are very powerful, but, like many powerful tools, they could inflict heavy damage if incorrectly used. Properly calculated, they provide a whale of vital information concerning how the current output of a process satisfies customer requirements. Incorrectly applied and/or interpreted, these indices can generate an abundance of

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297

misinformation that may confuse practitioners and cause them to carry out incorrect decision. 15.2 A brief review of various approaches Below we shall summarize several different methods for measuring process capability indices appearing in the literature. 15.2.1 Probability

plotting

approach

Probability plotting is an alternative to a histogram that can be used to determine the shape, center, as well as the shape of the distribution. It has the advantage that it is unnecessary here to divide the range of the variable into class intervals, and it often produces reasonable results when using moderately small samples. A probability plot is a graph of the ranked data versus the sample cumulative frequency on a special paper with a vertical scale chosen so that the c.d.f. of the assumed type becomes a straight line. A widely accepted approach for PCI computation is to use the popular normal probability plot (1994) so that the normality assumption can be verified simultaneously. Analogous to the normal probability plot, where the "natural" process width is between the 0.135 percentile and the 99.865 percentile, surrogate PCI values may be obtained via appropriately selected probability plots c

USL - LSL USL - LSL ~ (upper 0.135% point) - (lower 0.135% point) ~ Up - Lp '

= v

where Up ( Lp ) is the 99.865 (0.135) percentile of observations respectively. These percentile points can easily be obtained from a simple computer code performing probability plots. Since the median is usually the preferable central value for a skewed distribution, the corresponding Cpu and Cpi are defined as:

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~

__

*~/ pu nil

_ p

USL — median (upper 0.135% point) — median

USL — median — median '

-XQ.99865

median — LSL _ median — LSL median — (lower 0.135% point) median — X0.00135

The index Cpk will then be calculated as the minimum of Cpu and Cpi, namely: Cpk = rnin | C p u , Cpl j USL — median median — LSL = mm' ^0.99865 - median ' median - X . 35 0 001 In the non-normal case, if we are able to find a better distributional form for the data, which provides a very satisfactory fit (this can be tested by means of goodness-of-fit tests), we can obtain more accurate measures of the three quantiles under consideration. This involves modeling the process data with alternative probability models, such as the Weibull or gamma ones (see e.g. Dudewicz and Mishra (1998), Kotz and Lovelace (1998)). However, an obvious disadvantage of probability plotting is that it is not a truly objective procedure. It is quite possible for two analysts to arrive at different conclusions using the same data. For this reason, it is often desirable to supplement probability plots with goodness-of-fit tests which possess more formal statistical foundations (see, e.g., Shapiro (1980)). A good introduction to these tests is in Shapiro (1980). Choosing the distribution to fit the data is also an important step in probability plotting. Sometimes one can use the available knowledge of the physical phenomenon or the past experience to suggest a choice of the distribution. Several earlier authors recommend to employ data transformations. Kane (1986) suggested transforming the data to

Process Capability Assessment for Non-normal Processes

299

create an approximately normal situation. Gunter (1989), among numerous other researchers and applied statisticians, has found empirically that in this general situation, results for transformed data are typically much better than for the original raw data! The well-known quantile transformation techniques were developed by Clements (1989), who utilized the Pearson curves to provide better estimates of the relevant quantiles. 15.2.2

Clements'

approach

For non-normal Pearsonian distributions (which include a wide class of "populations" with non-normal characteristics), Clements (1989) has proposed a novel method for calculating estimators of Cp and Cpk , which uses Pearson curves to provide more accurate estimates of -XQ.00135 i -^0.50 (= median) and •^0.99865 • His modifications of Cp and Cpk are appealing since they do not require mathematical transformation of the data, have a transparent meaning which is easy to comprehend, for nonstatisticians and also can be estimated manually with miniature hand-held calculators. Furthermore, Pearn and Kotz (1994) applied Clements' method to obtain estimators for the other two basic indices: Cpm and Cpmk. To refresh the memory, we cite the relevant expressions: - _ USL - LSL ~ U -L. P n V

Cp

Cpk = min r>

—

USL-M Up-M

M-LSL ' M - Lp

USL - LSL Up-Lp

+ (M -Tf

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c,pmk

min

Up

USL-M -M? + (M

M M-Lp]

-Tf

-LSL + (M

-Tf

where Up is the 99.865 percentile and Lp is the 0.135 percentile determined from the "universal" tables of Gruska et al. (1989) for the particular values of mean, variance, skewness, and kurtosis calculated from the sample data. Clements' estimators for Cp and Cpm regardless of a specific distribution are obtained by replacing the 6cr by Up — Lp . For the indices Cpk and Cpmk , the process mean \x is estimated by the median M , and the two 3a s are estimated by Up — M and M — Lp respectively for the right and left-hand sides. Clements (1989) uses the classical estimators of skewness (/? 3 ) and kurtosis ( (3$ ) based on the third and fourth moments respectively, which are often unreliable for small sample sizes. To improve the accuracy, Pearn and Chen (1995) consider the following modification replacing the a by (Up — Lp) / 6 in all cases regardless of whether it is on the right-hand or left-hand sides. Thus, Clements' estimators become: o„ —

Cpk

USL - LSL U„ — L„

USL - M M - LSL = mm [Up-Lp}/r\Up-Lp}/2\

Process Capability Assessment for Non-normal Processes

r' ^pm

Cpmk —

m

i

n

'

-

301

USL - LSL Up — Lp

+ (M

-Tf

USL-M Up — Lp

+ {M

M Up — Lp

-Tf

-LSL + (M

-Tf

See also the discussions of the estimator of C„ C C ^pk 5 ^pm and Cvmk for various particular situations presented in previous chapters. To illustrate how the modified estimators outperform the original Clements' estimators, Pearn and Chen (1995) consider an example of three processes: one on-target and the other two offtarget. In their example original Clements' estimators show little sensitivity to the departure of the process median from the target value, while the modified estimators clearly differentiate the ontarget process from the other two (severely) off-target ones. 15.2.3 Box-Cox

power transformation

approach

Box and Cox (1964) (B.C) proposed a useful and by now widespread family of power transformations of a (necessarily) positive response variable X . The B.C. times transformation is given as:

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Encyclopedia and Handbook of Process Capability Indices vA _ -i

x^ = •

, for A * 0 A

lnX,

for A = 0.

This (continuous) family depends on a single parameter A which can easily be estimated by the method of maximum likelihood. Indeed first, a value of A from a pre-assigned range is collected. Next, we evaluate Anax = ~ 2 l n ( ? 2 +ln
= -ilnX 8 j=i

where J(A, X) = II ^ i=l OX

= II X?-1

for all A .

i=l

(Thus lnJ(A,X) = (A - l ) ^ " = i l n Z i •) T h e e s t i m a t e o f &2 f o r a fixed A is a 2 = 5(A)/n , where 5(A) is the residual sum of squares in the analysis of variance of X^ . After calculating Anax (A) for several values of A within the range, we plot Lmax(X) against A . The maximum likelihood estimator of A is then obtained as the value of A that maximizes Lmax(\) . With the optimal A value, each of the X data specification limits are transformed to a normal variate using the above equation. Thus, the corresponding PCIs can be calculated from the mean and standard deviation based on the transformed data. 15.2.4 Johnson

transformation

approach

Over 50 years ago, N.L. Johnson (1949) proposed-by now a classical system of distributions based on the method of moments,

Process Capability Assessment for Non-normal Processes

similar to the Pearson system. transformation is given by

The

general

form

303

of

the

z = 7 + T]T(X; e, A)

= 7 + Sg(x) T) > 0 , A > 0, —oo < 7 < oo, —oo < e < oo , X = 9~1

or

(Z-l\ V

Here z is the variable to be fitted by a Johnson distribution and Z is a standard normal variate. The four parameters, 7 , rj , £ and A are to be estimated and g is a function which may take one of the following three forms: the lognormal system (Si), the unbounded system (SJJ) and the bounded system (SB) (see, e.g., Tang and Than (1999)). For example, the Sg (bounded system) involves transformation of the form g(x) = \og{x/(I — x)} . (Z is a standard normal random variable with the p.d.f. /(z) = (l/V2^)e-* 2 /2 ,

zeR,)

and the system Sg results in the p.d.f. of X :

^l7,n) = ^ ^ 15.2.5 Other quantile

r

^ e x

P

transform

-H

7 + 7?1 g

\2

° ((T^

approaches

Gilchrist (1993) has also developed a quantile transform technique that is based on the same philosophy as Clements' method but uses the so-called standard distributions for the transform rather than Pearson's curves. Gilchrist's technique utilizes an inverse transform function to define Xp in terms of p for the modeled distribution (see, e.g., Banks (1996)). As an

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example, suppose the process data are "reasonably" exponentially distributed with the p.d.f. f(x) = Xe~Xx , x > 0, A > 0. In this case, the function Xp = —(1/A)ln(l — p) can be used to estimate ^0.00135 , -Xo.50 a n d -^Q).99865 > where 1/A is the mean of the exponential distribution and also 1/A = X . Large sample sizes are required in order to ensure a satisfactory distributional fits. Chang and Lu (1994) have also extended Clements' technique by proposing a percentile method for calculating PCIs without assuming normality of the distribution, using appropriate statistical tables, or calculating skewness and kurtosis of the data. This approach guarantees that both the indices Cp and Cpk > 1 if all the observations in the sample fall within the specification limits (and the process is admissible). The technique assumes that the sample dataset reflects the entire variation range of the population, in other words, it is a "perfect" sample (which rarely appears in real-world situations). The algorithm developed by Hill et al. (1976) is used to match the first four moments of X to the above distribution families. Then the values of PCIs can be calculated using the transformed data. 15.2.6 Distribution-free

tolerance

intervals

approach

Chan et al. (1988) adopted a distribution-free tolerance interval approach to compute Cp and Cpk for non-normal processes using _ USL - LSL _ USL - LSL _ USL - LSL ~ 6a ~ f (4a) ~ 3(2a) "

v

Process Capability Assessment for Non-normal Processes

305

This results in p

_ USL - LSL _ USL - LSL _ USL - LSL w | ^2 3u; 3 min | USL — n, n — LSL [

where w is the width of the tolerance interval with the 99.73% coverage 95% of the time (corresponding to 3cr in each direction in the normal case), w2 is the width of the tolerance interval with the 95.46% coverage 95% of the time(corresponding to 2a in each direction) and w3 is the width of the tolerance interval with 68.26% coverage 95% of the time(corresponding to 1
15.2.7 Flexible

index Cjkp

Johnson et al. (1994) introduced a "flexible" PCI Cjkp (the letters jkp indicate the first letter of the authors names) which takes into account possible differences in the variability above and below the target value. The C ^ index also provides an advantage of changing its value (slightly) when the process distribution changes its shape (i.e. becoming skewed), something that the other process capability indices do not accomplish. CU^ and CL^ are defined as the capability of the upper and lower halves of the process data, respectively, CjAp being the minimum of the two. That is

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Cjkp = min|C77#p, CLjkp j = min

USL-T T -LSL 3W+ V2 ' 3W_V2

where W2 = EX>T[(X-T)2] and Wl = EX
= E[(X-T)2 -T)2} = E[(X -T)2

\X>T]PT(X>T),

\ X < T]Pr(X

< T).

Johnson et al. (1994) have used the multiplier 1/(3V2) instead of 1/3 (used in other PCIs) because, for a symmetrical distribution with variance a2 and the expected value T EX>T [(X - Tf} = EX
Note that if we have T = (USL + LSL) /'2 = m , so that USL-T = T -LSL = d, where d = (USL -LSL)/ 2, then Cjkp reduces to: Cjkp = ^ = m a x { %

T

{(X - T)2\, EX
T)2}}^.

A natural estimator of C^p in the general case is then Cjkp = r - ^ m i n

3V2

where s+ = J2Xl>T^ that

~ T?

USL-T T -LSL\ yjs+ / n ' yjs_ I n J '

and s

~ = T.Xi
~ T? •

E[8+] = nE[(X - Tf | X > T] P r ( X > T)

Note

Process Capability Assessment {or Non-normal Processes

307

and E[s+] = nE[(X

-Tf

| X > T ] P r ( X > T)

Consequently s+ / n and s_ / n are unbiased estimators of EX>T [(X - Tf ] and EX
15.2.8 The Wright's

Cs index

The index Cpmk provides warnings about the increase of the process variation and the process departure from the target, but is sensitive to the changes in the shape of the distribution, particularly its skewness. A modification of Cpmk proposed by Wright (1995) incorporates a skewness term in the denominator to

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reduce the index value when non-symmetry is present. Utilizing the third central moment /x3 = E(X — /x)3 as a measure of skewness, Wright's index Cs possesses the Cpm^ characteristic of applicability to processes whose mean may not be centered between the specification limits and moreover may not be centered at the target. The index Cs is defined as follows: min{t/SL-/x, (i - LSL] 3yja2 +(n-Tf

+\nz/°\

d-\fi-T\ 3Vc72+(M-r)2+|/x3/a|

(d-\»-T\)/a 3jl+

[(fi-T)/a}2

+1^'

where as usual d = (USL — LSL) / 2 is the half-interval of the specification range, T = (USL + LSL) / 2 is the target value, and fcj = /i 3 / a 3 is the skewness coefficient. The term / j 3 is divided by a to ensure that the skewness term is expressed in the same units as the other terms in the denominator and the absolute value of this term guarantees that a negative skewness will also incur a penalty. In the numerator, the well known identity min(x, y) = (x + y)/2 — \x — y\/2, x, y e R is utilized. In practice, it is required to estimate the unknown process mean fj,, the process standard deviation a, and the third central moment fj,3 , to calculate Wright's Cs. Usually these are estimated from a sample of size n. A natural estimator of Cs is proposed by Wright (1995) to be: d-\X C, =

Tf +

n2m3 (n - l)(n - 2)

n

x

n —if 1

rri2 ~2~ ci

-1/2

where r ^ = Y^i=SXi ~ x? I n , m3 = ^ " = i ^ ~ Xf In a r e the sample central moments and c4 = [2/(n — l)] 1 ' 2 x

Process Capability Assessment for Non-normal Processes

309

r ( n / 2 ) r [ ( r a - l ) / 2 ] " 1 is the corrector for the bias Wright (1995) investigated this estimator of Cs and studied its bias and variance using simulation for normal distributed processes. By evaluating the percentage bias in Cs and Cpm\., he has discovered that Cs is a more accurate estimator than C,prafc for small samples from offtarget processes, when significant shifts in the mean have been occurred. For skewed distributions, Pearn and Chang (1997) examined the bias of Cs, which turn out to be quite substantial. Chan and Kotz (1996) investigate the distributional properties of Wright's index and reach the conclusion that the asymptotic behavior of the estimator Cs is actually sensitive to skewness, and the contribution to overall variability of the skewness correction factor is additive and non-interactive. These authors also substantiate Wright's conclusion that the asymptotic distribution of Cs is normal when /i ^ T and /ix3 ^ 0 , which makes it a natural choice for process capability assessment for large sample sizes, regardless of the underlying distribution of the data. In general the Cs index provides a statistically sound approach to process capability analysis under the three (common) conditions: (1) USL-^^fi-LSL (i.e. n * M = (USL + LSL)/2 ); (2) \i ^ T ; (3) /i 3 ^ 0 . In our discussion, the process is assumed to be exactly centered ( \x = T ) which implies that Cs = Cpk = Cpmk for symmetric distributions.

15.2.9 A superstructure

capability

indices

CNp(u, v)

As it was mentioned in previous Chapter, Vannman (1995) constructed a superstructure of indices Cp(u,v) which includes the four basic indices, Cp , Cpk , Cpm and Cpmk as special cases. However, the indices Cp(u,v) seems to be appropriate only for the normal and near-normal processes. To accommodate those cases where the underlying distributions may not be normal, Pearn and Chen (1997) considered a generalization of Cp(u,v) defined below which is called CNp (u, v), suitable for application to processes with

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arbitrary distributions Recall that d — u\ fi — m\ • '

CP(u, v) =

Sja2

+v(fi-T)2

,

(15.1)

Pearn and Chen (1997) modification is: d - u\M - m\ ^

CNp(u,v) = —f= r

99.865

—

^0.135

(15.2) 2

+ v(M - T)

Here Fa is the a th percentile of the underlying distribution, M is the median of the distribution, m = (USL + LSL) / 2 is the mid-point between the upper and the lower specification limits, JJL and a are the process mean and the process standard deviation, respectively and u, v > 0 . In developing this generalization C?fp(u,v), Pearn and Chen (1997) replaced the process mean /i by the process median, M (a more robust measure for process central tendency, particularly for skewed distributions with long-tails), and the process standard deviation a by (F 99865 — F0,135)/6 , calculated from the distributional percentiles in the definition of the original index Cp(u, v). The idea behind such replacements is to mimic the property of a normal distribution that the tail probability that the process is outside the ±3
Process Capability Assessment for Non-normal Processes

311

which can explicitly be expressed as: USL - LSI CfJr, 'Np —

•^99.865

—

(15.3)

-^0.135

USL-M

CNVk = min

M - LSI r

•99.865 ~~ ^0.135

99.865

—

, (15.4)

^0.135

to mimic the property of a normal distribution that the tail probability that the process is outside ±3
^Npm

and C Npmk

~

~

4

w

[ ^99.865

—

^0.135

(18.8)

I2 + (M - Tf

6

USL-M -^99.865

—

^0.135

2

6

+ (M - Tf

M -LSL

si

-^99.865 ~ -^0.1 35

6

(15.6)

2

+ (M - Tf

Zwick (1995) and Schneider et al. (1995) consider two generalizations of Cp and Cpk , which are similar to C^p and Cpfpk but using process mean /z rather than process median M in the basic definitions. Extending their definitions to include the indices Cpm and Cpmfc , a superstructure:

CiNp{u,v) =

d — u\fi — m\ ^99.865 ~ ^0.135

6

+ «(/x - T)

(15.7) 2

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could be constructed, designated C'Np(u,v) (Chen and Pearn (1997)). Note that (15.3) is a variant of the original CP(u,v). Chen and Pearn (1997), Pearn and Kotz (1994), and Pearn and Chen (1995) propose estimators for calculating the indices Cp(u,v) assuming that the underlying distributions are of the Pearson types. These estimators essentially apply Clements' method (1989) (described above) by replacing the unknown 6CT distance by Up — Lp , which is calculated based on the available sample data obtained from a stable process using the estimates of the mean, standard deviation, skewness and kurtosis. These four parameters determine the type of the Pearson distribution curve; hence the Fa percentiles of the Pearson curves as a function of skewness and kurtosis can be calculated utilizing, for example, the tables provided by Gruska et al. (1989). These estimators are of the form CNp(u,v) =

d — u\M — m Up — Lp

l2

+ v(M - Tf

where Up estimates the 99.865 percentile F 99 865, Lp — the 0.135 percentile F0 135, and M is an estimator of the median. To obtain the values of Up , Lp and M Gruska et al. (1989) tables supplemented by certain interpolation are required. Based on sample percentiles, Chang and Lu (1994) consider a different method for calculating the -F99.865, ^0.135 > a n d the median M for non-normal distributions. Their method is essentially based on sample percentiles which can be calculated using only interpolations, and do not require the use of in Gruska et al. (1989) tables. Applying this method one can obtain the percentile estimators for C^p(u,v)&s

Process Capability Assessment for Non-normal Processes

CNp(u,v)

d — u\M — m\

=

•^99.865 ~~ -^0.135

+ v(M - Tf

6 where

313

F 99 . 865 = X{[Rl]) + {fy - [i^]} x {X{[Ri]+1) - X{[Rl]) } , ^0.135 = ^([i%]) + {-^2 - [-^2]} X {-^([iJj]+l) - X(.[Ri])}> M

=

X

([R3]) + iR3

_ /99.865n + 0.135 \ \ 100 /

~ [^3]}

x

{^([iJ3]+i) -

_ /0.135n + 99.865 \ \ 100 }'

x

([i? 3 ])} '

_/n + T ° \ 2 I

The notation [R] signifies the greatest integer less than or equal to the number R , and X^ is as usual defined as the i-th order statistic.

15.2.10

The Cpc

index

Luceno (1996) introduced the Cpc index, which is designed as to consider both the process location and the spread, but the other indices also provides for the confidence bounds that are insensitive to departures from normality. This index is defined as „

_

USL-

LSL

' pc

6j^E\X~m\ where m = (USL + LSL) / 2 . Luceno (1996) mentions that the second subscript in Cpc stands for confidence, to indicate that the confidence intervals based on Cpc are reliable. This statement however ought to be interpreted rather cautiously. The author also uses the factor 6-7^/2 = 7.52 in the denominator, to equate

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it to 6cr in the case of the normal distribution N(fi,a2) . Note, however, that the Q-JTT/2 factor does not depend on the normality. Luceho (1996) developed a 100(1 — a)% confidence interval for Cpc by first evaluating | X — m | , whose expected value is estimated by the sampling analog: _ 1 " c = - Y] I Xi - m |, thus leading to the estimator n

—

USL - LSL < #

A 100(1 — a)% confidence interval for E\X — m\ is given by T +t u

-1- ua, n - 1

A. /— )

where ta/2,n~i i s the a/2 quantile (i.e. the value of that is exceeded with probability a/2 by the Student's t distribution with n — 1 degrees of freedom) and

;

l

A„

, _,,

l

= ^rE(l^- m l- c ) 2 =^ri ElXi — m

|2

—nc2

Therefore, a 100(1 — a)% confidence interval for Cpc is given by n

Process Capability Assessment for Non-normal

Processes

315

Lucefio (1996) comments that the confidence intervals are insensitive to departures from normality since they are based on the Central Limit Theorem. In addition, approximations are available for this index for n large. For instance, when X is distributed as N(n, a2) , the quotient sc/~c is close to ( T T / 2 - 1 ) 1 / 2 » 0.7555 , so that the 100(1 - a)% confidence interval for Cpc can be compactly approximated by Cpc/(l±0.7555iQ/2,n_1Vn). A critical region for the hypothesis H0 : Cpc = Cpc0 versus Hi : Cpc < Cpca at a level of significant is ~c-Jn C,pc -1 CBCO

15.2.11 The (general)

Weighted

-^ ''a, n—1 •

Variance

(WV)

method

The weighted variance (WV) method, already alluded to in an earlier Chapter, was first introduced by Choobineh and Ballard (1987) to construct control charts when the underlying population is skewed. The idea is based on the "semivariance" approximation of Choobineh and Branting (1986). Their method provides asymmetric control limits on X and R charts for skewed distributions. Bai and Choi (1995) also proposed a simple heuristic method to construct X and R charts using the weighted variance method with no assumptions on the population. This method provides asymmetric control limits in accordance with the direction and the degree of skewness estimated from the sample data by using different variances in calculating upper and lower control limits for skewed or asymmetric populations. When the population is symmetric, these charts are reduced to the Shewhart control charts. Choobineh and Ballard (1987) conclude that the weighted variance method gives the same limits as the Shewhart method in the case of a symmetric underlying distribution. When

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the underlying distribution is skewed, however, the new limits are adjusted in accordance with the direction of the skewness. The study of Bai and Choi (1995) also indicates that for symmetric populations these control charts are equivalent to the Shewhart charts. However, when the underlying population is Weibull or Burr, the "heuristic charts" are found to be superior over Shewhart charts especially as the skewness increases. Bai and Choi (1997) have utilized the weighted variance method to adjust capability index values in order to account for the degree of skewness of non-normal process data. This technique computes separately the standard deviations above and below the mean. The Cp and Cpk indices based on the weighted variance method, C^v and C^, are defined as: C?v

=min\

USL - LSL 6axy[2Px

USL - LSL -mm 6cr T

and

^Pk

— min-j

USL - LSL 6^V2(1 - Px) 1

1

m^2{\-px)\

USL — nx 3ax^2K'

nx — LSL 3axj2(l-Px)

where Wx = 42 max (4TX, 41 - Px ) = J1+ \1-2PX\ Px=Pi(X<^).

where

Similarly, the Cpm and C*m indices (defined by Chan et al. (1988)) can be modified as CpVf = min •

USL - LSL USL - LSL 6aTj2fy ' 6aTj2(l - PT)

USL - LSL -min V 2 ^ ' 6(7T

c,pm yl2(l-PT)

Wr

Process Capability Assessment for Non-normal Processes

and

Cpm

= min-

USL-T 6<JTJ2F£,

317

T-LSL §oTJ2(l-PT)y

where, now PT = P r ( X < T ) , WT = Jl+ | 1 - 2PT | and aT — yjal + (n — T)2 . To evaluate these indices, the parameters \ix , ax and
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a similar formula to the one used to calculate the sample variance in this situations as: 2^/(Xl-Xf Si = - ^ 2nx - 1

2^(Xt-Xf and Sf = -i=i ' 2«2 - 1

On the other hand, the sample standard deviations from the target T, STi and ST2 , can be expressed by the following equations using the Taguchi's unbiased estimator ( ST = Y^l=l(Xi -Tf /n) as given in Boyles (1991):

2Yj{Xi-Tf S k = - ^

2

=

2ni

-^Sl+{X-T)\

2rii

2J2(X1-Tf

and

Sh- «

=^SUtX-T?.

For the original weighted variance method (as used by Bai and Choi (1995)), the expression for Sf , 5 | , STi a n d ST2 can be modified to be:

5X2(B&C)

E(^-^)2

= 2(1 - Px)^

lb

E(^-^)2

,Sf (B&C) = 2PX ^

—-

i

X

E(^-r)2 $ i ( B & C ) = 2(1 - PT)-1^ n n

E(^-T) 2 ST2 (B&C) = 2P T i=i

l

l

.

Process Capability Assessment for Non-normal Processes

319

where, as above, Px = P r ( X < /x) and PT =?x{X
=

UoL — LSL

fiwv

3(5 +s)

pk

= mm

USL-X

X - LSL

002

ob\

Moreover, the denominator ST in the estimators Cpm and Cpmk is replaced by Sn and ST2 using the expressions of the WV method. Then, the estimators C^ and C 1 ^ c a n be expressed as:

^wv

. \USL-T

C„ = mm — J m pm I

and

C^k = min

OOx2

USL-X 0&T2

T-LSL]

,——

OOTI

)

X - LSL ODyj

The advantage of using the WV method to modify normally-based process capability indices is that the weighted variance-based capability indices reflect the values of the skewness and kurtosis of a distribution, since their sample standard deviations, Si and S2 ,

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and the sample standard deviations from the target, S^i and ST2, are adjusted by means of different skewness and kurtosis values. Moreover, the samples of size nx and no. vary as the skewness and kurtosis undergo changes. On the other hand, when the distribution is symmetric, S\ and S2 are equivalent to S and so are Sx\ and 5 , r 2 , while n\ and no, are approximately equal to n / 2 . Consequently for symmetric distributions the weighted variance-based capability indices turn out to be the original process capability indices. 15.2.12

The Weighted

Standard

Deviation

(WSD)

method

Chang and Bai (2001) suggested a WSD method to construct control charts in accordance with the degree of skewness of the underlying population. Similarly to the WV method (Section 15.2.1) the WSD method is based on the idea that ox can be divided into upper and lower deviations, ofj and oY , which represent the degrees of the dispersions of the upper and lower sides from \ix , respectively. An asymmetric p.d.f. f(x) can be approximated by two normal p.d.f. s

fu(x) = irw Zov

Vx

2ov

and

fdx) =

W

4,

%

l-t>x

~2aT

with the same mean \ix but different standard deviations lay and loY i i-e-j t n e upper and lower sides of f(x) are approximated by the upper side of fu(x) and the lower side of fi(x) , respectively. The WSDs off and oY are obtained as o$ = Pxox and oY = (1 — Px)ox , where, as above, Px = P r ( X < \ix) . Details can be found in Chang and Bai (2001).

Process Capability Assessment for Non-normal Processes

321

The WSD method can be used to adjust the values of PCIs according to the degree of skewness of the underlying distribution by using different standard deviations when computing the upper and lower PCIs. Namely, 2ajj and 2oY can be used to construct PCIs for skewed populations. The Cp based on the WSD method, CfSD, is defined as (see, e.g., Chang et al. (2002)): C™D = min

~

USL - LSL 6-2^

USL - LSL '

. [ USL -LSL l Q-2Pxax '

m m

_ USL - LSL Qax

6-2af USL - LSL 6-2(1-Px)ax

. |J_ 1 \ _CP l 2 P / 2(1 - Px)\ ~ D,'

m m

where Dx = 1+ | 1 — 2PX \. In this definition, the quantities 2ojf and 2aY are used in place of ax to reflect the degree of skewness. If the underlying distribution is symmetric, Px = 0.5 and Dx = 1.0 , i.e. C^SD = Cp . For skewed distributions, however, Dx > 1.0 and thus C^0 < Cp. Chang et al. (2002) have also considered the situation when only a single specification limit exists. In this case using the WSD method, the upper and lower capability indices stemming from Cpk are defined as: n

WSD pku

_ USL — [ix _

3 • 2aJf

USL — [ix 6PX"xaX

and riWSD _ Vx ~ LSL _ \ix — LSL Pkl

O-W 3Q • 2<

6Pxax

The capability index for the case of two-sided specifications becomes

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Cpk

f USL — fJLx fix — LSL

- min(C p f a , Cpkl ) - m m |

Qp^

,

_

For a symmetric population, CJiJt'51' reduces to Cpj. , and whenever \ix is located at the center of the two specification limits, C^kD reduces to CfSD . To apply these indices in practice, \ix, ax , and Px ought to be estimated. Let a random sample Xx, X2 , ..., X„ of size n be available, /i^ and ax can then be estimated by the sample mean X and the sample standard deviation Sx . Since Px is the probability that X will be less than or equal to \xx (i.e. Px — Pr (X < /J,X ) ) , it can naturally be estimated by using the number of observations less than or equal to X : 1 "

n J=I ., where the characteristic function I(x) = 1 for x > 0 and I(x) = 0 for x < 0 . Thus, the proposed PCIs C\fD and Cf OT can be estimated as WSD

C.

USL — LSL 6 • DXSX

and ^Pk

— rnin|Opfo, , o p H j — m m

USL6PXSX

X

X - LSL 6(1 - PX)SX

where D, = 1+ I 1 - 2PIt is well known that Sx underestimates ax , hence the estimate of a PCI overestimates the true value, and the bias correction factor will be (as noted in earlier Chapters):

Process Capability Assessment for Non-normal Processes

323

r[(n-l)/2]

^ = £?Tr [ ( n - 2 ) / 2 ] which is applicable whenever n is small. Note that b(n) is obtained under the normality assumption, and thus cannot be directly used for skewed populations. Therefore, a bias correction factor that also reflects the degree of skewness ought to be determined for skewed populations. The WSD method approximates an asymmetric distribution by means of two normal distributions, and it is assumed that nPx observations among the n observations are obtained from the lower side of the p.d.f., fi{x) , and n(l — Px) observations from the upper side of fu(x) . Hence, if 2nPx and 2n(l — Px) observations are viewed as random samples from fL (x) and fv (x), respectively, the bias correction factors bYSD = b(2nPx) and ^WSD _ fr(2n(l — Px)) could be used to reduce the bias. Using these correction factors, the estimated WSD PCIs finally become: nWSD

USL — LSL

r<WSD _ ___•

^pk

WSD

WSD

2PX

2(1-PJ

[CWSDfiWSD

— mill 10(7

Opfoj , 0L

iWSDr<WSD\

Opfc;

|,

where bfSD = b(2nPx) and bJ[SD = 6(2n(l - Px)). As Px increases, so does bYSD, while bjjSD decreases. Consequently the skewness is reflected in the correction factors. In addition, the performance of these WSD PCIs compared with the various PCIs, was analyzed, and finite sample properties of these estimates were also investigated by Chang et al. (2002) utilizing Monte Carlo simulations. Numerical results show that substantial improvements over the existing methods can be achieved by using the WSD method when the underlying distributions are skewed.

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Ahmed and Abdul Kader capability index

(2004) consider the

following

„ _ allowable process spread or tolerance actual process spread or variation where, as above, LSL = lower specification limit and USL = upper specification limit and o is the standard deviation associated with the process. Let R be the specification range, i.e., R = USL — LSL. The formula above can be rewritten as Ct=^,

to

o>0,

where t is a positive constant. This constant is called the tuning factor and is chosen so that the probability Pr{^i — to < Y < JJL + to} will be close to 1. This index is less sensitive to departure from normality. For t — 6 , Ct = Cp. The number 6 associated with the assumption of a normal distribution for Y , provides a satisfactory approximation. For t = 6 , a value of Ct greater than or equal to 1 implies that the tolerance band (USL — LSL) is at least 6a of the process distribution. If \i is the location parameter of the process population, then for t = 6 , by the Chebyshev inequality at least 89% of the products are within the specification limits. On the other hand, if the process distribution is normal, then by the properties of the normal distribution approximately 99.73% of the products will be within the specification limits. Pearn et al. (1992) discuss the problem of dealing with the above index whose interpretation (unlike that of the other standard indices such as Cp). ) is less sensitive to departures from normality. The capability index as defined above, is a multiple of Cp and has a quite similar interpretation provided the samples are drawn from a normal population. For example, for t = 5.15 , C5.15 = 1.17Cp . Thus, one can paraphrase the interpretation of Cp as given by Ekvall and Juran (1974) and assert that the values of C 5 1 5 less than 1.17 should be classified as inadequate, the values between 1.17 and 1.56 are designated as

Process Capability Assessment for Non-normal Processes

325

adequate, and the values greater than 1.56 are considered to be even better than adequate. However, a practical interpretation of C515 is not fully apparent for data populations that are substantially skewed.

Chapter 16

Multivariate Process Capability Indices

16.1 Introduction 16.2 Multivariate PCIs 16.3 Concluding remarks 16.1 Introduction This Chapter deals with some more recent developments in the field of Process Capability when several characteristics of a process are involved. For completeness we shall present again some basic facts related to the subject matter under consideration, in our attempt to render this Chapter to be as self-contained as possible. Various quality measures have been proposed to evaluate a process's performance. In general, there are several issues to be considered when assessing product quality. First and foremost is the issue of process capability indices. A process capability index (PCI) is a numerical summary that compares the behavior of a product or processes characteristics related to engineering specifications. Its convenience is due to the reduction of complex information about the process to a single number. Specifically, process capability is universally defined as the range over which the measurements of a process vary when the process variation is due only to random causes. In this case, process capability indices do provide an effective measure of process capability. By using a process capability index, we may obtain a conclusion whether the product meets its specifications and is stable. Currently, numerous 326

Multivariate Process Capability Indices

327

customers request their suppliers to record capability indices for product characteristics on a regular basis. In the majority of companies it became a key index to evaluate product quality. A large number of quality engineers and statisticians have proposed methodologies for assessing product/process quality. However, the bulk of the studies associated with analyzing the quality and efficiency of a process, are so far limited to discussing one single quality specification. In most processes, the products possess multiple quality characteristics rather than a single one. Multiple characteristics processes are by now so common that our studies to capability indices can't be restricted to the univariate domain. The multivariate relationship among the quality characteristics may or may not be reflected in the engineering specifications. For instance, USL 's and LSL 's may be given separately for each quality characteristic. In two-dimension cases, these tolerance ranges compose a rectangular tolerance region. In higher dimensions, they form a hyper cube. For more complex engineering specifications, the tolerance region could be quite complicated. For processes with multiple characteristics, Bothe (1992) proposed a simple measurement of a tolerance region by taking the minimum of the measure for each single characteristic. Consider a v -characteristic process with v yield measures (percentage of conformities) Pu P 2 , ..., and v . The overall process yield is measured as P = minjPx, P 2 , ..., u}. However this approach does not accurately reflects the real situation. Suppose the process has five characteristics ( v = 5) with equal characteristic yield measures P1 = P2 = P3 = Pi = Pt = 99.73%. Using the approach considered by Bothe (1992), the overall process yield is calculated as P = min{i\, P2, P 3 , P 4 , P 5 } = 99.73% (or equivalently 2700 PPM of non-conformities). Assuming that the five characteristics are mutually independent, the actual overall process yield should be calculated as P = Px x P 2 x ... x P 5 = 98.66% (or 134273 PPM

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of non-conformities), which is significantly less than calculated by Bothe (1992) method. When these variables are related characteristics, the analysis ought to be based on a multivariate statistical technique. Chan et al. (1991), Taam et al. (1993), Pearn et al. (1992), Chen (1994), Karl et al. (1994), Shariari et al. (1995), Boyles (1996), Wang and Du (2000), Wang et al. (2000) and others have developed and presented multivariate capability indices for assessing capability. Wang and Chen (1998) and Wang and Du (2000) proposed multivariate extentions for Cp , C^ , Cpm and Cpmk based on the principal component analysis, which transforms numbers of original related measurement variables into a set of uncorrected linear functions. A comparison of three novel multivariate methodologies for assessing capability are illustrated (and their usefulness is discussed) in Wang et al. (2000). 16.2 Multivariate PCIs Some basic knowledge of the classical multivariate analysis based on multivariate normal distribution is required to fully appreciate the theory behind the Multivariate PCIs. For the multivariate case, it is even more dangerous to use a single number as a measure for the capability of process than in the univariate case. In such cases, it has been usually assumed (up till very recently) that the observations X posses a multivariate normal distribution Nv([i, E) , where v is the dimension of the variables, fi is the mean vector and E represents the variancecovariance matrix of X . Chan et al. (1991) introduced a version of the multivariate index Cpm . They have considered measurements of a set of v characteristics Xi, x2, ,xv . For specification region, all points lying on or within the specification boundary satisfy the inequality (X -T)'A-\X

-T)

< c2,

(16.1)

Multivariate Process Capability Indices ./

__

r r r

,

_ ,

329

j

where X = [x\, ,xv] , T = [Tlf ,TV] and A is a « x « positive define matrix. Here T and ^4 completely determine the shape and center of the ellipsoid, while c determines the coverage of the ellipsoid. The coverage is the percentage of observations expected to be within the boundaries. Chan et al. (1991) proposes a multivariate process capability analog of Cpm defined by Cpm

'^[(Xi-TyA-^Xi-T)]'

(16 2)

'

For a known A, an unbiased estimator of the denominator of C2pm is

\Y.{Xi-TfA-\Xi-T), n .,

and a natural estimator of the multivariate Cpm becomes: r

_ ,

nv

fciXi-Tj'A-^Xi-T) *=i

where Xt denotes the i th vector of dimension v ( v is the number of variables used in assessing capability), n is the sample size, T is the target value and A is a matrix representing the covariance structure determined from the specification limits. The numerator of Cpm is the product of the sample size n and the numbers of variables v (used in evaluating capability). This value indicates the degrees of freedom associated with the denominator representing the sum of the observed the so-called Mahalanobis distances from the target. For processes with observations clustered around the target, the denominator of Cpm will be smaller in magnitude than that of the same process, with more scattered and with a center of mass located not at the target value. For a fixed nv observations, the value of Cpm will diminish when

330

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the denominator grows. Smaller values of Cpm indicate that the process is unable to meet the specifications, while larger values indicate that the process is indeed capable of meeting its specifications. Based on equation (16.2), Pearn et al. (1992) have introduced two multivariate PCI which are viewed to be more natural generalizations of Cpm proposed by Chan et al. (1991). Setting H = T and £ = A , the value of c2 in the specification requirements leading to the proportion 0.0027 of non-confirming items becomes the upper 0.9973 quantile of a chi-squared distribution with v degrees of freedom. This is a basic property of the multivariate normal distribution presented in any text on multivariate analysis (e.g. Johnson and Wichern (2004)). Indeed: let T2 = XTS^X, where X and S are independently distributed random variables with X ~ Nv([i,Yl) and fS ~ Wv(/,]C), / > p • Then XT £ ~ : X ~ X2(P,T2) with r 2 = fiT £ ~ 1 fi . Denoting this 2 2 value by c (c = xi 0.9973 ) one defines 2 VC P

=4

(16.3)

cv as a generalization of Cp where c2 is determined by (16.1). Analogously one can define C:2 vCpm 1

<ji-T)'A-\n-T) v

as a generalization of Cpm , where Vol. ((X - ii)'A-\X v

p

Vol. ((X - n)'A~\X

- /x) < K2) - M) < xl 0.9973)

(16 4)

"

331

Multivariate Process Capability Indices

For /J, = T , E = A and „ C2p = 1, the expected proportion of NC items is 0.0027 as in the univariate case. It should be noted that for [i •*• T and E ^ A, the value cl satisfying P[(X - T)'A-\X

-T)
= 0.9973

— under the multivariate normal assumption — depends on /i, T, E, and A in a rather complicated manner and /z and E ought to be estimated. Taam et al. (1993) proposes a multivariate capability index defined as a ratio of two volumes n MCpm

_ Vol. (Rx) _ Vol. (modified tolerance region) ~ Vol. (ifc) _ Vol. ((X - p)W(X -fi)< K{q))'

, (

. j

where R1 is a modified tolerance region (to be defined below) and Rq is a scaled 99.73% percent region. Rq is an elliptical region under the normality assumption. A modified tolerance region is the largest ellipsoid that is centered at the target completely within the original tolerance region. When the process of multiple characteristics is in statistical control so that the process mean is on target and the process variation is small relative to the engineering specifications, the multivariate MCpm should be close to 1. An estimator for MCpm can be expressed as: Q

MCpm = -J- , where G„v —

Vol. (tolerance region) Vol. (estimated 99.73% process region)

(16.6)

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Encyclopedia and Handbook of Process Capability Indices

Vol. (tolerance region) ISI 1 / 2 (7rK(v)y/2[T(v J2 + l)]- 1 ' and

77

-

,

-

l1/2

D = 1 + — — (X - T) S-\X - T) n —1 Note that | S | 1 / 2 (?rii'(7j))''/2[r(7j/2 + l)]^ 1 is a generalization of the volume of a sphere and from elementary geometry where K(v) is the 99.73% percentile of the x 2 distribution with v degrees of freedom and |-| denotes the determinant. When the process mean vector is equal to the target vector and the index has the value 1, then 99.73% of the process values lie within the modified tolerance region. The numerator of Cp is analogous to Cp in the univariate case. Namely value of Cp greater than 1 implies that the process has a smaller variation than it is allowed by the specification limits with a certain confidence level; on the other hand, a value less than 1 implies a larger variation. Analogously, 0 < 1/ D < 1 measures the closeness between the process mean and the target; a larger 1/ D would imply that the mean is closer to target. Chen (1994) proposes a multivariate process index using the concept of a tolerance zone. A tolerance zone is expressed as V = {XeRv

: h(X - ^) < r0},

(16.7)

where h(-) is a nonnegative homogeneous scalar function satisfying the condition h(tx) = th(x) for all t > 0 and r0 is a positive number. A process is considered to be capable if P(X e V) > 1 - a , i.e., P(h(X - /x0) < r0) > 1 - a , where a is the allowable expected proportion of nonconforming production from a process (conventionally, a — 0.27% (about 1/4 of percent)). Let now r = min{c : P(h(X — fi0) < c) > 1 — a} . It should be noted that if the c.d.f. of h(X — /j,0) is increasing in a neighborhood of r , then r is simply the unique root of the equation P(h(X - /i0) < r) = 1 - a. The process is deemed to be

333

Multivariate Process Capability Indices

capable if and only if r < r0 , that is, (r0 / r) > 1. This allows us to express the multivariate process capability in the form MCV=^.

(16.8) r

Here, the value of MCP corresponds to the expected proportion of conforming products of 1 — a . Namely, MCp = 1 indicates that the expected proportion of nonconforming products is exactly a . The index MCp is well-defined, since a tolerance zone V (Equation (16.7)) is specified by the function h uniquely up to a scale constant and MCP is scale-invariant. The definition of MCp provides the following advantages: (i) the specifications are general as given by V (ii) there is no assumption about the underlying particular distribution, such as the normal one, and (iii) the arbitrariness of a permits flexibility in setting a criterion for the capability of a process. In the case when the function h is of the form: h(X -T)

= [(X - TJA~\X

-

T)fl2

with a v x v positive definite matrix A, V (the tolerance zone) is an ellipsoidal specification given by { X e Rv : {X - T)'A-\X

-T)
}.

where r0 determines the coverage of the ellipsoid. For this tolerance zone and a = 0.0027 , MCp reduces, under the normality assumption to the multivariate PCI proposed by Pearn et al. (1992). Setting ft = T and E = A , MCp becomes in this case. r0 /(o"Xu,a) > (where Xv,a is the 100(1 — a ) t h percentile of the X2 distribution with v degrees of freedom). Here a2 =(n-l)-1J2ni=l(X-X)'A~\X-X) so that MCp can be estimated by r0 /{aXv,a) •

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Furthermore, considering that a rectangular solid tolerance zone: V = {XeRv:\Xi-^ii\
,i = 1,

2,...,v},

where /% is the process mean f i = 1, 2,...., v) and rt is a positive constant, we arrive at the multivariate process index MCp 1 / r , where r is such that: P(max{|Xj -/Mal/n,

i = 1, 2,....,v} < r) = 1 - a.

Let F be the c.d.f. of the variable h(X - fi0) = max{|Xj - ^ o l / n , * = 1, 2,...,«}, Setting r = F _ 1 ( l — a ) , it follows immediately that for any y > 0 F(y) < mm{P\Xi

-fio\/rty,

i = 1, 2,...,v}.

If the value of MCp is greater than or equal to 1, the process is capable at a certain confidence level. Thus, a necessary condition for a process to be capable over a rectangular solid zone is that each individual univariate process is capable with the corresponding specification limits. This seems to be an appropriate condition. Indeed, this fact is the starting point of the Kocherlakota and Kocherlakota (1991) bivariate generalizations of Cp . Karl et al. (1994) introduce applications of multivariate capability index in geometric dimensioning and tolerance (GDT). Their paper presents three cases of GDT drawings of the same product and the corresponding capability indices evaluations. Shariari et al. (1995) proposes a multivariate capability vector, consisting of three components. The first one is a ratio of areas or volumes, analogous to the ratio of lengths in the univariate Cp .

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Multivariate Process Capability Indices

The numerator is the area (in the two dimensional case) or the volume (in three or more dimensions) defined via the engineering tolerance region. The denominator is the area or the volume defined by the "modified process region", which is the smallest region, similar in its shape to the engineering tolerance region, circumscribed about a specified probability contour. The number of dimensions of the process data is accounted for by taking the u t h root of the ratio (see also Wang et al. (2000)). Thus, the first component, labeled CPM , is defined as Vol. of engineering tolerance CPM

Vol. of modified process region

1/v

(16.9)

This approach yields a "modified process region" by drawing the smallest rectangle around the ellipse. The edges of the rectangle are defined as the lower and upper process limits {LPLt and UPLi where i = 1, 2,..., v) and are determined by solving the equations of first derivatives, with respect to each xt , of the quadratic forms such as (X-Lio)'m-l^) = xla,

(16.10)

where X is a v x n sample matrix, v being the number of product quality characteristics measured on a given part and n is the number of parts measured (see also Nickerson (1994)). Namely, each column in the matrix represents the v measurements recorded for a sampled part. These n observations are assumed to be independent and represent a sample drawn from a multivariate distribution with a correlation among the v variates. The two solutions to this equation, for each dimension, provide the upper and lower limits

UPLi = m+ A

—,-—

V det^" 1 )

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Encyclopedia and Handbook of Process Capability Indices

and

LPLi = ^

det^" 1 )

'

, i = l, 2,...,v,

(16.11)

where Xv,a is the upper a quantile of a \2 distribution with v degrees of freedom associated with the probability contour and det(S^ 1 ) is the determinant of E^ 1 (a matrix obtained from E^ 1 by deleting the ith row and column). As a result, the CPM can now be expressed as i/t)

HiUSL, - LSh) CpM —

i=\

(16.12)

Y[(UPLt - LPL,) j=i

If this value is greater than 1, this indicates that the circumscribed "modified process region" is smaller than the engineering specified region, namely the part is "acceptable". Evidently, the modified process region is influenced by the shape of the elliptical contour (represented by the variance-covariance matrix) and the size of the contour (based on the chosen probability level). The second component of the Shariaried al.(1995) index is the significance level of the observed value with the Hotelling's T2 statistic T2 = n(X - n)'S-l{X

- fi)

(16.13)

expressed as PV = Pr

F

>

n ~ v T2 v(n — 1)

(16.14)

Multivariate Process Capability Indices

337

where Fv
if modified process region is contained within the tolerance regions

LI = 0

otherwise.

Consequently, the multivariate capability vector has three components, [ CPM , PV, LI } providing a comparison of the volumes of the regions, locations of the centers and location of the regions. Wang and Du (2000) proposes an useful method using the principal component analysis (PCA) for describing the process performance for a multivariate data. As a rule, multivariate PCIs assume the multivariate normal distribution. But for an application of the PCA this assumption is not required. Advantages of this approach are that the principal component analysis is able to not only to transform high-dimensional problems into lower dimensional ones, but also can transform correlated data into independent data, which simplifies calculations of a multivariate PCI. In addition, the corresponding confidence intervals are also derived. We shall expand on the structure of this method below: Assume that X is a v x n sample data matrix, where as usual in this chapter v is the number of product quality

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characteristics from one part and n is the sample size of the measured part. X is the sample mean vector ( v x 1 ) of the observations and S is a v x v symmetric matrix representing the covariance between the observations. Engineering specifications are given for each quality characteristic. Here LSL and USL are the v vectors of the lower and upper specification limits, respectively. The T vectors (v xl) represent the target values of the v quality characteristics. By using the spectral decomposition theorem, we can obtain a matrix D = UTSU , where D is a diagonal matrix. The elements of D , Aj, A2, A„ , are the eigenvalues of S and the columns of U , ux, it?, •••,uv are its eigenvectors. Consequently, the i th principal component (PCi) is expressed as PCi = uTx, i = 1,2,...,«,

(16.15)

where x s are u x l vectors of the original observations on the variables. The engineering specifications and target values of PCt s are LSLPC. = ujLSL • USLpC. = ufUSL,

i = 1, 2, ...,v ,

(16.16)

TPCi = ujT. Similarly, the relevant sample estimators, S and X , of PC^s can defined as SpCi

XpCl = ufX,

=

\)

i = 1,2,...,«.

It is known (see e.g. Johnson and Wichern (2004)) that the ratio of each eigenvalue to the sum of the eigenvalues is the proportion

Multivariate

Process

Capability

Indices

339

of variability associated with each principal component variable. Namely, the proportions of variability are Ai/j>,

Vi = l,2,...,V.

(16.17)

j=i

Usually, only a few principal components contribute to the bulk of the total variability (about 70%~90%). By using just this subset of the components, the dimension of the multivariate quality characteristic problem can be reduced. Jackson (1980) proposes a X2 test for identifying the significant components. Using this method, we can easily choose the suitable number of PC^s. There are of course more recent method to identity significant components (see, e.g. Johnson and Wichern (2004)) but Jackson's method seems to quite straightforward. Subsequently in the paper, the procedure of obtaining multivariate capability indices for the normal data (as well the non-normal data) is described in the following manner (see Wang, and Du (2000)). a. Multivariate normal data Define

MCV

m

MCpk

(16.18)

P;PC,

«=i I v

and

l/v

l/v

C

Yl Pk\PCi Vi=l

where ~

a

vk\PCi

USLPC. — LSLPC.

min[USL PCi - XPCi, XPCi - LSLPCi }

(16.19)

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Encyclopedia and Handbook of Process Capability Indices

Hence an approximate 100(1 — a)% confidence interval for MCP is —

\l/v

Xl-g/2,n-l

IK

p-PC,

n-1

i=l

<MCp

<

n

l/v

A

jXa/2,n-l

(16.20)

Analogously an approximate 100(1 — a)% confidence interval for MC„i. turns out to be l/v

1 1 Cpk;PCi

1-Z,

all

1 1 9nC.pk;PC-t + 2(n - 1)

< MC,p i l/u

<

C

Yl pk;PC, l + Za / 2

1 1 2(n - 1) 9nC,pk\PCi- +

(16.21)

where in (16.20) X^-i, a / 2 a n d X^-i,i- Q /2 a r e t h e u P P e r a / 2 a n d the 1 — a j 2 quantiles of a chi-squared distribution with n — 1 degrees of freedom, ZQ/2 is the upper a/2 quantile of the standard normal distribution and the Hilfiey-Wilson approximation to the x2 distribution is utilized (c./. Section 3.2.3). The estimators Cpm-pct and Cpmk.PCi can be developed in a similar fashion. b. Non-Multivariate normal data As it was mentioned earlier, Luceno (1996) proposed a process capability index, Cpc , designed to account for both the process location and spread, as well as to provide for confidence bounds that are insensitive to departures from a normal data. Wang and

Multivariate Process Capability Indices

341

Du (2000) suggest that for this type of data the process capability could be estimated as l/v

(16.22)

MCVc = [ [ Cpc-Pd

where Cpc.
C.pc;Pd

jM

< MCpc < 1

+ tcn-iSci / (CiJii) d <

l/v

C, ^pc;FCj

rr

(16.23)

where

and

_

1 "

Sn. =

1 n-1

PC;,

(USLPCi

E rL/ii

—

+LSLPCi)

(USLPCi

+LSLPCi) nc:

Here tan_\ is the upper a / 2 quantile of the Student t distribution with n — 1 degrees of freedom. Wang et al. (2000) compare the three process capability indices: a multivariate capability vector [ CpM , PV , LI ] proposed by Shariari et al. (1995), a multivariate capability index MCpm proposed by Taam et al. (1993), and a multivariate capability index MCp proposed by Chen (1994). They conclude that in general, the multivariate indices can be obtained from (a) the ratio of a tolerance region to a process region (b) the probability of the nonconforming product, and (c) other approaches using loss functions or vector

342

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representation. The purpose of Wang et al. (2000) paper is to illustrate the distinctions among the various meanings of capability in the multivariate domain. Through the use of several graphical and computational examples, the information summarized by these methodologies is illustrated and their usefulness is discussed. 16.3 Concluding remarks It is evident that in the multivariate case specification limits and the actual process spread is more difficult to define than their univariate counterparts. A difficult issue involving multivariate PCI is also the interpretation of the modified tolerance transferred from the tolerance region. This is an important issue and worth discussing. In general, the specification region is assumed to be ellipsoidal in most multivariate process capability indices due to the assumption of the multivariate normality. Chen (1994) proposes a multivariate process index over a tolerance zone, which makes the definition of the engineering specifications more general and does not require a particular distribution. However, one should note that if the observation are not from the multivariate normal distribution, calculations of these multivariate PCIs will be quite involved and even may not be obtained. Looking back at the Cpm as defined in equation (16.2), this capability index considers proximity to the target and dispersion around the target while will retaining some reasonably attractive statistical properties. Using these statistical properties, one is able to carry out a further analysis and to test (using a hypothesis testing) whether the process capability is capable. Here a benchmark of 1 for assessing the process is suggested. However, we know that the process should be capable but can't as yet determine what the nonconforming percentage will be when this index is equal to 1. The two multivariate PCIs, mentioned in equations (16.3) and (16.4) respectively, seem to be more natural generalizations of the univariate Cp and Cpm. When T = \x and

Multivariate Process Capability Indices

343

A = £ , equations (16.3) and (16.4) are equivalent, and the value 1 in equation (16.3) indicates that the nonconforming portion in the process is 0.27%. The MCpm presented in equation (16.5) has the same property as a univarite PCI, when the process is centered at the target and the capability index equals 1, it should indicate that the 99.73% of the process values lie inside the tolerance region. The definition of engineering specification of MCp as presented in (16.8) provides some flexibility in setting a criterion for the capability of a process. When the values of MCP is 1, it corresponds to the expected proportion of conforming product being 1 — a . Recall that the three components of a multivariate capability vector, CPM , PV and LI, provide a comparison of volumes of the regions, locations of the centers, and location of regions. These generate additional versions of the process capability indices. When the Cpy[ is higher than 1, it implies that the circumscribed modified process region is smaller than the engineering specified region, that is, the index is "capable". The value 1 of PV indicates that the center of the process is close to the engineering target value. And finally the value 1 for LI implies that the modified process region is contained within the tolerance region. Wang as we have mentioned above, and Du (2000) propose an application of the PCA to the process capability indices for multivariate data. This approach does not require the assumption of normality. It reduces the dimension of the data and also transfers correlated data into independent data. Consequently calculation of the indices becomes much easier. Although the use of PCA when calculating the multivariate process indices may result in some loss of information, it is nevertheless an adequate approach for assessing the multivariate process capability. No index is the best to cover all the aspects of the structure of multivariate process capability indices. Some methods may possess letter properties than others in some fields, but the others may

344

Encyclopedia and Handbook of Process Capability Indices

have good properties that are superior to those enjoyed by the original method. To summarize, the proposed approaches all have some reasonable qualities, but we still can't obtain a fully consistent method for assessing the multivariate PCIs. This requires a concentrated further study of the issues related to multivariate PCIs. Up untie very recently, the research in the theory and practice of multivariate process capability indices has been very sparse in comparison to the research dealing with the univariate case. At present, for the multivariate capability indices, consistency in the methodology for evaluating this capability is still absent. Moreover, it is quite difficult to obtain the relevant statistical properties needed for a more detailed inference about multivariate PCI. Moreover there are, of course, still intrinsic difficulties in trying to assess the value of multivariate systems by mean of a single number. Obviously, further investigations in this field are strongly desirable. This completes our surrey of the process Capability Indices. (up dated up to 2004). The dynamic area-inspite of obstacles has grown tremendously during the 20 years of its existence. The unprecedented technological advances, we are witnessing at present, will no doubt provide a fertile grown for new developments in the theory, methodology and application of the PCIs.

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Index

Clements' method, 299 process yield, 41 sampling plan, 237 weakness, 66

Assignable cause variation, 289 Asymmetric tolerances, 181-217 Autoregressive model, 281, 292 Beazley, 78, 80, 262 Binomial distribution, 235 Bissell's approximation, 54 Boyles, R.A., asymmetric, 181-91 Cpk, 42, 66 Cpm, 69, 73, 75-94 Spk, 135, 163

confidence intervals, 160 estimator, 161 process yield, 147 n asymmetric tolerances, 197-202 confidence intervals, 78-80 estimators, 88 expected loss, 68 hypothesis testing, 86 measurement errors, 263 multivariate, 329 non-normality, 299-301 Clements' method, 301 Capability, 1-6 Capability Potential Index, see Cp Capability ratio (CR), 8, 144, 220 Central Limit Theorem, 137-8, 315 Chebyshev inequality, 324

confidence intervals, 13 estimator, 9-10 hypothesis testing, 12-3 measurement errors, 252 Clements' method, 299 weakness, 30 chi-square distribution, 50 confidence intervals, 51-4, Bayesian interval estimate, 62-4 Bayesian-like, 64 hypothesis testing, 55 measurement bias and error, 258-62 375

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Encyclopedia and Handbook of Process Capability Indices

Chi-square distribution, 21, 24, 49, 52, 89-90, 122, 129, 202-4, 218, 223-5, 270 non-central, 78, 264 Clements, J.A., 295, 299 Clements' technique, 304 Column vectors, 224 Contaminants, accumulation of, 284 Control charts, 23, 26, 128, 281, 315, 320 Shewhart, 162, 284, 315-6, 320 Correlated data, 282, 337, 343 Cumulative distribution function (CDF), 9 Data collection, 14, 20, 83, 87, 93, 128, 154 Data transformation techniques, non-normal process, 295 Deleryd, M., 163, 172 Drift, process, 284 Expected relative loss, 72, 96, 1034, 108 Expected values, 120, 138 Exponential distribution, 304 Folded normal distribution, 33, 36, 48,49 Ford Motor Company, 8 Franklin, L.A., 14, 50, 84, 185, 295, Gamma function, 11, 283 Gauge capability, 249-252, 255, 258 Gilchrist, W.G., 303 Guirguis, G., 48 Gunter, B.H., 294

Gupta, A.K., 218 Heavlin, W.D., 13, 50 Hobbs, J.R., 120 Hoffman, L.L., 52 Hsiang,. T.C., 57, 95, 177 Hsu, N.-F., 120 Huang, D.-Y., 218 Hubele, N.F., 58, 80, 129, 154, 263 Hurley, P., 50, 80, 185 Hypothesis testing, Cp, 14, 18, 23 Cpk, 54 Cpm> 86 Cpm/b 125 Spk, 143

Johnson, N.L., 11, 41, 47, 70, 96, 176, 182, 258, 295, 303 Kane, V.E., 7, 97, 134, 146, 181, 220, 298 Kirmani, S.N.U.A., 15, 58, 87, 154 Kotz, S., 11, 41, 70, 114, 181, 258, 284, 295 Kurtosis, 300, 304, 312, 319-20 Kushler, R, 50, 80, 185 Lee, R.F., 137, 218 Leone, F.C, 33, 48, Li, H., 20, 57, 154 Lovelace, C.R., 26, 284, 295 Lu, K., 305, 313 Marcucci, M.O., 78, 263 Mean square error (MSB), 100, 115, 202, 210, 254, 285, 291 Measurement system, 250-1 Median, 297, 299-301, 310-2 Montgomery, D.C. Cp, 8, 23

377

Index Cpk, 44, 132 CPU/CPL, 148

MPPAC, 165 non-normal 293, 295 gauge error, 250-1 sampling plans, 234 tool wear, 282, 289 Nagahata, H., 50-1 Nagata, Y., 50-1 Non-central chi-square distribution, 74, 76, 78-9, 83, 89, 201, 202, 265 Nonconformities (NC), 43, 72, 114, 147, 165, 172, 232, 268, 277 Normal distribution, 9, 34, 42, 77, 99, 111, 134, 148, 182, 219, 235, 249, 280, 294, 328 folded, 33, 36, 49, 131 multivariate data, 329-30, 338, 343 process spread, 30 specification limits, 10 Oakland, J. S., 281 One-sided specifications, 148 Operating characteristic (OC) curve, 232, 235 Owen, D.B., 10, 47, 149, 181, 198, 203, 235, 273 Pearn, W.L., Cp, 11 C 0 ,32 Cpk, 47 Cpm, 74 C-pmk: H O

Pearson curves, 299, 303, 312 Pearson system, 303

Percentage nonconformities, see Nonconformities Porter, L.J., 281 Probability density function (pdf), Cp, 10 G„, 33 Cpm, 74

Process capability analysis, 7, 21, 111, 157, 172, 216, 280 Process capability indices (PCIs), 1,2,5 basic structure, 7-8 non-normal populations, 296 Process Capability Ratio, 8, 148, 220 Process capability studies, 8, 146, 158, 279-280 Process drift, 284 Process yield, 41-44 p-value, 19, 28 Quadratic form, 335 Quantiles, 12, 298-9, 340 Relative loss, 72, 96, 103-4, 108 Relative worth, 96 Rodriguez, R.N., 48, 51, 296 Sample sizes, 14, 22, 238, 245, 300 Sampling plan, 231-248 Shapiro, S.S., 298 Shapiro-Wilk test, 262, 268, 277 Shewhart, W.A., 162, 315-6 Shewhart control charts, 315 Skewness, 294, 296, 307-9, 312, 315-6 Special cause, 281 Spiring, F.A., 28-9, 61, 92, 109, 181, 288, 191

378

Encyclopedia and Handbook of Process Capability Indices

Standard deviation, 8-10, 15, 20 Statistical control, 8, 39, 46, 59, 117, 169, 280, 331 Statistical process control (SPC), 9, 282 Subbaiah, P., 80 Systematic assignable causes, 28890 Taam, W., 80, 237, 331, 341 Taguchi, G., 67, 95, 176, 198 Taguchi method, 167 Target values, 71, 170, 243, 338 Taylor expansion, 137, 145 Temperature change, 284

Tolerances, 45 Tool wear, 279-292 Tukey.J.W., 76 Unbiased estimators, 36, 128, 149, 160 Vannman, K., 76, 78 Vectors, 337 Wallgren, E., 292 Wasserman, G.S., 51, 185, 295, 307 Weibull distributions, 294 Wesolowsky, G.O., 69 Wright, P.A., 120, 307-9 Zhang, N.F., 47, 50, 88

Encyclopedia and Handbooh of Process Capability. Indices R Comprehensive Exposition of Quality Control Measures

'. i

tt

This unique volume pro\ ides an up-to-date and detailed description ol the various process capabilitv indices widely (and sometimes misleadingly) used in the applications at production sites. The authors. who are internationally recognized experts in this area with numerous contributions to the field, provide a lucid exposition, which

covers a l l

the m a i n a s p e c t s , d e v e l o p m e n t s

and

advances.

The concept of Process Capability Index (PCI) is barely 20 years o l d . but the multitude of available versions can overwhelm even the most seasoned practitioner. The organized and self-contained presentation of the material starling from I980's primitive indices (Cp and Cpk) up to the newly proposed indices for the cases of multiple dependent characteristics results in an authoritative and indispensable reference. A proper balance between theoretical investigation and "rule-of-lhumb" practical procedures is maintained in order to eliminate the tensions among various methodologies of assessing the capabilitv of industrial processes.

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