A Bayesian approach for assessing process precision based on multiple samples

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Production, Manufacturing and Logistics

A Bayesian approach for assessing process precision based

on multiple samples

W.L. Pearn

*

_{, Chien-Wei Wu}

Department of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan Received 2 October 2003; accepted 3 February 2004

Available online 14 August 2004

Abstract

Using process capability indices to quantify manufacturing process precision (consistency) and performance, is an essential part of implementing any quality improvement program. Most research works for testing the capability indices have focused on using the traditional distribution frequency approaches. Cheng and Spiring [IIE Trans. 21 (1) 97] proposed a Bayesian procedure for assessing process capability index Cp based on one single sample. In practice,

manufacturing information regarding product quality characteristic is often derived from multiple samples, particu-larly, when a routine-based quality control plan is implemented for monitoring process stability. In this paper, we consider estimating and testing Cpwith multiple samples using Bayesian approach, and propose accordingly a Bayesian

procedure for capability testing. The posterior probability, p, for which the process under investigation is capable, is derived. The credible interval, a Bayesian analogue of the classical lower conﬁdence interval, is obtained. The results obtained in this paper, are generalizations of those obtained in Cheng and Spiring [IIE Trans. 21 (1), 97]. Practitioners can use the proposed procedure to Cheng and Spiring determine whether their manufacturing processes are capable of reproducing products satisfying the preset precision requirement.

Keywords: Process capability indices; Bayesian approach; Credible interval; Posterior probability; Decision making; Quality control

1. Introduction

Understanding process and quantifying process performance are essential for any successful quality improvement initiative. Process capability analysis has become an important and integrated part in applying statistical process control to continuously improve process quality and productivity. The rela-tionship between the actual process performance and the speciﬁcation limits may be quantiﬁed using appropriate process capability indices. Process capability indices (PCIs), the purpose of which is to provide

*

Corresponding author. Tel.: +88635714261; fax: +88635722392. E-mail address:roller@cc.nctu.edu.tw(W.L. Pearn).

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numerical measures of whether or not the ability of a manufacturing process meets a predetermined level of production tolerance, have received considerable research attention and increased usage in process assessments and purchasing decisions in the automotive industry during last decade. The ﬁrst process capability index appearing in the literature was the precision index Cp, and deﬁned as (Kane, 1986):

Cp¼

USL LSL 6r ;

where USL is the upper speciﬁcation limit, LSL is the lower speciﬁcation limit, and r is the process standard deviation. The numerator of Cp gives the range over which the process measurements are

acceptable. The denominator gives the width of the range over which the process is actually varying. The index Cp was designed to measure the magnitude of the overall process variation relative to the

manu-facturing tolerance, which is to be used for processes with data that are normally distributed, independent, and in statistical control. Clearly, the index Cp measures the precision/consistency of a process, or the

potential to reproduce acceptable products, which provides actual performance measure for centered processes. The index Cp is particularly useful when reducing process variation is the guiding principle for

process improvement.

The use of the capability indices was ﬁrst explored within the automotive industry. Ford Motor Company (1984) has used Cp to track process performance and to reduce process variation. Recently,

manufacturing industries have been making an extensive effort to implement statistical process control (SPC) in their plants and supply bases. Process capability measures derived from analyzing SPC data have received increasing usage not only in process performance assessments, but also in the evaluation of purchasing decisions. Capability indices have become the standard tools for quality reporting, particularly, at the management level around the world. Proper understanding and accurate estimation of the capability indices is essential for the company to maintain a capable supplier. The usual practice of judging process capability by evaluating the point estimates of process capability indices, have a flaw since there is no assessment of the error of these estimates. Therefore, a simple point estimate of the index is highly unre-liable in making decision in assessing process capability since the estimate does not represent the true index value. When the estimate is greater than a pre-specified value w, say 1.00, or 1.33, it does not guarantee that the index is greater than w and vice versa. It is therefore preferable to obtain an interval estimate for which we can assert, with a reasonable degree of certainty, that it contains the true index value. Existing methods for testing the capability indices have focused on traditional distribution frequency approaches. Examples include Chou and Owen (1989), Chou et al. (1990), Li et al. (1990), Kirmani et al. (1991), Kocherlakota (1992), Pearn et al. (1998), Kotz and Lovelace (1998) and Pearn and Yang (2003). Kotz and Johnson (2002) presented a thorough review for the development of process capability indices in the past 10 years and Spiring et al. (2003) consolidated the research findings of process capability analysis for the period 1990– 2002.

Bayesian statistical techniques are an alternative to the frequency approach. These techniques specify a prior distribution for the parameter of interest, in order to obtain the posterior distribution of the parameter. We then could infer about the parameter by only using its posterior distribution given the sample data. It is not diﬃcult to obtain the posterior distribution when a prior distribution is given. Even in the case where the form of the posterior distribution is complicated one can always use numerical methods or Monte Carlo methods (Kalos and Whitlock, 1986) to obtain an approximate point estimate or interval estimate. This is the advantage of the Bayesian approach over the traditional distribution frequency approach.

Cheng and Spiring (1989) proposed a Bayesian procedure for assessing process capability index Cpbased

on one single sample. In practice, manufacturing information regarding product quality characteristic is often derived from multiple samples rather than one single sample, particularly, when a routine-based quality control plan is implemented for monitoring process stability. In this paper, we consider estimating and testing Cp with multiple samples using Bayesian approach, and propose accordingly Bayesian

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proce-dure for capability testing. The posterior probability, p, for which the process under investigation is capable, is derived. The credible interval, a Bayesian analogue of the classical lower conﬁdence interval, is obtained. The results obtained in this paper, generalize those obtained in Cheng and Spiring (1989). Practitioners can use the proposed procedure to determine whether manufacturing processes are capable of reproducing products satisfying the preset precision requirement.

2. Estimating Cp based on multiple samples

If one single sample is given asfx1; x2; . . . ; xng, we may consider the natural estimator ^Cpof Cpdeﬁned as

^ Cp¼ USL LSL 6s ; where s¼ ½Pn i¼1ðxi xÞ 2

=ðn 1Þ1=2 is the estimator of the process standard deviation r, which can be obtained from a stable process. Under the assumption of normality, Chou and Owen (1989) obtained the probability density function (PDF) of the natural estimator ^Cp, which can be expressed as the following, for

y >0: fðyÞ ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn 1Þ=2 p Cp n1 C½ðn 1Þ=2 y n_{exp½ðn 1ÞðC} pÞ 2 ð2y2_Þ1 :

Pearn et al. (1998) obtained an unbiased estimator eCp ¼ bn1Cbp where the correction factor

bn1¼ ½2=ðn 1Þ1=2fC½ðn 1Þ=2=C½ðn 2Þ=2g and CðkÞ ¼R01tk1etdt is the gamma function. Pearn

et al. (1998) also showed that the estimator eCp is the uniformly minimum variance unbiased

estima-tor (UMVUE) of Cp, which is asymptotically eﬃcient, consistent, and that n1=2ð eCp CpÞ converges to

Nð0; C2

p=2Þ in distribution.

For cases where data are collected as multiple samples, Kirmani et al. (1991) considered m samples each of size n and suggested the following estimator of Cp, where xi is the ith sample mean, and si is the ith

sample standard deviation: b C_p¼ðUSL LSLÞdp 6 ; where dp ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mðn 1Þ 1 mðn 1Þ s emðn1Þ1 sp ; emðn1Þ1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 mðn 1Þ 1 s C mðn 1Þ 2 C mðn 1Þ 1 2 1 ; s2_p¼ 1 mðn 1Þ Xm i¼1 ðn 1Þs2 i ¼ 1 m Xm i¼1 s2_i;

noting that under normality assumption sp=r is distributed as vmðn1Þ1=½mðn 1Þ 1 1=2 . Therefore, the estimator bC_p is distributed as b C_p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mðn 1Þ 1 p emðn1Þ1 ffiffiffiffiffiffiffiffiffiffiffiffiffi v2 mðn1Þ q Cp:

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The estimator bC_pis unbiased, and its probability density function can be obtained as the following, for y >0, where k¼ ½mðn 1Þ 1e2

mðn1Þ1Cp2, which can be expressed as a function of Cp:

gðyÞ ¼ 2k mðn1Þ=2 2mðn1Þ=2_{C½mðn 1Þ=2}y ½mðn1Þþ1_exp k 2 1 y2 :

Furthermore, Pearn and Yang (2003) investigated some statistical properties of bC_pand showed that bC_p is the UMVUE of Cp, which is also asymptotically eﬃcient andðmnÞ

1=2

ð bC_p CpÞ converges to N ð0; C2p=2Þ

in distribution. It is easy to verify that bC_p is consistent. The variance of bC_p can be calculated as the following (Kirmani et al., 1991):

Varð bC_pÞ ¼ E½ð bC_pÞ2 ½Eð bC_pÞ2¼ ðUSL LSLÞ2e2_mðn1Þ1½mðn 1Þ 1 36mðn 1Þ Eðs 2 pÞ 1_C2 p ¼ C2 p ðe 2 mðn1Þ2Þ 1 n 1o:

For multiple samples with variable sample size, we can consider the generalized estimator of Cpdeﬁned

below. We show that the generalized estimator bC_p obtained from m samples each of size ni, remains

unbiased. In fact, it can be shown that the unbiased estimator bC_pis indeed the UMVUE of Cpin the case of

multiple samples: b C_p_{¼ bP}m i¼1 ðni1Þ USL LSL 6sp ; s2_p¼ Pm i¼1 ðni 1Þs2i Pm i¼1ðni 1Þ ; and b_Pm i¼1 ðni1Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pm i¼1 ðni 1Þ v u u u t C Pm i¼1 ðni 1Þ 2 0 B B @ 1 C C A C Pm i¼1 ðni 1Þ 1 2 0 B B @ 1 C C A 2 6 6 4 3 7 7 5 1 :

3. Bayesian approach for testing Cp

Cheng and Spiring (1989) proposed a Bayesian procedure for assessing process capability index Cp.

Shiau et al. (1999) applied a similar Bayesian approach to index Cpm. Shiau et al. (1999) also applied

Bayesian method for testing the index Cpk but under the restriction that the process mean l equals to the

midpoint of the two speciﬁcation limits, M . In this case Cpkreduces to Cp. However, these research works

only focused on cases with one single sample. A common practice of process capability estimation in the manufacturing industry is to ﬁrst implement a daily-based or weekly-based sample data collection plan for monitoring/controlling the process stability, then to analyze the past ’’in control’’ data. It is more practical to develop a procedure for assessing process capability for cases with multiple samples. Therefore, in the following we consider the problem of estimating and testing Cp with multiple samples based on Bayesian

approach, and propose accordingly a Bayesian procedure for testing process precision. The posterior probability, p, for which the process under investigation is capable, is derived. A 100p% credible interval is the Bayesian analogue of the classical 100p% conﬁdence interval, where p is the conﬁdence level for the interval. The credible interval covers 100p% of the posterior distribution of the parameter (Berger, 1980). Assuming that the m samples are randomly taken from independent and identically distributed (i.i.d.) Nðl; r2_{Þ, a normal distribution with mean l and variance r}2_{. Denote the measures of the ith sample as}

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xi¼ fxi1; xi2; . . . ; xinig with variable sample size ni, and X¼ fxij; i¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; nig. Then, the

likelihood function for l and r can be expressed:

Lðl; r j XÞ ¼ ð2pr2_Þ P m i¼1 ni 2 _exp 8 > > > < > > > : Pm i¼1 Pni j¼1 ðxij lÞ 2 2r2 9 > > > = > > > ; :

The first step for the Bayesian approach is to find an appropriate prior. Usually, when there is little or no prior information, or there is only one parameter, one of the most widely used non-informative priors is the so-called reference prior, which is a non-informative prior that maximizes the difference between infor-mation (entropy) on the parameter provided by the prior and by the posterior. In other words, the reference prior allows the prior to provide information as little as possible about the parameter (see Bernardo and Smith, 1993 for more details). Several priors have been considered in the literature. In practical situation, however, the choice of prior information is hard to justify. Therefore, in this paper we adopt the following non-informative reference prior chosen by Cheng and Spiring (1989):

hðl; rÞ ¼ 1=r; 1 < l < 1; 0 < r <1:

We note that the parameter space of the prior is infinite, hence the reference prior is improper, which means that it does not integrate to one. However, it is not always a serious problem, since the prior incorporated with ordinary likelihood will lead to proper posterior. Furthermore, the credible interval obtained from a non-informative prior has a more precise coverage probability than that obtained from any other priors. The posterior probability density function (PDF), fðl; r j XÞ of ðl; rÞ may be expressed as the following: fðl; r j XÞ / Lðl; r j XÞ hðl; rÞ / r P m i¼1 niþ1 exp 8 > > < > > : Pm i¼1 Pni j¼1 ðxij lÞ 2 2r2 9 > > = > > ; : Also Z 1 0 Z 1 1 r P m i¼1 niþ1 exp 8 > > < > > : Pm i¼1 Pni j¼1 ðxij lÞ 2 2r2 9 > > = > > ; dl dr ¼ Z 1 0 r P m i¼1 niþ1 exp 1 br2 Z 1 1 exp 0 B B @ 2 6 6 4 Pm i¼1 niðl xÞ 2 2r2 1 C C Adl 3 7 7 5dr ¼ ffiffiffiffiffiffiffiffiffiffiffi_p 2P m i¼1 ni v u u t CðaÞba:

In order to satisfy the integration property that the probability over PDF is 1, a coefficient of fðl; r j XÞ can be obtained through some algebraic manipulations. Consequently, the posterior PDF of ðl; rÞ can be expressed as fðl; r j XÞ ¼ 2 ffiffiffiffiffiffiffiffiffi Pm i¼1 ni s ffiffiffiffiffiffi 2p p CðaÞbar P m i¼1 niþ1 exp 0 B B @ Pm i¼1 Pni j¼1 ðxij lÞ 2 2r2 1 C C A;

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where a¼ ðPm_i¼1ni 1Þ=2, b ¼ ½P_i¼1m Pni_j¼1ðxij xÞ 2

=21, x¼Pm_i¼1 Pni_j¼1xij=Pni_j¼1xij. As mentioned

ear-lier, it is natural to consider the quantity Prfprocess is capable j Xg in the Bayesian approach. We need the posterior probability p¼ PrfCp > wj Xg for some ﬁxed positive number w. Therefore, given a pre-speciﬁed

precision level w > 0 and denote a¼ ðUSL LSLÞ=ð6wÞ, the posterior probability for index Cp based on

multiple samples that a process is capable is given as

p¼ PrfCp> wj Xg ¼ Pr USL LSL 6r > w X ¼ Pr r <USL LSL 6w j X ¼ Pr rf < a j Xg ¼ Z a 0 fðr j XÞ dr ¼ Z a 0 2r P m i¼1 ni CðaÞba exp 1 br2 dr:

By changing the variable, let y¼ ðbr2_Þ1

, then dy¼ 2ðbr3_Þ1

dr, the above posterior probability p expression may be rewritten as

p¼ Z 1 1=t r P m i¼1 ni3

CðaÞba1 expðyÞ dy ¼ Z 1 1=t ya1 CðaÞ expðyÞ dy ¼ Cða; 1=tÞ CðaÞ ð1Þ or, equivalently, p¼ 1 Gð1=t; a; 1Þ; ð2Þ

where Cða; 1=tÞ is the value of the incomplete gamma function of 1=t with parameter a, Gð1=t; a; 1Þ is the cumulative probability at 1=t for the gamma distribution with parameters a and 1, and

t¼ 2c Pm i¼1 ðni 1Þ b C_p wb_Pm i¼1 ðni1Þ 0 B B @ 1 C C A 2 ; c¼ Pm i¼1 Pni j¼1 ðxij xiÞ 2 Pm i¼1 Pni j¼1 ðxij xÞ 2 ¼ Pm i¼1 ðni 1Þs2p Pm i¼1 ðni 1Þs2pþ Pm i¼1 niðxi xÞ 2 ; b_Pm i¼1 ðni1Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pm i¼1 ðni 1Þ v u u u t C Pm i¼1 ðni 1Þ 2 0 B B @ 1 C C A C Pm i¼1 ðni 1Þ 1 2 0 B B @ 1 C C A 2 6 6 4 3 7 7 5 1 :

Note that the posterior probability p depends on m, ni, c, w and bCp only through m, ni, c and bCp=w.

Denoted C_{¼ b}_C

p=w. There is a one-to-one correspondence between p and Cwhen m and ni, are given, and

by the fact that c and bC_pcan be calculated from the process data, we ﬁnd that the minimum value of C

required to ensure the posterior probability p reaching a certain desirable level, can be useful in assessing process capability. Denote this minimum value as C_{ðpÞ. Then, the value C}_{ðpÞ satisﬁes}

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p¼ PrfCp> wj Xg ¼ Pr Cp ( > Cb p C_ðpÞj X ) :

Therefore, a 100p% credible interval for Cpis½ bCp=CðpÞ; 1), where p is a number between 0 and 1, say 0.95,

for 95% conﬁdence interval, which means that the posterior probability that the credible interval contains Cp is p. We say that the process is capable in a Bayesian sense if all the points in this credible interval are

greater than a pre-speciﬁed value of w, say 1.00 or 1.33. When this happens, we have p¼ PrfCp> wj Xg. In

other words, to see if a process is capable (with capability level w and conﬁdence level p), we only need to check if bC_p> C_{ðpÞ w. For the single sample, that is, m ¼ 1, c ¼ 1, and s}

p¼ s, the results obtained in this

paper can be reduced to those obtained in Cheng and Spiring (1989).

4. Decision making for testing Cp

In current practice, process precision is said to be inadequate if Cp <1:00; it indicates that the process is

not adequate with respect to the production tolerances. Process precision is said to be marginally capable if 1:00 6 Cp<1:33; it indicates that caution needs to be taken regarding the process consistency and some

process control is required (usually using R or S control charts). Process precision is said to be satisfactory if 1:33 6 Cp<1:67; it indicates that process consistency is satisfactory, material substitution may be allowed,

and no stringent precision control is required. Process precision is said to be excellent if 1:67 6 Cp<2:00; it

indicates that process precision exceeds satisfactory. Finally, process precision is said to be superior if CpP2:00.

In recent years, many companies have adopted criteria for evaluating their processes based on process capability objectives that are more stringent than those recommended minimums above. For instance, the ’’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. Thus, in eﬀect, requires that the process capability ratio will be at least 2.0 (Harry, 1988).

Therefore, it would be desirable to determine a bound which practitioners would be expected to ﬁnd the true value of the process capability no less than the bound value with certain level of conﬁdence. For users’ convenience in applying our Bayesian procedure, we tabulate the minimum values C_{ðpÞ of b}_C

p=w, for

various c with m¼ 2ð2Þ10; 15, ni¼ n ¼ 10ð5Þ30 in Tables 1–3 to ensure p ¼ 0:99, 0.975, and 0.95,

respectively. For example, if w¼ 1:33 is the minimum capability requirement, then for p ¼ 0:95, with m¼ 10 of each sample size ni¼ n ¼ 10 and c ¼ 0:90, we can ﬁnd CðpÞ ¼ 1:1297 by checking Table 3.

Thus, the minimum value of bC_prequired for a capable process is C_{ðpÞ w ¼ 1:1297 1:33 ¼ 1:5026. That}

is, if bC_pis greater than 1.5026, we say that the process is capable in Bayesian sense.

As a result, to judge if a given process meets the capability requirement, we first determine the pre-specified capability level w, and the confidence level p or the a-risk for the interval. Check the appropriate table or solve Eq. (1) or (2), we may obtain the minimum value of C_{ðpÞ based on given values of p, m}

sub-samples of size niand c calculated from samples. If the estimated value bCpis greater than the critical value

C_{ðpÞ w, then we may conclude that the process meets the capability requirement (C}

p > w). Otherwise, we

do not have suﬃcient information to conclude that the process meets the present capability requirement. In this case, we would believe that Cp6w. Therefore, the practitioners can easily use the procedure on their

in-plant applications to obtain reliable decisions.

We remark that the process must be stable in order to produce a reliable estimate of process capability. If the process is out of control, it will be unreliable to estimate process capability. In these cases the priority is to ﬁnd and eliminate the assignable causes of variability in order to bring the process in-control.

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Table 1

The minimum values of C_{ðpÞ of b}_C

p=w, with m¼ 2ð2Þ10; 15, n ¼ 10ð5Þ30 required to ensure p ¼ 0:99

m c(n¼ 10) c(n¼ 15) c(n¼ 20) c(n¼ 25) c(n¼ 30) 0.7 0.8 0.9 1 0.7 0.8 0.9 1 0.7 0.8 0.9 1 0.7 0.8 0.9 1 0.7 0.8 0.9 1 2 1.7577 1.6442 1.5502 1.4706 1.6297 1.5244 1.4373 1.3635 1.5601 1.4593 1.3759 1.3053 1.5151 1.4173 1.3362 1.2676 1.4831 1.3873 1.3079 1.2408 4 1.5168 1.4188 1.3376 1.2690 1.4566 1.3625 1.2846 1.2187 1.4208 1.3290 1.2530 1.1887 1.3964 1.3062 1.2316 1.1684 1.3785 1.2895 1.2158 1.1534 6 1.4296 1.3373 1.2608 1.1961 1.3908 1.3010 1.2266 1.1637 1.3665 1.2782 1.2051 1.1433 1.3494 1.2622 1.1900 1.1290 1.3366 1.2503 1.1788 1.1182 8 1.3821 1.2928 1.2189 1.1564 1.3543 1.2668 1.1943 1.1330 1.3359 1.2496 1.1781 1.1177 1.3227 1.2373 1.1665 1.1067 1.3127 1.2279 1.1577 1.0983 10 1.3514 1.2641 1.1918 1.1306 1.3303 1.2444 1.1732 1.1130 1.3158 1.2308 1.1604 1.1008 1.3051 1.2208 1.1510 1.0919 1.2968 1.2131 1.1437 1.0850 15 1.3061 1.2217 1.1518 1.0927 1.2946 1.2110 1.1418 1.0831 1.2856 1.2025 1.1338 1.0755 1.2785 1.1959 1.1275 1.0697 1.2728 1.1906 1.1225 1.0649 Table 2

The minimum values C_{ðpÞ of b}_C

p=w, with m¼ 2ð2Þ10; 15, n ¼ 10ð5Þ30 required to ensure p ¼ 0:975

m c(n¼ 10) c(n¼ 15) c(n¼ 20) c(n¼ 25) c(n¼ 30) 0.7 0.8 0.9 1 0.7 0.8 0.9 1 0.7 0.8 0.9 1 0.7 0.8 0.9 1 0.7 0.8 0.9 1 2 1.6272 1.5221 1.4350 1.3614 1.5361 1.4369 1.3547 1.2852 1.4848 1.3889 1.3095 1.2422 1.4509 1.3572 1.2796 1.2140 1.4266 1.3345 1.2581 1.1936 4 1.4435 1.3503 1.2731 1.2077 1.4011 1.3106 1.2357 1.1723 1.3748 1.2860 1.2124 1.1503 1.3566 1.2689 1.1964 1.1350 1.3429 1.2562 1.1843 1.1236 6 1.3752 1.2863 1.2128 1.1505 1.3487 1.2616 1.1894 1.1284 1.3312 1.2452 1.1740 1.1137 1.3185 1.2334 1.1628 1.1032 1.3089 1.2243 1.1543 1.0951 8 1.3374 1.2510 1.1795 1.1189 1.3193 1.2341 1.1635 1.1038 1.3064 1.2220 1.1521 1.0930 1.2968 1.2131 1.1437 1.0850 1.2893 1.2061 1.1371 1.0787 10 1.3128 1.2280 1.1578 1.0984 1.2999 1.2159 1.1464 1.0876 1.2900 1.2067 1.1377 1.0793 1.2824 1.1996 1.1310 1.0729 1.2764 1.1939 1.1256 1.0679 15 1.2762 1.1938 1.1255 1.0677 1.2708 1.1887 1.1208 1.0632 1.2652 1.1835 1.1158 1.0586 1.2605 1.1791 1.1117 1.0546 1.2566 1.1755 1.1082 1.0514 Table 3

The minimum values C_{ðpÞ of b}_C

p=w, with m¼ 2ð2Þ10; 15, n ¼ 10ð5Þ30, required to ensure p ¼ 0:95

m c(n¼ 10) c(n¼ 15) c(n¼ 20) c(n¼ 25) c(n¼ 30) 0.7 0.8 0.9 1 0.7 0.8 0.9 1 0.7 0.8 0.9 1 0.7 0.8 0.9 1 0.7 0.8 0.9 1 2 1.5268 1.4282 1.3465 1.2774 1.4622 1.3678 1.2896 1.2234 1.4246 1.3326 1.2564 1.1919 1.3992 1.3089 1.2340 1.1707 1.3808 1.2916 1.2177 1.1552 4 1.3850 1.2956 1.2215 1.1588 1.3561 1.2685 1.1960 1.1346 1.3372 1.2508 1.1793 1.1188 1.3237 1.2382 1.1674 1.1075 1.3135 1.2287 1.1584 1.0989 6 1.3310 1.2450 1.1738 1.1136 1.3141 1.2292 1.1589 1.0995 1.3020 1.2179 1.1482 1.0893 1.2929 1.2094 1.1402 1.0817 1.2858 1.2027 1.1340 1.0758 8 1.3008 1.2168 1.1472 1.0883 1.2903 1.2070 1.1380 1.0796 1.2818 1.1991 1.1305 1.0725 1.2752 1.1928 1.1246 1.0669 1.2698 1.1878 1.1199 1.0624 10 1.2810 1.1983 1.1297 1.0718 1.2746 1.1923 1.1241 1.0664 1.2685 1.1866 1.1187 1.0613 1.2634 1.1818 1.1142 1.0570 1.2592 1.1778 1.1105 1.0535 15 1.2514 1.1706 1.1036 1.0470 1.2509 1.1701 1.1032 1.0466 1.2482 1.1676 1.1008 1.0443 1.2454 1.1650 1.0984 1.0420 1.2429 1.1627 1.0961 1.0399 W.L. Pearn, C.-W. Wu / European Journal of Operational Research 165 (2005) 685–695

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5. Application example

Liquid crystals have been used in various configurations for display applications. Most of the current displays involve the use of either Twisted Nematic (TN) or Super Twisted Nematic (STN) liquid crystals. The STN-LCD products are used in making PDAs (personal digital assistants), notebook computers, word processors and other peripherals. Due to the advancement of modern manufacturing technology STN-LCD and relatively low production cost, STN-STN-LCDs maintained a competitive advantage in the market. To illustrate the practicality of our proposed Bayesian approach, we present a case study on a STN-LCD manufacturing process, which located in the Science-Based Industrial Park, Taiwan. This factory manu-factures various types of the LCD. For a particular model of the STN-LCD investigated, the upper specification limit, USL, of a glass substrate’s thickness is 0.77 mm, the lower specification limit, LSL, of a glass substrate’s thickness is 0.63 mm, and the target value is T ¼ 0:70 mm. If the product characteristic does not fall within the tolerance (LSL, USL), the lifetime or reliability of the STN-LCD will be discounted. The collected sample data (15 samples each of size 10) are displayed in Table 4.

A 100p% credible interval means the posterior probability that the true capability index lies in this interval is p. Let p be a high probability, say, 0.95. Suppose for this particular process under consideration to be capable, the process index must reach at least a certain level w, say, 1.33. Now, from the process data, we compute the lower bound of the credible interval for the index. The resulting Bayesian testing procedure is simple. That is, if bC_p> C_{ðpÞ w, then we say that the process is capable.}

As noted earlier, in order to make the estimation of these capability indices meaningful, we would check if the manufacturing process is under statistical control and the distribution is normal. For those 15 samples of size 10 each, the Shapiro–Wilk test for normality conﬁrms this with p-value > 0.1. That is, it is reasonable to assume that the process data collected from the factory is normally distributed. We then construct the X S charts to check if the process is in control. The X S charts based on the collected samples are displayed in Figs. 1 and 2. The X S control charts show that the process seems to be in-control since all the sample points are within the control limits without any special pattern. Therefore, the basic assump-tions are justiﬁed so we could proceed with the capability calculaassump-tions. The calculated sample mean xiand

the sample variance s2

i for the ﬁfteen samples are summarized in the last two columns of Table 4. Thus, we

have s2 p ¼ Pm i¼1s2i=m¼ 0:000158, x¼ 0:6998 and c ¼ 0:869, bC p¼ bPm i¼1ðni1Þ ðUSL LSLÞ=ð6spÞ ¼

1:8459. We run the computer program by solving equation (1) (which is available from authors) to

Table 4

The 15 samples of 10 observations with calculated sample statistics Sample i Observations xi s2i 1 0.727 0.701 0.678 0.694 0.713 0.699 0.695 0.696 0.733 0.703 0.7039 0.000267 2 0.677 0.712 0.686 0.689 0.682 0.683 0.709 0.687 0.698 0.699 0.6922 0.000139 3 0.692 0.687 0.685 0.698 0.687 0.698 0.707 0.717 0.702 0.717 0.6990 0.000140 4 0.701 0.702 0.695 0.703 0.682 0.696 0.692 0.720 0.687 0.686 0.6964 0.000120 5 0.700 0.719 0.699 0.697 0.714 0.697 0.683 0.688 0.693 0.714 0.7004 0.000139 6 0.693 0.690 0.709 0.707 0.713 0.701 0.706 0.684 0.695 0.688 0.6986 0.000099 7 0.699 0.722 0.714 0.706 0.694 0.700 0.699 0.704 0.683 0.704 0.7025 0.000113 8 0.708 0.712 0.703 0.721 0.692 0.691 0.678 0.698 0.712 0.713 0.7028 0.000170 9 0.711 0.693 0.677 0.710 0.708 0.702 0.680 0.713 0.711 0.694 0.6999 0.000177 10 0.703 0.686 0.720 0.727 0.714 0.713 0.698 0.713 0.693 0.685 0.7052 0.000208 11 0.693 0.724 0.715 0.708 0.722 0.705 0.710 0.715 0.714 0.694 0.7100 0.000109 12 0.700 0.712 0.686 0.707 0.683 0.699 0.705 0.705 0.691 0.727 0.7015 0.000169 13 0.708 0.696 0.718 0.704 0.678 0.703 0.713 0.694 0.684 0.681 0.6979 0.000188 14 0.686 0.688 0.678 0.701 0.718 0.694 0.688 0.691 0.704 0.689 0.6937 0.000128 15 0.690 0.693 0.673 0.678 0.711 0.684 0.712 0.714 0.694 0.686 0.6935 0.000210

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obtain the critical value bCðpÞ w ¼ 1:4938 based on p ¼ 0:95, m ¼ 15, n ¼ 10. The calculated sample estimator bC_p¼ 1:8456 is greater than the critical value bCðpÞ w ¼ 1:4938 and the lower conﬁdence bound of Cp is obtained as bCp=CðpÞ ¼ 1:8459=1:1231 ¼ 1:6346. Therefore, we may conclude, with 95%

conﬁ-dence level, that the process meets the capable precision requirement ‘Cp>1:33’ in this case.

6. Conclusions

Process capability indices establish the relationships between the actual process performance and the manufacturing speciﬁcations. Statistical properties of the estimated Cp based on one single sample, have

been investigated extensively, but not for multiple samples. For applications where a routine-based data collection plans are implemented, a common practice on process control is to estimate the process precision by analyzing past ’’in control’’ data. Therefore, the manufacturing information regarding product quality characteristic should be derived from multiple samples rather than one single sample. In this paper, we considered estimating and testing Cp with multiple samples using Bayesian approach, and propose

accordingly Bayesian procedure for capability testing. The posterior probability, p, for which the process

Sample number 2 4 6 8 10 12 14 0.68 0.69 0.70 0.71 0.72 UCL LCL X-bar

Fig. 1. X control chart of the process.

Sample number 2 4 6 8 10 12 14 0.005 0.010 0.015 0.020 UCL LCL S

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under investigation is capable, is derived. The credible interval, a Bayesian analogue of the classical lower conﬁdence interval, is obtained. The results obtained in this paper, are generalizations of those obtained in Cheng and Spiring (1989). To make this Bayesian procedure practical for in-plant applications, we tabu-lated the minimum values of C_{ðpÞ for which the posterior probability p reaches various desirable}

conﬁ-dence levels. Subsequently, a real-world case on the STN-LCD manufacturing process, is also investigated using the proposed approach to data collected from the factory.

Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments and careful read-ings, which signiﬁcantly improved the paper.

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