Production quality and yield assurance for processes with multiple independent characteristics

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Production, Manufacturing and Logistics

Production quality and yield assurance for processes

with multiple independent characteristics

W.L. Pearn

a,*

, Chien-Wei Wu

b

a_{Department of Industrial Engineering and Management, National Chiao Tung University, 1001 Ta Hsueh Road,}

Hsin Chu 30050, Taiwan, ROC

b

Department of Industrial Engineering and Systems Management, Feng Chia University, 100 Wenhwa Road, Taichung 40724, Taiwan, ROC

Received 19 January 2004; accepted 7 February 2005 Available online 23 May 2005

Abstract

Process capability indices have been widely used in the manufacturing industry providing numerical measures on process potential and process performance. Capability measure for processes with single characteristic has been inves-tigated extensively, but is comparatively neglected for processes with multiple characteristics. In real applications, a process often has multiple characteristics with each having different specifications. Singhal [Singhal, S.C., 1990. A new chart for analyzing multiprocess performance. Quality Engineering 2 (4), 397–390] proposed a multi-process per-formance analysis chart (MPPAC) for analyzing the perper-formance of multi-process product. Using the same technique, several MPPACs have been developed for monitoring processes with multiple independent characteristics. Unfortu-nately, those MPPACs ignore sampling errors, and consequently the resulting capability measures and groupings are unreliable. In this paper, we propose a reliable approach to convert the estimated index values to the lower confidence bounds, then plot the corresponding lower confidence bounds on the MPPAC. The lower confidence bound not only gives us a clue minimum actual performance which is tightly related to the fraction of non-conforming units, but is also useful in making decisions for capability testing. A case study of a dual-fiber tip process is presented to demonstrate how the proposed approach can be applied to in-plant applications.

Keywords: Bootstrap sampling; Lower conﬁdence bound; MPPAC control chart; Process capability indices

1. Introduction

During the last decade, numerous process capa-bility indices (PCIs), including Cp, Ca, Cpu, Cpl,

* _{Corresponding author. Tel.: +886 35714261; fax: +886}

35722392.

E-mail address:[email protected](W.L. Pearn).

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and Cpk, have been proposed in the manufacturing industry to provide numerical measures on process performance, which are eﬀective tools for quality/ reliability assurance (see Kane, 1986; Chan et al., 1988; Pearn et al., 1992, 1998; Kotz and Lovelace,

1998; Kotz and Johnson, 2002 for more details).

These indices are deﬁned as Cp¼ USL LSL 6r ; Cpu¼ USL l 3r ; Cpl¼ l LSL 3r ; Ca¼ 1 jl mj d ; Cpk ¼ min USL l 3r ; l LSL 3r ;

where USL and LSL are the upper and the lower specification limits, respectively, l is the process mean, r is the process standard deviation, m = (USL + LSL)/2 is the mid-point of the specifica-tion interval and d = (USL LSL)/2 is half the length of the specification interval. For normally distributed processes, Cp, Caand Cpkindices are appropriate measures for processes with two-sided specifications. The index Cp measures the overall process variation relative to the specification toler-ance, therefore only reflects process potential (or process precision). The index Cameasures the de-grees of process centering, which alerts the user if the process mean deviates from its center. There-fore, the index Ca only reflects process accuracy. The index Cpktakes into account process variation as well as process centering, providing process per-formance in terms of yield (proportion of confor-mities). Given a fixed value of Cpk, the bounds on process yield p can be expressed as 2U(3Cpk) 1 6 p 6 U(3Cpk) (Boyles, 1991), where U(Æ) is the cumulative distribution function of the standard normal distribution. For instance, if Cpk= 1.00, then it guarantees that the yield will be no less than 99.73%, or equivalent to no more than 2700 parts per million (ppm) of non-conformities. On the other hand, the indices Cpu and Cpl have been designed particularly for processes with one-sided manufacturing specifications, which measure the-smaller-the-better and the-larger-the-better process capabilities, respectively. For normally distributed processes with one-sided specification limit, USL or LSL, the relationship between the one-sided capability indices and the process yield

can be calculated as pu= P(X < USL) = U(3Cpu) and pl= P(X > LSL) = U(3Cpl).

In factory applications a product usually has multiple characteristics with each having different specifications, which need to be monitored and controlled hence is a difficult and time-consuming task for factory engineers. A multi-process perfor-mance analysis chart (MPPAC) proposed by

Sing-hal (1990), which evaluates the performance of a

multi-process product with symmetric bilateral specifications. Singhal (1991) further presented a MPPAC with several well-defined capability zones by using the process capability indices Cpand Cpk for grouping the processes in a multiple process environment into different performance categories on a single chart. Using the same technique, sev-eral modified control charts have been developed for monitoring processes with single or multiple independent characteristics. Pearn and Chen

(1997)proposed a modiﬁcation to the CpkMPPAC

combining the more-advanced process capability indices, Cpm or Cpmk, to identify the problems causing the processes failing to center around the target. By combining Singhals MPPAC with asymmetric process capability index Cpa, Chen

et al. (2001)introduced a process capability analy-sis chart (PCAC) to evaluate process performance for an entire product composed of multiple char-acteristics with symmetric and asymmetric speciﬁ-cations. Pearn et al. (2002)introduced a MPPAC to the chip resistors applications based on the inca-pability index Cpp. Chen et al. (2003) also devel-oped a control chart for processes with multiple characteristics based on the generalization of yield index Spk proposed by Boyles (1994). We should note that the process mean l and the process var-iance r2are usually unknown in practice. In order to calculate the index value, sample data must be collected and a great degree of uncertainty may be introduced into capability assessments due to sampling errors. However, those existing research works on MPPAC are restricted to assuming the value of l and r2are known or obtaining quality information from one single sample of each pro-cess ignoring sampling errors. The information provided from the existing MPPAC, therefore, is unreliable and misleading resulting in incorrect decisions. In this paper, we propose a reliable

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approach to obtain the lower confidence bounds and apply it to the modified Cpk MPPAC. A real-world application to the dual-fiber tips manu-facturing process is presented for illustration.

2. Capability measure for multiple characteristics 2.1. Processes with multiple dependent

characteristics

Process capability analysis often entails char-acterizing or assessing processes or products based on more than one engineering speciﬁcation or quality characteristic (variable). When these vari-ables are related characteristics, the analysis should be based on a multivariate statistical technique.

Chen (1994) and Boyles (1996) and others have

presented multivariate capability indices for assessing capability.Wang and Chen (1998–1999)

and Wang and Du (2000) proposed multivariate

equivalents for Cp, Cpk, Cpm and Cpmk based on the principal component analysis, which trans-forms numbers of original related measurement variables into a set of uncorrected linear functions. Moreover, a comparison of three recently pro-posed multivariate methodologies for assessing capability are illustrated and their usefulness is dis-cussed in Wang et al. (2000). However, those indices and methodologies were appropriate for products with either multiple unilateral speci-fications or multiple bilateral specifications exclusively. For practical applications, most mul-ti-process products are composed of numerous unilateral specifications and bilateral specifica-tions, and customers are satisfied when all quality characteristics of an entire product meet preset specifications. Therefore, neither univariate pro-cess capability indices nor multivariate propro-cess capability indices can meet the needs for the requirements.

2.2. Processes with multiple independent characteristics

For processes with multiple characteristics,

Bothe (1992)considered a simple measure by taking

the minimum of the measure of each single

characteristic. For example, consider a m character-istics process with m yield measures P1, P2, . . ., and Pm, the overall process yield is measured as P = min{P1, P2, . . . , Pm}. We note that this approach does not reflect the real situation accurately. Sup-pose the process has five characteristics (m = 5), with equal characteristic yield measures P1= P2= P3= P4= P5= 99.73%. Using the approach con-sidered by Bothe (1992), the overall process yield is calculated as P = min{P1, P2, P3, P4, P5} = 99.73% (or 2700 ppm of non-conformities). Assum-ing that the five characteristics are mutually inde-pendent, then the actual overall process yield should be calculated as P = P1· P2· · P5= 98.66% (or 134,000 ppm of non-conformi-ties), which is significantly less than that calculated byBothe (1992). Generally, the quality characteris-tics of a product can be classified into three types: the-nominal-the-best, the-smaller-the-better and the-larger-the-better types. Cpk, Cpu and Cpl are three indices to evaluate the process capabilities on the MPPAC. For a multi-process product, as-sume there are nkprocesses of the-nominal-the-best type evaluated by Cpkj, j = 1, 2, . . . , nk, nu the-smal-ler-the-better processes evaluated by Cpuj, j = 1, 2, . . . , nu, and nl processes of the-larger-the-better type evaluated by Cplj, j = 1, 2, . . . , nl. Thus, as described earlier, the general form of pro-cess yield can be calculated for unilateral character-istics as puj= U(3Cpuj), j = 1, 2, . . . , nu or plj= U(3Cplj), j = 1, 2, . . . , nl and pkjP 2U(3Cpkj) 1, j = 1, 2, . . . , nkfor bilateral specifications.

Assume the individual process yields are inde-pendent, the entire process yield pT can be calcu-lated as p_T ¼Y i2G Yni j¼1 p_ij;

where G = {k, u, l}. Furthermore, utilizing the inequality U(x) P 2U(x) 1, the above relations of process yield can be rewritten as: pijP 2U(3Cpij) 1, i 2 {k, u, l}, j = 1, 2, . . . , ni. Then, the overall process yield pTcan be described as p_T ¼Y i2G Yni j¼1 p_ijPY ni j¼1 ½2Uð3CpkjÞ 1 Ynu j¼1 Uð3CpujÞ Y nl j¼1 Uð3CpljÞ P Y i2G Yni j¼1 ½2Uð3CpijÞ 1.

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In general, the overall process yield of a multi-process product is lower than any individual pro-cess yield, namely, pT6pij. Similarly, when the overall process yield (or entire product capability) is preset to satisfy the required level, the individual process yield (or individual process capability) should exceed the preset standard for the entire product. Based on the above analysis, if each char-acteristic is mutually independent and normally distributed, the process yield can be evaluated in terms of an integrated process capability index CTin the following: CT ¼ 1 3U 1 Y i2G Yni j¼1 ½2Uð3CpijÞ 1 þ 1 !, 2 " #! . Some minimum capability requirements have been recommended in the manufacturing industry (seeMontgomery, 2001), for speciﬁc process types, which must run under some more designated strin-gent quality conditions.

3. A reliable modiﬁed MPPAC for capability control

Process capability index measures the ability of the process to reproduce products that meet spec-ifications. However, the fact that process capabil-ity indices combine information about closeness to target and process spread, and express the capa-bility of a process by a single number, may in some cases also be held as one of their major drawbacks. If, for instance, the process is found non-capable, the operator is interested in knowing whether this non-capability is caused by the fact that the pro-cess output is off target or that the propro-cess spread is too large, or if the result is a combination of these two factors. In order to circumvent this shortage of process capability indices, recent re-search suggests that different graphical methods be used to support the improvement initiative aimed at accomplishing more capable processes (see, e.g., Gabel, 1990; Boyles, 1991; Tang et al.,

1997; Deleryd and Va¨nnman, 1999). The modiﬁed

CpkMPPAC proposed by Pearn and Chen (1997) is shown inFig. 1, with ﬁve capability zones corre-sponding to the ﬁve process conditions for Cpk= 1.00, 1.33, 1.67, and 2.00.

In this modified MPPAC, Cpuand Cplrepresent the X-axis and Y-axis, respectively. Whereas Cpis the average of Cpu and Cpl, namely, Cp= (Cpu+ Cpl)/2 and Cpk is the minimum value of the X- and Y-axes, namely, Cpk= min{Cpu, Cpl}. Thus, based on CpkMPPAC, the vertical and hor-izontal axes of the chart are to evaluate the-larger-the-better and the-smaller-the-larger-the-better characteris-tics, respectively. Furthermore, a few subsidiary lines of Cacan be added on MPPAC for precisely controlling the process centering. Off-diagonal subsidiary lines are plotted when Ca are 0.500, 0.750 and 0.875 inFig. 1. Note that we will assume the preset target value T at the mid-point of the specification (i.e. m = T). Ca< 0.875 indicates that the process is not accurate; actions to shift the process mean closer to the process target are re-quired. Namely, CaP0.875 indicates a process with good accuracy. In general, Cacannot be too small since a smaller Caimplies the process mean shifts farther away from the process target and re-sults in much process loss. Let r =jl mj/d, then the values of l are m + r· d and m r · d for each Ca. The slope of the corresponding subsidiary line is (1 + r)/(1 r) when the process mean is greater than the process target, and the slope of the corresponding subsidiary line is (1 r)/ (1 + r) when the process mean is smaller than the process target. Table 1 briefly displays the

Fig. 1. The modiﬁed Cpk MPPAC with capability zones for

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values of Cawith the corresponding l, r, and slope of the subsidiary lines.

In quality improvement, reduction of variation from the target is as important as increasing the process yield and reduction process spread. A modified CpkMPPAC put the two concepts: close-ness to target and small spread in more efficient way than by using a process capability index alone. If the exact values of l and r are known, then the modified CpkMPPAC can easily be used. That is, if the corresponding value of (Cpu, Cpl) is inside the capability region, then the process is defined to be capable, and if the value is outside, the process is defined as non-capable. In practice, though, we never know the true values of capability indices. In the next section we will develop the procedure to construct the lower confidence bounds of indi-ces Cpk, Ca, Cpu, and Cpl for each characteristic type. These lower confidence bounds can be simul-taneously plotted on a single chart to check if the process output is off target or that the process spread is too large, or if the result is a combination of these two.

4. Lower conﬁdence bounds for production yield assurance

As noted before, several MPPACs have been developed for monitoring processes with multiple characteristics. In current practice of

implement-ing those charts, practitioners simply plot the esti-mated index values on the chart then make conclusions on whether processes meet the capa-bility requirement and directions need to be taken for further capability improvement. Such ap-proach is highly unreliable since the estimated in-dex values are random variables and sampling errors are ignored. A reliable approach is to first convert the estimated index values to the lower confidence bounds then plot the corresponding lower confidence bounds on the MPPAC. Using lower confidence bounds, the MPPAC applica-tions become more efficient and the results are not misleading.

4.1. Lower conﬁdence bounds on Cpk

Construction of the exact lower conﬁdence bounds on Cpk is complicated since the distribu-tion of bCpk involves the joint distribution of two non-central t distributed random variables, or alternatively, the joint distribution of the folded-normal and the chi-square random variables, with an unknown process parameter even when the samples are given (Pearn et al., 1992). Numerous methods for obtaining approximate conﬁdence bounds of Cpkhave been proposed, including

Bis-sell (1990),Chou et al. (1990),Zhang et al. (1990),

Porter and Oakland (1991), Kushler and Hurley

(1992), Rodriguez (1992), Nagata and Nagahata

(1994),Tang et al. (1997)and many others. Under

Table 1

The values of Cawith the corresponding l, r, and slope of lines

Ca l r Slope 1.000 m 0.000 1.000 0.875 m + 0.125· d 0.125 1.286 m 0.125 · d 0.125 0.778 0.750 m + 0.250· d 0.250 1.667 m 0.250 · d 0.250 0.600 0.500 m + 0.500· d 0.500 3.000 m 0.500 · d 0.500 0.333 0.250 m + 0.750· d 0.750 7.000 m 0.750 · d 0.750 0.143 0.000 USL 1.000 1 LSL 1.000 0.000

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the assumption of normality,Pearn and Lin (2004)

obtain an exactly explicit form of the cumulative distribution function of the natural estimator bCpk as F bCpkðyÞ ¼ 1 Z bpffiffin 0 G ðn 1Þðb ffiffiffi n p tÞ2 9ny2 ! /ðt þ n pffiffiffinÞ þ /ðt npffiffiffinÞdt;

for y > 0, where b = d/r, n = (l m)/r, G(Æ) is the cumulative distribution function of the chi-square distribution with n 1 degrees of freedom, v2

n1, and /(Æ) is the probability density function of the standard normal distribution. Hence, given the sample of size n, the conﬁdence level 1 a, the esti-mated value bCpk and the parameter n, using numerical integration technique with iterations, the 100(1 a)% lower conﬁdence bounds for Cpk, LC, and bL= 3LC+jnj, can be obtained by solving the following equation,

Z bL ffiffin p 0 G ðn 1ÞðbL ffiffiffi n p tÞ2 9n bC2_pk 0 @ 1 A /ðt þ n pffiffiffinÞ þ /ðt npffiffiffinÞdt¼ a. 4.2. Lower confidence bounds on Ca

On the assumption of normality, Pearn et al.

(1998) showed that the natural estimator bCa¼

1 jx mj=d of the process accuracy index Ca, is the maximum likelihood estimator, consistent, asymptotically efficient and pffiffiffinð bCa CaÞ con-verges to Nð0; 1=ð3C2

pÞÞ in distribution. And the statisticpffiffiffinjx mj=r has a folded normal distribu-tion as defined by Leone et al. (1961). Therefore, owing to pffiffiffinð bCa CaÞ converging to N ð0; 1= ð3C2 pÞÞ, 3 ffiffiffi n p eCpð bCa CaÞ converges to N(0,1) in distribution, An approximate 100(1 a)% confi-dence interval of Cacan be established as

b Ca za=2 3p eCffiffiffin p ; bCaþ za=2 3p eCffiffiffin p " # ; where Cep¼ bn1Cbp, bn1= (2/(n 1))1/2· C[(n 1)/2]/C[(n 2)/2], and za is the upper ath quantile for the standard normal distribution. While a 100(1 a)% lower confidence bound of

Ca, LCa, can be constructed using only the lower

limit as bCa za=ð3p eCffiffiffin pÞ.

4.3. Lower conﬁdence bounds on Cpuand Cpl

Chou and Owen (1989)showed that under

nor-mality assumption the estimators bCpuand bCpl are distributed asð3pffiffiffinÞ1tn1ðdÞ, where tn1(d) is dis-tributed as the non-central t distribution with n 1 degrees of freedom and the non-centrality parameter d¼ 3pffiffiffinCpu and d¼ 3 ffiffiffin

p

Cpl, respec-tively. A 100(1 a)% lower confidence bound LC for Cpu satisfies Pr (CpuP LC) = 1 a. It can be written as Pr USL l 3r P LC ¼ Pr tðn1ðd1Þ 6 t1Þ ¼ 1 a; where t1¼ 3 bCpu ffiffiffin p and d1¼ 3 ffiffiffin p LC. Thus, LCcan be obtained by solving the above cumulative distri-bution function of tn1(d1). Similarly, a 100 (1 a)% lower confidence bound for Cpl can be obtained by solving Pr(CplP LC) = 1 a.

5. Bootstrap conﬁdence bound for overall capability testing

Statistical hypothesis testing used for examining whether the process capability meets the custom-ers demands, can be stated as follows: H0: CT6C versus H1: CT> C. The null hypothesis states that the overall process capability is no greater than the minimum capability level C. We conclude that the entire product capability satisﬁes the required level if the sample statistic bCT is greater than the critical value (or p-value < a) or the lower conﬁdence bound of CTis greater than the capability require-ment C. Otherwise, we reverse the conclusion. Unfortunately, the exact sampling distribution of

b

CT is intractable. Efron (1979, 1982) introduced a non-parametric, computational intensive but eﬀective estimation method called the ‘‘Boot-strap’’, which is a data based simulation technique for statistical inference. Efron and Tibshirani

(1986) developed three types of bootstrap

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(SB) confidence interval, the percentile bootstrap (PB) confidence interval, and the biased corrected percentile bootstrap (BCPB) confidence interval.

Efron and Tibshirani (1986)indicated that a rough

minimum of 1000 bootstrap resamples is usually sufficient to compute reasonably accurate confi-dence interval estimates. We apply these three bootstrap methods to the entire product capability measure CT to obtain the confidence bounds. In order to obtain more reliable results, B = 10,000 bootstrap resamples are taken and these 10,000 bootstrap estimates of CT are calculated and or-dered in ascending order. The notations bCT and

b

C_TðiÞ will be used to denote the estimator of CT and the associated ordered bootstrap estimates. For instance, bC_Tð1Þ is the smallest of the 10,000 bootstrap estimates of CT. For each single charac-teristic, the Cpu, Cpl, and Cpk values can be esti-mated by their estimators _C_bpuj¼ ðUSLj xjÞ=sj, j = 1, 2, . . . , nu, Cbplj¼ ðxj LSLjÞ=sj, j = 1, 2, . . ., nl, and Cbpkj¼ ðdj jxj mjjÞ=ð3 sjÞ, j = 1, 2, . . . , nk, where xj and sj are the sample mean and standard deviation of the j-th characteristic. Thus, the bootstrap estimates of CTare deﬁned as

b CT¼ 1 3U 1 Y i2S Yni j¼1 2Uð3 bCpijÞ 1 h i þ 1 ! " , 2 #! .

Based on the SB method, the 100(1 a)% lower conﬁdence bound for CT is bCT za Sb_C

T,

where SbCT is the sample standard deviation of

b

CT. From the ordered collection of bC

TðiÞ, the a percentage and the (1 a) percentage points are used to obtain 100(1 2a)% PB conﬁdence inter-val for CT is ½ bC

TðaBÞ, bC

Tðð1 aÞBÞ. While a 100(1 a)% lower conﬁdence bound can be con-structed by using only the lower limit bC_TðaBÞ.

That is, for a 95% lower confidence bound for CT based on the PB method with B = 10,000 would be obtained as bC_Tð500Þ. For the BCPB method, it calculates the probability p₀¼ Pð bC_T 6_Cb_T_{Þ and computes the inverse of the} cumulative distribution of a standard normal based on p0 as z0= U1(p0), pL= U(2z0 za). The 100(1 a)% BCPB lower confidence bound can be obtained as Cb_TðpLBÞ. Therefore, to determine whether the total product capability is capable or not, the minimum requirement level C and the significant level a-risk are first decided. And if the lower confidence bound of CT, is greater than the capability requirement C, we conclude that the entire product capability sat-isfies the required level. Otherwise, we reverse the conclusion.

6. A case study

In the following, we consider a case study to demonstrate how the modified Cpk MPPAC and the lower confidence bound can be used in analyz-ing processes with multiple characteristics. The case we investigate involves a process manufactur-ing the dual-fiber tips, which is used in makmanufactur-ing fiber optic cables. For a particular model of the dual-fiber tips, the specifications of characteristics are presented in Table 2, which is taken form a optical communication manufacturing factory lo-cated on Science-based Industrial Park in Taiwan, devoted to the optical fiber component module products, such as single-fiber tips for collimators, isolators, switches, WDM, circulators, etc., and dual-fiber tips for WDM, hybrid isolators, com-pact circulators, etc. The applications of these fiber

Table 2

Speciﬁcations of characteristics for the dual-ﬁber tips

Characteristic Type LSL Target USL Capillary diameter Nominal-the-best 1.795 mm 1.800 mm 1.805 mm Capillary length Nominal-the-best 6.00 mm 6.25 mm 6.50 mm Wedge Nominal-the-best 7.5 8 8.5 Core diameter Nominal-the-best 126 lm 127 lm 128 lm Return loss Larger-the-better 60 dB – – Polishing direction Smaller-the-better – – 5

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tips are fabricated with high performance optical ﬁber ends and precise glass capillary.

The key quality characteristics of a dual-fiber tip include (1) capillary diameter, length, wedge and core diameter, which are nominal-the-best specifications, (2) return loss, which is the-larger-the-better specification, (3) polishing direction, which is the-smaller-the-better specification. Cus-tomers expect all of the quality characteristics of a dual-fiber tip to meet or exceed expected levels.

Fig. 2 shows a sample of the dual-ﬁber tips. We

take a random sample of size 60, for the dual-fiber tips from a stable (under statistical control) pro-cess in the factory, and measure the six product quality characteristics, the capillary diameter (I), length (II), wedge (III), core diameter (IV), return loss (V), and polishing direction (VI). For these 60 measurements of each characteristics, under the Shapiro–Wilk test for normality, the result con-firms that all the p-value >0.1. That is, it is reason-able to assume that the process data collected from the factory are normally distributed. The calcu-lated sample mean, sample standard deviation, the estimated PCIs, bCpu, bCpl, bCp, bCa, bCpk, a 95% lower confidence bound of Ca, and lower confi-dence bound of Cpk(Cpuor Cpl), LC, are summa-rized in Table 3.

The modiﬁed CpkMPPAC for the six processes based on the estimated PCI values and the lower conﬁdence bound listed in Table 3, are displayed

in Figs. 3 and 4, respectively. Table 4 displays

the manufacturing quality and capability group-ings for the six dual-fiber tips processes using the estimated values (unreliable) and the lower confi-dence bounds (reliable) associated with the corre-sponding non-conformities (NC) expressed in ppm (with asterisks * indicating incorrect group-ings). Therefore, from these figures and tables, an

approach widely used in current industrial applica-tions based on the estimated PCI values only, we note that such MPPAC obviously conveys unreli-able information and is misleading, which should be avoided in real applications.

Hence, based on the analysis of this chart as

Fig. 4, it provides directions and priority for pro-cesses important to mining process defect. We can make some conclusions and recommendations to these six processes in the following:

Fig. 2. A sample of the dual-ﬁber tips.

Table 3

The calculated sample mean, sample standard deviation, the estimated capability indices and lower conﬁdence bound Code Characteristic x s Cbpu Cbpl Cbp Cba LCa Cbpk LC I Capillary diameter 1.8009 0.00097 1.412 2.032 1.722 0.820 0.748 1.412 1.184 II Capillary length 6.255 0.04035 2.024 2.107 2.065 0.980 0.908 2.024 1.706 III Wedge 7.99 0.0959 1.773 1.703 1.738 0.980 0.908 1.703 1.433 IV Core diameter 126.8 0.2458 1.627 1.085 1.356 0.800 0.728 1.085 0.904 V Return loss 63.6 0.9547 – 1.257 – – – 1.257 1.051 VI Polishing direction 4.2 0.3027 0.881 – – – – 0.881 0.728

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(a) The plotted points IV and VI are not located within the contour of Cpk= 1.00. It indicates that the process has a very low capability. For the point IV, since the lower confidence bound of Cais 0.728, that is, the process of core diameter represents that the process mean is towards the lower specification limit (process mean is smaller than target value), and the poor capabilities are mainly contributed by the significant process depar-ture from target. Thus, both characteristics core diameter and polishing direction are candidates for high-priority quality improvement effort focus. Under the six-sigma program, the quality improvement effort could focus on the reduction of process variability and the decrease of the process mean from the target to improve the process quality.

(b) The plotted points I and V lie within the con-tours of 1.00 6 Cpk< 1.33. The point I lies inside the area, which is to the left of the 45 target line (slope = 1) represents pro-cesses where the process mean is towards the upper specification limit (process mean is greater than the target value). On the other hand, for the point V, the lower confidence bound of Cpuis 1.051, the process is capable and the corresponding non-conformities are about 800 ppm. Thus, quality improvement effort for these processes should be first focused on reducing their process departure from the target value T for the process of capillary diameter, then the reduction of the process variance.

(c) Process wedge (III) lies inside the contours of 1.33 6 Cpk< 1.67, the process is ‘‘Satisfac-tory’’. And the lower conﬁdence bound of Ca is close to the 45 target line (Ca= 1). Thus, the quality improvement eﬀort for pro-cess wedge could be focused on the reduction of the process variation.

(d) The plotted point II lies inside the contours of 1.67 6 Cpk< 2.00, and the lower conﬁ-dence bound of Ca is greater than 0.875. The corresponding non-conformities of pro-cess are 0.309 ppm only. Thus, stringent con-trol for characteristic capillary length could be reduced since the process is ‘‘Excellent’’.

6.1. Overall process yield analysis

The sample estimates of CTand three bootstrap lower conﬁdence bounds of CT for the dual-ﬁber tips can be calculated from the sample. Table 5

Fig. 4. The modiﬁed CpkMPPAC for the LCB.

Table 4

Estimated value and lower conﬁdence bound (LCB) of capability indices with their groupings for the six characteristics Code Characteristic Estimated index NC Grouping LCB NC Grouping I Capillary diameter 1.412 22.75 Satisfactory* 1.184 383.32 Capable II Capillary length 2.024 0.0013 Super* 1.706 0.309 Excellent III Wedge 1.703 0.324 Excellent* 1.433 17.16 Satisfactory IV Core diameter 1.085 1133.9 Capable* 0.904 6687.9 Incapable V Return loss 1.257 81.30 Capable 1.051 799.74 Capable VI Polishing direction 0.881 4108.8 Incapable 0.728 14481 Incapable

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displays the manufacturing quality and its corresponding ppm of non-conformities for the dual-ﬁber tips processes using the estimated bCT values and the lower conﬁdence bounds of CT based on the three bootstrap methods.

Based on the analysis ofTable 5, we could find that the modified product capability obtained using three bootstrap methods are certainly more reliable than the estimated bCT index values, since the sampling errors are considered in the LCB ap-proach. In fact, as the sample estimate bCT may overestimate the true capability (overall process yield), it conveys unreliable and misleading infor-mation, which should be avoided in factory appli-cations. The lower confidence bound not only gives us a clue on the minimal actual performance of the process which is tightly related to the frac-tion of non-conforming units, but is also useful in making decisions for capability testing.

7. Conclusions

Process capability indices establish the relation-ship between the actual process performance and the manufacturing specifications, which quantify process potential and process performance, are essential to any successful quality improvement activities and quality program implementation. Capability measure for processes with single char-acteristic has been investigated extensively, but is comparatively neglected for processes with multi-ple characteristics. In real applications, a process often has multiple characteristics with each having different specifications. MPPAC can be used for evaluating the performance of a multi-process product, sets the priorities among multiple pro-cesses for capability improvement and indicates if reducing the variability, or the departure of the

process mean should be the focus of improvement. However, existing applications on MPPAC con-trol charts simply look at the estimated indices val-ues and then make a conclusion on which the given process is classified, is highly unreliable and mis-leading since they didnt considered sampling errors. We proposed a reliable approach to vert the estimated index values into the lower con-fidence bounds, then plot the corresponding lower confidence bounds on the MPPAC. The lower confidence bound obtained by analytical method gives us reliable performance measure for each sin-gle characteristic and overall production yield. Based on the proposed approach, the practitioners can make reliable decisions for capability testing and monitoring the performance of all process characteristics simultaneously.

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