DOI 10.1007/s00170-004-2323-0 O R I G I N A L A R T I C L E Int J Adv Manuf Technol (2006) 27: 1223–1227
G.H. Lin
Process performance assessment based on sub-samples –
a large sample approach
Received: 8 April 2004 / Accepted: 1 July 2004 / Published online: 5 March 2005 ©Springer-Verlag London Limited 2005
Abstract Process incapability index has been introduced to pro-vide quantitative measures on process performance. Contribu-tions of the estimated incapability index based on sub-samples have been proposed under the normality assumption. In this paper, investigations based on sub-samples are considered under general conditions having fourth central moment. The limiting distribu-tion of the estimated incapability index based on sub-samples is derived. An approximate 100(1−α)% upper confidence bound of the considered incapability index is constructed. A demonstrated example is also provided to illustrate how the proposed approxi-mate upper confidence bound may be applied for judging whether the process runs under the desirable quality requirement. Keywords Approximate· Asymptotic · Incapability index · Sub-samples
1 Introduction
Process capability indices, whose purpose is to provide numeri-cal measures on whether a manufacturing process is capable of reproducing items satisfying the quality requirements preset by the engineer or the product designer, have received substantial research attention in the quality control and statistical literature. The three basic capability indices Cp, Ca and Cpk, have been defined as (see Kane [1] and Pearn et al. [2]):
Cp= USL− LSL 6σ , (1) Ca= 1 −|µ − m| d , (2) Cpk= min USL− µ 3σ , µ − LSL 3σ , (3) G.H. Lin
Department of Transportation & Logistics Management, National Penghu Institute of Technology,
Makung, Penghu, Taiwan 88042 E-mail: [email protected] Tel.: +886-6-9264115 ext. 3622 Fax: +886-6-9260373
where USL and LSL are the upper and lower specification lim-its preset by the process engineers or product designers,µ is the process mean,σ is the process standard deviation, m = (USL + LSL)/2 and d = (USL − LSL)/2 are the mid point and half length of the specification interval, respectively.
The index Cpreflects only the magnitude of the process vari-ation relative to the specificvari-ation tolerance; therefore, it is used to measure process potential. The index Ca measures the degree of process centering (the ability to cluster around the center) and is referred to as the process accuracy index. The index Cpktakes into account process variation as well as the location of the pro-cess mean. The natural estimators of Cp, Ca and Cpk can be obtained by substituting the sample mean ¯X=ni=1Xi/n for µ and the sample variance Sn2−1=ni=1Xi− ¯X2/(n − 1) for σ2 in Eqs. 1, 2, and 3. Chou et al. [3], Kotz et al. [4], Pearn et al. [2], Lin et al. [5] and Lin [6] investigated the statistical properties and the sampling distributions of the natural estimators of Cp, Ca and Cpk.
Boyles [7] noted that Cpk is a yield-based index. In fact, the design of Cpkis independent of the target value T , which can fail to account for process targeting (the ability to cluster around the target). For this reason, Chan et al. [8] developed the index Cpm to take the process targeting issue into consideration. The index
Cpm is defined as the following:
Cpm=
USL− LSL
6σ2+ (µ − T )2 . (4)
For processes with asymmetric tolerance (T = m), Chan et al. [8] also developed index C∗pm, a generalization of Cpm, which is defined as:
C∗pm= min{DL, DU}
3σ2+ (µ − T )2 , (5)
where DL= T − LSL, and DU= USL − T. The index C∗pm re-duces to the original index Cpm if T= m (processes with sym-metric tolerance). Unfortunately, the statistical property of the natural estimator of C∗pmis rather complicated.
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where zαrepresents the upperα-th quantile of the standard nor-mal distribution.
4 A demonstrated example
Nowadays PCBs (printed circuit boards) are widely and numer-ously used on electronic products. Factories that produce these products (like computers and peripherals) are classified in “PCB industry” because the core components inside these products are exactly PCBs. The process of PCB making mainly consist of a series of chemical relations, and the chemical properties de-termine the functions of a PCB. The operation, solder resist, in the post process of PCB making is not a chemical procedure but discussed particularly below. The effects of solder resist are to protect the metal ingredients inside circuits from oxidizing and also protect a board itself from exterior harm when embedding specific electronic components afterwards. Besides these two ef-fects above, the even surface of a PCB is seriously considered in quality control (QC).
The operation on the post process, solder resist, is the key to well-covering in PCB making. The method to conclude the PCB’s quality in its flat degree after solder resist is to measure its thickness. It concerns the uneven parts including caves and towers of a PCB. By measuring the thickness, we can have the even degrees of a PCB’s surface through control chart and PCI. The example investigated is taken from a PCB factory in Taiwan. This factory manufactures multi-layer printed circuit boards. For a particular model of PCBs investigated, the USL (upper
specifi-Table 2. Sample data with twenty five groups of size five Sample No. m Observations in sample of size n= 5 1 21.1758 19.4366 20.2354 18.5574 22.0713 2 18.9208 19.9780 17.4147 19.5567 18.9967 3 19.9721 21.5262 19.3949 20.1640 20.6530 4 19.4589 22.1350 20.1844 19.3834 19.5513 5 20.8238 20.2521 20.1918 20.8280 18.2577 6 21.1231 19.8394 18.4441 19.2193 19.4813 7 19.7417 18.4638 20.4202 19.4540 19.4766 8 20.0561 21.5216 18.2109 18.5913 18.2237 9 19.0534 19.8950 21.9965 18.7421 19.9436 10 21.1135 20.3691 19.9685 20.6190 21.7438 11 18.8809 19.5083 19.5352 20.4534 19.3626 12 19.0530 21.5973 20.4530 18.4731 20.3519 13 20.6335 19.6728 19.0526 19.6712 19.6513 14 19.9313 19.5785 19.8529 21.2596 19.9415 15 21.3278 18.7063 21.4218 20.5918 21.3464 16 19.0740 18.1585 19.3593 18.1629 21.7231 17 18.5812 19.1523 19.7809 18.3939 19.4081 18 20.1241 21.8232 19.3684 17.6704 18.0449 19 19.1862 21.8639 21.9595 20.5186 20.7390 20 19.9460 20.7021 20.6625 20.5912 20.5803 21 20.4751 20.5889 18.6714 20.0420 19.5398 22 19.5181 21.5813 20.7783 20.5295 19.4935 23 19.7671 19.8046 20.1934 19.8638 19.5769 24 20.9100 22.0304 20.9615 20.1611 20.8591 25 18.6383 19.4095 20.6956 18.9444 18.6633
cation limit) of a PCB’s thickness is set to 25.0 µm, and the LSL (lower specification limit) of a PCB’s thickness is set to 15.0 µm. The target value is 20.0 µm.
Twenty five samples of five parts each from a stable pro-cess are shown in Table 2. If the true Cppvalue fell into Eq. 17, we conclude that the underlying process reaches the desirable quality requirement under the given confidence level, other-wise, the process is regarded as incapable of the time. The data gives that the half length of the specification interval
d= (USL − LSL)/2 = 5.0, D = min {USL − T, T − LSL}/3 =
1.6667. The sample mean ¯¯X = 19.9393, smn= 1.0511, ˜Cpp= 0.3991, M3 = 8.2073 × 10−4, and M4 = 3.0693. Applying
Eq. 16 we obtain ˜σpp2 = 0.2417. From Eq. 17, an approximate 95% one-sided confidence interval of Cpp is (0, 0.3268). Re-ferring to Table 1, we claim that this process meets the capa-bility requirement at least “excellent” of the time with 95% confidence.
5 Conclusions
Process incapability index has been introduced to provide quan-titative measures on process performance. Contributions of the estimated incapability index based on sub-samples have been proposed under the normality assumption. In this paper, inves-tigations based on sub-samples were considered under general conditions having fourth central moment. The limiting distribu-tion of the estimated incapability index based on sub-samples was derived. An approximate 100(1 − α)% upper confidence bound of the considered incapability index was constructed. A demonstrated example was also provided to illustrate how the proposed approximate upper confidence bound may be applied for judging whether the process runs under the desirable quality requirement.
Acknowledgement The author would like to thank the anonymous referees for their helpful comments, which significantly improved the paper.
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