DOI 10.1007/s00170-003-1807-7
O R I G I N A L A R T I C L E Int J Adv Manuf Technol (2005) 25: 392–395
G.H. Lin
Process reliability assessment with a Bayesian approach
Received: 28 February 2003 / Accepted: 27 May 2003 / Published online: 16 June 2004
Springer-Verlag London Limited 2004
Abstract Based on the recommended maximum value of the
incapability index, previous research has provided a traditional procedure to evaluating actual performance of a production process. In this paper, an alternative procedure based on the Bayesian point of view is implemented for normally distributed processes. Useful minimum values required to ensure the pos-terior probability reaching a certain desirable level based on the incapability index are tabulated. A Bayesian procedure for judg-ing whether the process satisfies the preset quality reliability requirement is also proposed for practitioners to use.
Keywords Bayesian· Process incapability index
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 an engineer or the product designer, have received substantial re-search attention in the quality control and statistical literature. The two basic capability indices Cpand Cpk, have been defined as the following [6]: Cp= USL− LSL 6σ , (1) Cpk= min USL− µ 3σ , µ − LSL 3σ , (2)
where USL and LSL are the upper and the lower specification limits, respectively,µ is the process mean, and σ is the process standard deviation. The index Cpreflects only the magnitude of the process variation relative to the specification tolerance, there-fore is used to measure process potential. The index Cpk takes G.H. Lin
Department of Transportation & Logistics Management, National Penghu Institute of Technology,
Penghu, Taiwan 88042, R.O.C. E-mail: ghlin@npit.edu.tw
into account process variation as well as the location of the pro-cess mean. The natural estimators of Cpand Cpkcan be obtained by substituting the sample mean ¯X=ni=1Xi/nfor µ and the sample variance S2n−1=ni=1(Xi− ¯X)2/(n − 1) for σ2 in the expressions Eq. 1 and Eq. 2. Chou and Owen [4], Pearn, Kotz, and Johnson [10] and Kotz, Pearn, and Johnson [7] investigated the statistical properties and the sampling distributions of the nat-ural estimators of Cpand Cpk.
Boyles [2] 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, Cheng, and Spiring [3] developed the index Cpm to take the process targeting issue into considera-tion. The index Cpmis defined as the following:
Cpm=
USL− LSL
6σ2+ (µ − T )2 . (3)
For processes with asymmetric tolerance (T= m, m = (USL +
L SL)/2 is the midpoint of the specification interval (LSL, USL)).
Chan, Cheng, and Spiring [3] also developed the index C∗pm, a generalization of Cpm, which is defined as:
C∗pm= min{DL, DU}
3σ2+ (µ − T )2 , (4)
where DL= T − LSL, DU= USL − T. The index C∗pmreduces to the original index Cpm if T= m (processes with symmetric tolerance). Unfortunately, the statistical property of the natural estimator of C∗pmis rather complicated.
In attempting to simplify the complication, Greenwich and Jahr–Schaffrath [5] introduced an index called Cp p, which is easier to use and analytically tractable. In fact, the index Cp p is a simple transformation of the index C∗pm, Cp p= (1/C∗pm)2, which provides an uncontaminated separation between informa-tion concerning the process accuracy and the process precision while such separated information is not available with the in-dex C∗pm. If we denote D= min{DL, DU}/3, then Cp p can be
395
may claim that the process is capable in a Bayesian sense with 90 or 95% confidence.
4 Application of the procedure
A simple procedure, based on Cp pin our Bayesian approach, for judging whether the process satisfies the preset quality reliability requirement, is presented in the following.
STEP 1: Decide the quality requirement C0 (1.00, 0.56, 0.44,
0.36 or 0.25) and the posterior probability p (0.90, 0.95, 0.975 or 0.99).
STEP 2: Calculate ˆCp p = ni=1(Xi− T )2/D2 and δ = ¯x− T/sn from a stable (under statistical control) process.
STEP 3: Referring to Tables 1–2 to obtain C(p), the maximum values of ˆCp p/C0required to ensure the posterior
prob-ability p reaching a certain desirable level C0.
STEP 4: Conclude that the process is capable of the time with 100p% confidence in a Bayesian sense if ˆCp p< C0C(p). Otherwise, we do not have enough
informa-tion to conclude that the process is capable.
To illustrate how the proposed procedure may be performed, we consider the example given by Lin [9] where the measure-ments were taken on printed circuit board (PCB) thickness. The USL and the LSL were set at 25.0 µm and 15.0 µm, respectively with a Target value T= 20.0 µm. Assume that the quality re-quirement C0= 1 and the posterior probability p = 0.95. The
collected sample data (a total of 100 observations chosen from a stable process) given that d= 5, ˆCp p= 0.667,¯x− T/sn= 1.00. From Table 2, we find that C(p) = 0.8045, which implies that the maximum value of ˆCp pequal to C0C(p) = 0.8045. Since
0.667 < 0.8045, we claim that this process is capable of the time in a Bayesian sense with 95% confidence.
Appendix
Derivation of Eq. 9: the posterior probability
p= Pr{Cp p< C0|x} = Pr (µ − T )2+ σ2/D2< C 0| x = Pr σ2+ (µ − T )2<DC 0 2 |x = b 0 T+g(σ) T−g(σ) f(µ, σ |x ) dµ dσ = b 0 T+g(σ) T−g(σ) 2n π exp[−1/(βσ2)] σn+1βαΓ(α) × exp −n 2 ¯x − µ σ 2 dµ dσ = b 0 2 exp[−1/(βσ2)] σnβαΓ(α) × √ n √ 2πσ T+g(σ) T−g(σ) exp −n 2 µ − ¯x σ 2 dµ dσ = b 0 2 exp[−1/(βσ2)] σnβαΓ(α) × Φ T− ¯x + g(σ) σ/√n − Φ T− ¯x − g(σ) σ/√n dσ , where b= D√C0, g(σ) = √ b2− σ2,α = (n − 1)/2, β = nS2 n,
Φ is the cumulative distribution function of the standard
nor-mal distribution. Let y= βσ2,β= 2/ni=1(xi− T )2, b1(y) =
√
2/yδ2/(1 + δ2), γ = 1 + δ2, δ =¯x− T/s
n, b2(y) =
√
n{[1/(ty)] − 1}, t = n ˆCp p/(2C0), then the posterior
probabil-ity is p= 1/t 0 exp[−1/(γy)] yα+1γαΓ(α)
× {Φ[b1(y) + b2(y)] − Φ[b1(y) − b2(y)]} dy .
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