DOI 10.1007/s11135-009-9243-x
R E S E A R C H N OT E
Estimating and testing process accuracy with extension
to asymmetric tolerances
Chien-Wei Wu · Ming-Hung Shu · W. L. Pearn · Yi-Chang Tai
Published online: 9 May 2009
© Springer Science+Business Media B.V. 2009
Abstract Pearn et al. (Commun. Stat. Theory Methods, 27(4):985–1000,1998) introduced the process accuracy index Cato measure the degree of process centering, the ability to clus-ter around the cenclus-ter. In this paper, we derive an explicit form of the cumulative distribution function for the estimator ˆCawith the case of symmetric tolerances. Subsequently, the distri-butional and inferential properties of the estimated process accuracy index Caare provided. Calculations of the critical values, P-values, and lower confidence bounds are developed for testing process accuracy. Further, a generalization of Ca for the case with asymmetric tol-erances is proposed to measure the process accuracy. Based on the results practitioners can easily perform the testing of the process accuracy, and make reliable decisions on whether actions should be taken to improve the process quality. An application is given to illustrate how we test the process accuracy using the actual data collected from the factory.
Keywords Asymmetric tolerances· Critical value · Process accuracy · Process centering 1 Introduction
In recent years, process capability indices (PCIs) have received substantial research attention in quality assurance and statistical literatures as well (seeKotz and Lovelace 1998;Kotz and Johnson 2002;Spiring et al. 2003;Pearn et al. 2004for more details). Those indices have become popular as unit-less measures on whether a process is capable of reproducing items
C.-W. Wu
Department of Industrial Engineering and Systems Management, Feng Chia University, Taichung, Taiwan M.-H. Shu (
B
)Department of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan, ROC
e-mail: [email protected] W. L. Pearn· Y.-C. Tai
Table 1 Cavalues and ranges ofμ
Cavalue Range ofμ Cavalue Range ofμ
Ca= 1.00 μ = M 0.25 < Ca< 0.50 d/2 < |μ − M| < 3d/4 0.75 < Ca< 1.00 0< |μ − M| < d/4 0.00 < Ca< 0.25 3d/4 < |μ − M| < d 0.67 < Ca< 0.75 d/4 < |μ − M| < d/3 Ca= 0.00 μ = LSL or μ = U SL 0.50 < Ca< 0.67 d/3 < |μ − M| < d/2 Ca< 0.00 μ < LSL or μ > U SL
meeting the quality requirement preset by the product designer. Based on analyzing the PCIs, a production department can trace and improve a poor process so that the quality level can be enhanced and the requirements of the customers can be satisfied.
The first, and the original, process capability index was Cp. The Cpindex reflects product consistency by considering the overall process variability relative to manufacturing toler-ances, which is designed to provide an indirect measure of potential ability to meet require-ments (Kane 1986). The Ca index measures the degree of process centering (the ability to cluster around the center), which can be regarded as a process accuracy index (seePearn et al. 1998). The indices Cpand Ca are defined as the following:
Cp =
USL− LSL
6σ , Ca= 1 −
|μ − m|
d ,
where LSL and USL are the lower and upper specification limit,μ is the process mean, σ is the process standard deviation, m= (USL + LSL)/2 is the mid-point between the upper and the lower specification limits, and d= (USL − LSL)/2 is the half length of the specification interval. For processes with two-sided specification limits, the percentage of non-conforming items can be calculated as 1− F(USL) + F(LSL), where F(·) is the cumulative distribution function (CDF) of the process characteristic X . On the assumption of normality, non-con-forming items (NC) can be expressed in parts per million (PPM) as:
NC= 1− USL− μ σ + LSL− μ σ × 106
where(·) is the cumulative distribution function (CDF) of the standard normal distribution. If the process is perfectly centered at the specification range(μ = m), then the percentage of non-conforming items can be expressed as 2(−3Cp). For example, Cp= 1.00 corresponds to NC= 2,700 PPM, and Cp= 1.33 corresponds to NC = 63 PPM. However, Cpdoes not refer to the mean of the process, it will not give an exact measure of the percentage of non-conforming items in the general case, i.e.,μ = m. Therefore, it provides a lower bound on NC with 2(−3Cp). On the other hand, the index Caprovides a quantified measure of ability to cluster around the center, which alerts the user if the process deviates from its midpoint. For example, Ca = 1 indicates that the process is perfectly centered (μ = m), Ca= 0 indicates that the process meanμ is located at one of the specification limits. Thus, when 0 < Ca < 1, the process mean is located between the mid-point and one of the specification limits. Obvi-ously, if Ca < 0 then it indicates that μ fall outside the specification limits (i.e., μ > U SL orμ < LSL), the process is severely off-center and it needs an immediate troubleshooting. Table1displays various Cavalues and the corresponding ranges of the departure magnitude ofμ.
We remark here that the process capability approach can be used only if the manufacturing process is under statistical control. If the process is out of control in the early stages of process capability analysis, it will be unreliable and meaningless to estimate process capability.
Proper understanding and accurate estimation of the capability index 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 flaws since there is no assessment of the sampling errors. As the use of the capability indices grows more wide-spread, users are becoming educated and sensitive to the impact of the estimators and their sampling distributions, learning that capability measures must be reported in confidence inter-vals or via capability testing (Kushler and Hurley 1992;Vännman and Kotz 1995;Zimmer et al. 2001;Pearn and Yang 2003). Critical values are usually used for making decisions in capability testing with a designated Type I errorα, the risk of misjudging an incapable process as a capable one. The P-values present the actual risk of misjudging an incapable process as a capable one. That is, if P-value< α then we reject the null hypothesis, and conclude that the process is capable with actual Type I errorα. Similarly, the lower confi-dence bounds convey the minimum capability of the process which is essential to quality assurance.
Therefore, in order to assess process performance and make decisions in the manufactur-ing capability testmanufactur-ing, the exact cumulative distribution functions (CDF) needs to be derived in advance. This paper is organized as follows. In Sect.2, we derive the explicit forms of the cumulative distribution functions (CDF) and the probability density function (PDF) for the estimator ˆCa with the case of symmetric tolerances. The distributional and inferential properties of the estimated process accuracy index Caare provided. The calculations of the critical value, P-value and lower confidence bound are developed for testing process quality. Furthermore, extensions to the case of asymmetric tolerances are discussed in Sect.3. The decision making rules are presented in Sect.4. Practitioners can use the proposed results to perform quality testing and determine how well the process can reproduce product items to meet the specified quality requirement. Finally, some concluding remarks are made in Sect.5.
2 Estimation of accuracy indexCafor symmetric tolerances
A process is said to have symmetric tolerances if the target value T is set to be the mid-point of the specification interval, i.e. T = m = (USL + LSL)/2. Most research in quality assurance literature has focus on cases in which the manufacturing tolerances are symmetric. Examples includeChan et al.(1988),Pearn et al.(1992), Vännman and Kotz
(1995),Spiring(1997),Zimmer et al.(2001),Vännman and Hubele(2003), and many oth-ers. To estimate the accuracy index Ca,Pearn et al.(1998) considered the natural estimator
ˆCaas
ˆCa = 1 −
¯X− m
d ,
where ¯X =ni=1xi/n is the conventional estimator of the process mean μ, which may be obtained from a stable process.
To derive the explicit form for the CDF of the estimator ˆCa, we first define D=√nd/σ ,
Z=√n( ¯X −m)/σ , and H =Z, where Zis distributed as N(√n(μ − m)/σ, 1) and H is
FˆCa(x) = P ˆCa≤ x = P 1− H D ≤ x = 1 − P {H ≤ D (1 − x)} = 1 − D(1−x) 0 fH(h)dh (1)
since H=Zis a folded normal distribution, we have
fH(t) = φ
t+ ξ√n+ φ t− ξ√n, (2)
whereξ = (μ − T )/σ and φ(·) is the PDF of the standard normal distribution N(0, 1). Substituting (2) into (1) gives
FˆCa(x) = 1 −
D(1−x)
0
φ t+ ξ√n+ φ t− ξ√ndt, for − ∞ ≤ x ≤ 1. (3) By differential with respect to x gives the PDF of ˆCaas
fˆCa(x) = Dφ D(1 − x) + ξ√n+ φ D(1 − x) − ξ√n, for − ∞ ≤ x ≤ 1. (4)
In order to determine whether a given process capability meets the customers’ demands and runs under the desired quality condition, the statistical hypothesis testing can be stated as follows:
H0: Ca≤ C (process is inaccurate),
H1: Ca> C (process is accurate).
We will reject the null hypothesis H0(Ca ≤ C), when ˆCa> c0with Type I errorα(c0), the chance of incorrectly concluding an inaccurate process(Ca ≤ C) as an accurate (Ca > C) process. Based on the CDF of ˆCaexpressed in (3), given values of capability requirement C, parameterξ, sample size n, and α risk, the critical value c0can be obtained by solving the Eq.5using available numerical methods.
b√ n(1−c0) 0 φ t+ ξ√n+ φ t− ξ√ndt= α, (5) where b= d/σ .
For users’ convenience in applying our proposed procedure, we tabulate the critical values of ˆCafor values ofα = 0.01 and 0.05 with n = 10, 25(25)100 in Table2for various values of C. For example, if C= 0.667 is the minimum capability requirement, then for α = 0.01, with sample size n = 50, ξ = 1.0 and d/σ we can find c0 = 0.777 from Table2. Thus, the critical value of ˆCarequired for the process capable is c0 = 0.777. That is, if ˆCa is greater than 0.777, we say that the process is capable.
Given Ca= C, b = d/σ expressed as b = |ξ|/(1 − C), the P-value corresponding to c∗, a specific value of ˆCacalculated from the sample data, is:
P-value= P( ˆCa ≥ c∗|Ca = C) =
b√n(1−c∗) 0
φ t+ ξ√n+ φ t− ξ√ndt. (6)
Furthermore, based on the CDF of ˆCa expressed in (3), given the sample of size n, the confidence levelγ , the estimated value ˆCa, andξ, then the lower confidence bounds CaLcan be obtained by solving the following Eq.7
b√n 1− ˆCa 0 φ t+ ξ√n+ φ t− ξ√ndt= 1 − γ, (7)
Table 2 Critical values c0for values ofα = 0.01 and 0.05 and n = 25, 50, and 75 |ξ| 0.5 1.0 1.5 n d σ Ca c0 Ca c0 Ca c0 α = 0.01 α = 0.05 α = 0.01 α = 0.05 α = 0.01 α = 0.05 25 2 0.750 0.973 0.914 0.500 0.733 0.664 0.250 0.483 0.414 3 0.833 0.982 0.943 0.667 0.822 0.777 0.500 0.655 0.610 4 0.875 0.987 0.957 0.750 0.866 0.832 0.625 0.741 0.707 5 0.900 0.989 0.966 0.800 0.893 0.866 0.700 0.793 0.766 50 2 0.750 0.914 0.866 0.500 0.664 0.616 0.250 0.414 0.366 3 0.833 0.943 0.911 0.667 0.777 0.744 0.500 0.610 0.578 4 0.875 0.957 0.933 0.750 0.832 0.808 0.625 0.707 0.683 5 0.900 0.966 0.947 0.800 0.866 0.847 0.700 0.766 0.747 75 2 0.750 0.884 0.845 0.500 0.634 0.595 0.250 0.384 0.345 3 0.833 0.923 0.896 0.667 0.756 0.730 0.500 0.590 0.563 4 0.875 0.942 0.922 0.750 0.817 0.797 0.625 0.692 0.672 5 0.900 0.954 0.938 0.800 0.854 0.838 0.700 0.754 0.738
Table 3 Lower confidence bounds CaLof ˆCa= 0.75 for |ξ| = 1.0(0.1)2.0 and n = 20, 30, 50, and 70 with
α = 0.05 |ξ| 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 n= 20 0.605 0.625 0.640 0.651 0.661 0.669 0.675 0.681 0.686 0.690 0.694 n= 30 0.643 0.656 0.667 0.675 0.682 0.688 0.692 0.696 0.700 0.703 0.706 n= 50 0.674 0.683 0.690 0.696 0.700 0.704 0.708 0.710 0.713 0.715 0.717 n= 70 0.689 0.696 0.701 0.706 0.709 0.712 0.715 0.717 0.719 0.721 0.723 where b= |ξ|/(1 − CL
a). Table3displays the lower confidence bounds CaL of ˆCa = 0.75, for various parameter values, withα = 0.05, |ξ| = 1.0(0.1)2.0, and n = 20(10)80.
Therefore, practitioners can use the proposed previous results to perform quality test-ing and determine if the process can reproduce product items to meet the specified process accuracy requirement.
3 Estimation of accuracy indexCafor asymmetric tolerances
Although cases with symmetric tolerances are common in practical situations, cases with asymmetric tolerances(T = m) often occur in the manufacturing industry. From the cus-tomer’s point of view, asymmetric tolerances reflect that deviations from the target are less tolerable in one direction than in the other (seeBoyles 1994andWu and Tang 1998). Usually they are not related to the shape of the supplier’s process distribution. However, asym-metric tolerances can also arise in situations where the tolerances are symasym-metric to begin with, but the process distribution is skewed or follows a non-normal distribution. Dealing with this, the data have been transformed to achieve approximate normality, as shown by
Fig. 1 Plots of Cavalues for processes with 10≤ μ ≤ 50 under(LSL, T, U SL) =
(10, 40, 50)
Chou et al. (1998) who have used Johnson curves to transform non-normal process data. Moreover, these indices presented above, are designed to monitor the performance for only normal and near-normal processes with symmetric tolerances, which are shown to be inap-propriate for cases with asymmetric tolerances (Boyles 1994;Pearn and Chen 1998). Unfor-tunately, there has been comparatively little research published on cases with asymmetric tolerances. Exceptions areBoyles(1994),Chen(1998),Pearn and Chen(1998),Jessenberger and Weihs(2000), andShu and Chen(2005).
To overcome the asymmetric cases(USL − T = T − LSL), we modify Caindex, denoted here as Ca:
Ca = 1 − A ∗ d∗,
where A∗ = max{d∗(μ − T )/Du, d∗(T − μ)/Dl}, d∗ = min{Du, Dl}, Du = USL −
T, Dl= T −LSL. Obviously, if T = m (symmetric tolerance), then d∗= Du= Dl= d, Ca reduces to the original index Ca. The factor A∗ensures that the new generalization Caobtains its maximal value atμ = T (process is on-target) regardless of whether the tolerances are symmetric(T = m) or asymmetric (T = m). Figure1displays the plots of Cafor processes with 10 ≤ μ ≤ 50 where (LSL, T, U SL) = (10, 40, 50) is an asymmetric tolerance.
Ca = 0 can be verified when the process mean is on the specification limit (μ = LSL
orμ = U SL). On the other hand, Ca > 0 when LSL < μ < U SL. Thus, given two processes E and F withμE > T and μF < T , satisfying (μE− T )/Du = (T − μF)/Dl (i.e., processes E and F have an equal departure ratio), the Ca values given to processes E and F are the same. For example, consider processes E and F withμE = 45 > T and
μF= 25 < T . Clearly, the corresponding departure ratios are 1/2 for both processes E and F (i.e.,(45 − 40)/10 = (40 − 25)/30 = 1/2). We have Ca = 0.50 for both processes E and F. In addition, the index Cadecreases when meanμ shifts away from target T in either direction. In fact, Ca decreases faster whenμ shifts away from T to the closer specification limit than that to the farther specification limit. This is an advantage since the index would respond faster to the shift towards “the wrong side” of T than towards the middle of the specification interval.
To estimate the generalization Ca, we can define the natural estimator ˆCaas: ˆC a = 1 − ˆA∗ d∗, where ˆA∗= maxd∗ ¯X− T/Du, d∗ T− ¯X/Dl
and ¯X =ni=1xi/n. We now define
D∗ = n1/2(d∗/σ) , Z = n1/2 ¯X− T/σ, Y = max{(d∗/Du) Z , − (d∗/Dl) Z}
2
λ = δ2, andδ = n1/2(μ − T )/σ . Then, the estimator ˆC
a can be rewritten as:
ˆC
a = 1 −
√ Y D∗.
Under the assumption of normality, Z is distributed as the normal distribution N(δ, 1). We note that the statistic Z2 follows a non-central chi-square distribution with one degree of freedom and non-centrality parameterλ = δ2.Chen(1998) defined the distribution of Y as a weighted non-central chi-square distribution with one degree of freedom and non-centrality parameterλ under the assumption of normality. The PDF of Y is derived as:
fY(y) = e−λ/2 2√π ∞ j=0 hj(λ) 1+ j 2 2 i=1 (−1)i j di2 fYj y di2 , y > 0. (8) whereλ = δ2, δ = n1/2(μ − T )/σ, hj(λ) = (2λ)j/2/( j!), d1= d∗/Dl, d2= d∗/Du and Yj is distributed asχ12+ j, the Chi-square distribution with degree of freedom 1+ j. Thus, the CDF of ˆCacan be obtained as follows:
FˆC a(x) = P ˆC a ≤ x = P 1− √ Y D∗ ≤ x = 1 − P Y ≤ [D (1 − x)]2 = 1 − [D∗(1−x)]2 0 fY(y)dy, for x < 1. (9)
By substituting (8) into (9) gives
FˆC a(x) = 1 − e−λ/2 2√π ∞ j=0 hj(λ) 1+ j 2 2 i=1 (−1)i j di2 × [D∗(1−x)]2 0 fYj y di2 d y , for x < 1. (10) Changing the variable with t = y/d2
i, then d y= di2dt for i = 1, 2, the CDF of ˆCacan be rewritten as FˆC a(x) = ⎧ ⎪ ⎨ ⎪ ⎩ 1−e−λ/2 2√π ∞ j=0hj(λ) 1+ j 2 2 i=1 (−1)i j [D∗(1−x)/d i]2 0 fYj(t)dt, x < 1 1, x≥ 1 (11)
Taking the derivative of the CDF of ˆCain x, the PDF of ˆCa will be obtained as fˆC a(x) = e−λ/2 √ π ∞ j=0 hj(λ) 1+ j 2 × 2 i=1 (−1)i j(1 − x) D∗ di 2 fYj (1 − x)D∗ di 2 , x < 1. (12) If the manufacturing tolerance is symmetric(T = m), then d∗ = Du = Dl = d, d1 =
d2= 1, the CDF of ˆCain (10) is reduced to:
FˆC a(x) = ⎧ ⎨ ⎩1− ∞ =0P (λ) [D(1−x)]2 0 fY2 (t)dt, x < 1 1, x ≥ 1 (13)
and the corresponding PDF is:
fˆC a(x) = ∞ =0 P (λ) D 2 +1(1 − x)2 ((2 + 1)/2) 2(2 −1)/2e−[D(1−x)] 2/2 , x < 1 (14) where D= n1/2(d/σ ) and P (λ) = e−λ/2(λ/2) /( !). It can be remarked that (13) and (14) are equivalent to (3) and (4), respectively.
4 Decision making rule for asymmetric tolerances
To make a decision for the case of asymmetric tolerances, we consider a testing hypothesis with the null hypothesis Ca ≤ C (the process is inaccurate) and the alternative hypothesis
Ca > C (the process accurate). The null hypothesis will be rejected if ˆCa> c0, where the critical value c0 is determined by P
ˆC
a > c0|Ca= C
= α. Hence, we can find c0 by solving Eq.15 P ˆC a > c0|Ca= C =e−λ/2 2√π ∞ j=0 hj(λ) 1+ j 2 2 i=1 (−1)i j di2 × [D∗(1−cα)]2 0 fYj y di2 d y = α. (15)
Table4displays the critical values c0 for d∗/σ = 2, 3, 4 and 5 with sample sizes n = 25, 50, and 75, |ξ| = 0.5(0.5)1.5, α = 0.01 and 0.05 and Dl: Du = 6 : 4.
Similarly, if the estimated index value is c∗, given the values of C, ξ and sample size n, then the P-value can be calculated as:
P-value= P ˆC a > c∗|Ca= C = e−λ/2 2√π ∞ j=0 hj(λ) 1+ j 2 2 i=1 (−1)i j di2 [D∗(1−c∗)]2 0 fYj y di2 d y , (16)
Table 4 Critical value c0of Cafor various parameter values, with Dl: Du= 6 : 4, α = 0.01 and 0.05, and n= 25, 50, and 75 |ξ| 0.5 1.0 1.5 n d ∗ σ Ca c0 Ca c0 Ca c0 α = 0.01 α = 0.05 α = 0.01 α = 0.05 α = 0.01 α = 0.05 25 2 0.750 0.965 0.909 0.500 0.732 0.664 0.250 0.482 0.414 3 0.833 0.977 0.939 0.667 0.821 0.776 0.500 0.655 0.609 4 0.875 0.982 0.954 0.750 0.866 0.832 0.625 0.741 0.707 5 0.900 0.986 0.963 0.800 0.893 0.865 0.700 0.793 0.765 50 2 0.750 0.914 0.866 0.500 0.665 0.616 0.250 0.414 0.366 3 0.833 0.942 0.910 0.667 0.776 0.744 0.500 0.609 0.577 4 0.875 0.957 0.933 0.750 0.832 0.808 0.625 0.707 0.683 5 0.900 0.965 0.946 0.800 0.866 0.846 0.700 0.765 0.746 75 2 0.750 0.884 0.844 0.500 0.634 0.595 0.250 0.387 0.345 3 0.833 0.922 0.896 0.667 0.756 0.730 0.500 0.591 0.563 4 0.875 0.942 0.922 0.750 0.817 0.797 0.625 0.693 0.672 5 0.900 0.953 0.937 0.800 0.853 0.838 0.700 0.754 0.738
Table 5 Lower Confidence Bounds CaLfor various parameter values, ˆCa = 0.75, Dl: Du = 7 : 3, α = 0.01, 0.05, n = 20(10)50, and |ξ| = 0.7(0.1)1.0 n |ξ| = 0.7 |ξ| = 0.8 |ξ| = 0.9 |ξ| = 1.0 α = 0.01 α = 0.05 α = 0.01 α = 0.05 α = 0.01 α = 0.05 α = 0.01 α = 0.05 20 0.027 0.474 0.286 0.538 0.408 0.578 0.479 0.605 30 0.365 0.563 0.468 0.600 0.527 0.625 0.566 0.643 40 0.474 0.603 0.538 0.630 0.578 0.648 0.605 0.663 50 0.529 0.626 0.576 0.648 0.606 0.663 0.628 0.675
Furthermore, given the true process capability levelγ and the estimated value ˆCaandξ, the lower confidence bounds CaLcan be obtained by solving the following Eq.17
e−λ/2 2√π ∞ j=0 hj(λ) 1+ j 2 2 i=1 (−1)i j di2 [D∗(1− ˆC a)]2 0 fYj y di2 d y = 1 − γ. (17) Table5displays the lower confidence bounds for the asymmetric case Dl : Du = 7 : 3, with ˆCa= 0.75 α = 0.01, 0.05, n = 20(10)50, and |ξ| = 0.6(0.1)1.0.
4.1 An example of testing the laser marking on IC packages
Integrated circuit (IC) packages are used for encapsulating the chips and connections against environmental, electrical and electromagnetic effects, and for establishing thermal path from the semi-conductor junction to the environment and electrical connection from the chip to the printed circuit board (PCB). In mask marking of IC packages, the marking area is determined by the output power of the laser, the cross-section of the beam and the properties
of the material being marked. Marked area of 75 mm2 at 10 J/pulse with pulse duration of about 0.69 ms has been reported for uncoated plastic packages. For the coated plastic pack-ages (e.g., marker ink or varnish), the marked area can reach 600 mm2per pulse with pulse energy ranging from 0.1 J to 0.75 J and a pulse repetition frequency of 30 Hz. For mask marking, a minimum character height of about 0.5 mm has been reported.
The quality of a mark is assessed by its legibility characteristics such as mark contrast, mark with, mark depth, spattering, and micro cracks. The characteristics are usually evalu-ated using complementary techniques such as optical microscopy, ultrasonics microscopy, electron microscopy, surface roughness measurement, and contrast evaluation devices. The acceptance of level of each of these characteristics generally depends on the manufacturer’s requirements. Mark width refers to the width of the line segment that forms a character. With the mask image marking, the mark width in the characters is essentially determined by the mask geometry and the lens imaging quality. It can be as small as a few micro-meters, which can only be read under a microscope. In beam deflected marking, the line width is mainly determined by the focused beam spot size, which varies between 20 and 100 mm.
The IC packages company monitors the process that laser marking on IC packages by measuring the mark width. According to the customer’s requirement, this company consid-ers the following normally distributed process with asymmetric tolerances LSL = 20 mm,
T = 26.5mm and USL = 32 mm for the mark width and the α − risk = 0.05. We calculate
that Dl = T − LSL = 6.5, Du= USL − T = 5.5, d∗= min {Dl, Du} = 5.5. The sample of
n= 100, the sample mean ¯X = 27.35 and the sample standard deviation S = 2.0. We can
calculate ˆA∗= maxd∗ ¯X− T/Du,d∗
T− ¯X/Dl
= 0.85, ˆξ = ¯X− T/S = 0.425
and ˆCa= 0.845. The corresponding P-value is 0.0532 from calculating (20). Because the α − risk = 0.05 is small than 0.0532, we do not have sufficient information to conclude that
the process meets the present accuracy requirement. The further improvement actions for the accuracy are needed in this laser marking process.
5 Conclusions
The use of indices to measure process capability and communicate information about pro-cesses and the specified requirements has become widespread. In this paper, the process accuracy index Ca is proposed to measure the degree of process centering (the ability to cluster around the center). An explicit form of the cumulative distribution function for the estimator ˆCawith the case of symmetric tolerances is derived. We provide the distributional and inferential properties of the estimated process accuracy index Ca. The calculations of critical value, P-value and lower confidence bound are developed for testing process quality. Furthermore, a new generalization of Ca for cases with asymmetric tolerances is proposed to measure the process accuracy. The obtained results are useful for the practitioners in performing the process accuracy testing, and make decisions whether improvement actions should be taken. For illustrative purpose, a real-world application is also given to show how to test process accuracy by using the actual data collected from the factory.
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