DOI 10.1007/s00170-003-1858-9 O R I G I N A L A R T I C L E
W.L. Pearn · C.W. Wu · K.H. Wang
Capability measure for asymmetric tolerance non-normal processes applied to
speaker driver manufacturing
Received: 10 March 2003 / Accepted: 27 June 2003 / Published online: 23 June 2004 Springer-Verlag London Limited 2004
Abstract Process capability indices, Cp(u, v), including Cp, Cpk, Cpm, and Cpmk, have been proposed in the
manufactur-ing industry to provide numerical measures on process potential and performance for normal processes. Earlier studies consid-ered a class of flexible capability indices, called CN p(u, v), for
processes with non-normal distributions where the tolerances are symmetric. In this paper we consider an extension of CN p(u, v),
called CN p(u, v), to handle non-normal processes with asym-metric tolerances. The extension takes into account the important property of the asymmetric loss function, which is shown to be more sensitive to process shift and more accurate than CN p(u, v)
in measuring process capability, hence provides better manufac-turing quality assurance. Comparisons between CN p(u, v) and
the extension CN p(u, v) are provided. We propose a sample per-centile estimator, and apply the bootstrap method to find the lower confidence bound for testing manufacturing capability. We also develop an integrated S-PLUS program to calculate the per-centile estimator and the corresponding lower confidence bound. As an illustration, the proposed approach is applied to capability testing of home-theater speaker systems.
Keywords Asymmetric tolerances· Bootstrap method · Lower
confidence bound· Non-normal processes · Percentile estimator
1 Introduction
Process capability indices Cp(u, v), which include the two
ba-sic indices Cp and Cpk[1], and the two more advanced indices, W.L. Pearn (u)
Department of Industrial Engineering & Management, National Chiao Tung University, Taiwan, ROC C.W. Wu
Department of Business Administration, Feng Chia University, Taiwan, ROC K.H. Wang
Department of Applied Mathematics,
National Chung Hsing University, Taiwan, ROC
Cpm and Cpmk [2, 3] as special cases, have been proposed in
the manufacturing industry to provide numerical measures on process potential and process performance. The superstructure indices Cp(u, v) are defined as the following [4]:
Cp(u, v) =
d− u |µ − m|
3σ2+ v(µ − T)2, (1)
whereµ is the process mean, σ is the process standard devia-tion, d= (USL−LSL)/2 is half of the length of the specification interval, m= (USL + LSL)/2 is the midpoint between the up-per and the lower specification limits, T is the target value, and u, v0. By setting u andv equal to 0 or 1, we obtain Cp(0, 0) = Cp, Cp(1, 0) = Cpk, Cp(0, 1) = Cpm, Cp(1, 1) = Cpmk. Those
four indices have been investigated extensively by Kane [1], Choi and Owen [5], Chan et al. [2], Pearn et al. [3, 6], and Kotz et al. [7].
Applications of those indices include the manufacturing of semiconductor products [8] head/gimbals assembly for mem-ory storage system [9], jet-turbine engine components [10], flip-chips and chip-on-board [11], rubber edge [12], wood prod-ucts [13], aluminum electrolytic capacitors [14], audio-speaker drivers [15], and Pulux surround [16]. Other applications of those indices include performance measures on processes with tool-wear problem [17], production process monitoring [18], and many others.
Flexible capability indices CN p(u, v)
The indices Cp(u, v) are appropriate for normal and near-normal
processes, but have been shown to be inappropriate for non-normal processes. Pearn and Chen [14] considered the general-ization of Cp(u, v) defined in the following, called CN p(u, v),
which can be applied to processes with arbitrary distributions: CN p(u, v) = d− u |M − m| 3 P99.865−P0.135 6 2 + v(M − T)2 . (2)
In developing the generalization CN p(u, v), Pearn and Chen [14] replaced the process meanµ by the process median, M (a more robust measure for process central tendency), and the process standard deviationσ by (P99.865− P0.135)/6 calculated from the distribution percentiles in the definition of the original indices
Cp(u, v). If the process follows the normal distribution, then clearly CN p(u, v) reduces to the basic indices Cp(u, v). Pearn and Chen [14] investigated the generalization CN p(u, v), and considered a sample percentile method to calculate CN p(u, v). But, their investigation was restricted to processes with symmet-ric tolerances, and therefore may not be applied to processes with asymmetric tolerances.
2 The subwoofer speaker system
In this section we present a case on investigating the free air resonance, Fo, of the speaker driver used in home-theater sub-woofer speaker systems. The case we investigate is taken from a speaker driver supplier in Taiwan, which manufactures various types of speaker drivers including 3-inch tweeters, 3-inch and 4-inch full-ranges, 5-inch mid-ranges, 6.5-inch woofers, 8-inch, 10-inch, 12-inch, 15-inch, and 18-inch subwoofers. A standard woofer or subwoofer driver, depicted in Fig. 1, consists of the following components: edge, cone, dust cap, spider (also called a damper), voice coil, lead wire, frame, magnet, front plate, and back plate (also called a T-york). The edge (on the top) and the spider (on the bottom) are glued onto the frame to hold the cone during the piston movements, and the dust cap is glued onto the center top of the cone, to cover the voice coil, which decouples the noise from the musical signals.
One characteristic that critically determines the bass per-formance, musical image, clarity and cleanness of the sound, transparence, and compliance (excursion movement) of the mid-range, full mid-range, woofer, and subwoofer driver units, is the free-air resonance, known as Fo. Some key factors determin-ing the Fovalues include the hardness, thickness, and weight of the damper, the hardness, thickness, and weight of the edge and
Fig. 1. A subwoofer driver
Fig. 2. A subwoofer system
Fig. 3. LMS drawing for impedance and phase curves
cone. Some typical ranges of Fo are 60–15 000 Hz for the full ranges, 500–5000 Hz for the mid-ranges, 1000–18 000 Hz for hard-dome tweeters, and 1500–20 000 Hz for soft-dome tweet-ers. A typical subwoofer system with a front reflex tube, as depicted in Fig. 2, plays an important role in most home-theater applications.
A standard home-theater system contains two main speaker systems (normally with tweeters, midranges, and woofers) for the front channels, one speaker (normally with full ranges, or tweeters and midranges) for the center channel, one sub-woofer system, and two speakers for the (rear) background chan-nels. One particular home-theater application we investigated uses a subwoofer system with a 10-inch subwoofer driver. The upper and lower specification limits, the USL and LSL, for this subwoofer driver are set at 35 Hz and 20 Hz, respectively, and the target value is set at T = 30 Hz. The LMS (Learn-ing management system) is the computer software commonly used in speaker driver manufacturing industry for measuring the impedance, phase curve and other characteristics of the drivers (shown in Fig. 3). This is the case where the manufacturing tol-erance is asymmetric.
3 Capability measure for asymmetric tolerances
For asymmetric tolerances, the work of several researchers in-cluding Kane [1], and Kushler and Hurley [19], simply shift thespecification limits to make them symmetric around the target value T. These methods obviously can severely understate or over-state process capability and thus reflect process performance inac-curately. To overcome the problem, Vännman [20] considered an alternative method by adding a new term,|µ − T|, in the numera-tor of the capability definitions, which has been defined as:
Cpa(u, v) = d− |µ − m| − u |µ − T| 3 σ2+ v(µ − T)2 , (3)
where u andv are non-negative parameters. The index Cpa(u, v) has the advantage of having its maximum whenµ = T, and it decreases asµ shifts away from T in either direction.
Recently, Pearn and Chen [15] developed a new method to generalize Cpk for asymmetric tolerance. Chen et al. [21], and Pearn et al. [22] have applied the method to indices Cpm, and Cpmk, respectively, for normal processes with asymmetric tol-erance. Pearn et al. [23] also applied the method to Clements’ formula for calculating capability indices for non-normal pro-cesses with asymmetric tolerance. The method takes into account the important property of the asymmetric loss function, which is shown to be superior to the other existing methods, and is an ap-propriate method for asymmetric tolerances. We note, however, that Clements’ formula requires the underlying process distri-bution to be of the Pearson family type, and the computation requires checking extensive percentile tables of the Pearson fam-ily distributions. In the following, we apply the method shown in [15] to extend the flexible indices CN p(u, v), which we refer to as CN p(u, v), to cover arbitrary distributions with asymmetric tolerances. The extension CN p(u, v) can be expressed as:
CN p(u, v) = d ∗− u A∗ 3 P99.865−P0.135 6 2 + vA2 , (4)
where A= max{d(M − T)/du, d(T − M)/dl}, A∗= max{d∗(M − T)/du, d∗(T − M)/dl}, du= USL − T, dl = T − LSL, d∗= min{du, dl}. We note that if T = m (tolerance is symmetric), then A= A∗= |M − T|, and the extension CN p(u, v) reduces to the original index CN p(u, v). By setting u and v equal to 0 or 1 we obtain CN p(0, 0) = CN p, CN p(1, 0) = CN pk, CN p(0, 1) =
CN pm, and CN p(1, 1) = CN pmk, which can be expressed as the
M CN p CN pk CN pm CN pmk M CN p CN pk CN pm CN pmk 100 1.482 0.000 0.220 0.000 110 1.482 0.741 0.426 0.070 130 1.482 0.000 0.220 0.000 125 1.482 0.741 0.426 0.070 102 1.482 0.148 0.244 0.005 112 1.482 0.889 0.520 0.117 129 1.482 0.148 0.244 0.005 124 1.482 0.889 0.520 0.117 104 1.482 0.296 0.273 0.013 114 1.482 1.037 0.663 0.198 128 1.482 0.296 0.273 0.013 123 1.482 1.037 0.663 0.198 106 1.482 0.444 0.310 0.024 116 1.482 1.185 0.889 0.331 127 1.482 0.444 0.310 0.024 122 1.482 1.185 0.889 0.331 108 1.482 0.593 0.359 0.042 118 1.482 1.333 1.233 0.516 126 1.482 0.593 0.359 0.042 121 1.482 1.333 1.233 0.516
Table 1. CN p(u, v) for processes A, B,
satis-fying(MA− T)/du= (T − MB)/d following: CN p= d ∗ 3 P99.865−P0.135 6 2, CN pk= d ∗− A∗ 3 P99.865−P0.135 6 2, CN pm= d ∗ 3 P99.865−P0.135 6 2 + A2 , CN pmk= d ∗− A∗ 3 P99.865−P0.135 6 2 + A2 . (5)
The merit of the extension is that it takes into considera-tion the asymmetric loss funcconsidera-tion that other existing methods have not implemented. For processes with asymmetric toler-ances, the corresponding loss function is also asymmetric to the target value T. Consider the popular quadratic loss function de-fined as L(x) = [(T − x)/(T −LSL)]2, for LSL< x ≤ T, L(x) = [(x − T)/(USL − T)]2, for T≤ x < USL, and L(x) = 1, other-wise. Then, for x1= (T + LSL)/2, and x2= (T + USL)/2, the corresponding loss can be calculated as L(x1) = L(x2) = 1/4. Obviously, x1 and x2 have the same departure ratio k= (T −
x1)/dl= (x2− T)/du= 1/2 (equal departure relative to the tol-erance). Thus, a desired property for a capability index with asymmetric tolerances is that the capability measures for shifted processes with equal departure ratios are the same. While we do not employ the quadratic loss function, the extension penalizes processes with equal departure ratios equally.
Justification of the extension
To justify that the extension indeed possesses the important property of the asymmetric loss function, we consider the follow-ing example with asymmetric tolerance (LSL, T, USL) = (100, 120, 130) and fixed process variations P99.865− P0.135= 0.9d,
P99.865− M = 0.55d, and M − P0.135= 0.35d, where 100 M130. Table 1 displays the capability measures of those pro-cesses using CN p(u, v). The proposed extensions, CN p(u, v),
obtain their maximal values at the target value T= 120. It is easy to verify that when M= T, we obtain that A = A∗= 0, and the equations defined in Eq. 4 are indeed maximized. Table 1 displays the index values of the extension CN p(u, v) obtained for processes with equal departure ratios satisfying the property (MA− T)/du= (T − MB)/dl). For example, con-sider processes A and B with MA= 124 and MB= 112. It is easy to verify that (MA− T)/du= (124 − 120)/10 = 2/5, and (T − MB)/dl= (120 − 112)/20 = 2/5, thus satisfying (MA− T)/du = (T − MB)/dl. Checking Table 1 the extensions give same values for A and B.
4 Performance comparisons
In the following, we show that the proposed extension CN p(u, v) outperforms the original index CN p(u, v) in detecting process shifting. We consider the following example with an on-target process A, and three shifted processes A1, A2and A3, where the manufacturing tolerance (LSL, T, USL) is set to (100, 120, 130). Table 2 displays the characteristics of the four processes A, A1,
A2, A3 the index values of CN p, CN pk, CN pm, CN pmk, and the extensions CN p, CN pk, CN pm, and CN pmk.
We note that the index CN pm detects process shifts of A1,
A2and A3. But, CN pmfails to differentiate the high quality pro-cess A2(with nearly 100% process yield) from the low quality process A3(with only 50% process yield), as CN pm= 0.140 for both A2, and A3. Therefore, we consider CN pm inaccurate. On the other hand, the extensions detect the shifts of A1, A2, A3, and differentiate process A2 from process A3 by giving larger values to A2 and smaller values to A3 (except for CN p which never takes into account the process median and the target value hence provides no sensitivity to process departure at all). We also note that for the extensions CN pk, CN pm, and CN pmk, the on-target process A receives larger index values than the other three off-target processes A1, A2, and A3. But, for the original in-dices CN pkthe off-target process A1receives larger index values than the on target process A, and the capability measure is con-sidered inaccurate. Since the generalization is proposed to
han-Table 2. Characteristics of processes A, A1, A2, A3, the corresponding in-dex values of the original indices and extensions
A A1 A2 A3 M 120 119 110 130 P99.865 130 129 120 140 P0.135 115 114 105 125 CN p 2.000 2.000 2.000 2.000 CN pk 1.333 1.467 1.333 0.000 CN pm 2.000 1.765 0.140 0.140 CN pmk 1.333 1.294 0.093 0.000 CN p 1.333 1.333 1.333 1.333 CN pk 1.333 1.267 0.667 0.000 CN pm 1.333 1.240 0.157 0.043 CN pmk 1.333 1.178 0.078 0.000
dle non-normal processes with asymmetric tolerances, it has to deal with a large number (theoretically an infinite number) of arbitrary shapes of distributions, it can only reflect process qual-ity approximately (often conservatively), and may not be very accurate.
5 Percentile estimator of C
Np(u, v)
Pearn and Chen [14] considered a sample percentile estimator to calculate the index CN p(u, v). The estimator essentially applies the sample percentile along with interpolation for calculating the sample percentiles, P99.865, P0.135, and the median M. The esti-mator can be expressed as the following:
ˆCN p(u, v) = d − u ˆM− m 3 ˆP 99.865− ˆP0.135 6 2 + vMˆ− T2 , ˆP99.865= X([R1])+ {R1− [R1]} × X([R1]+1)− X([R1]), ˆP0.135= X([R2])+ {R2− [R2]} × X([R2]+1)− X([R2]), ˆ M= X([R3])+ {R3− [R3]} × X([R3]+1)− X([R3]), R1= 99.865n + 0.135 100 , R2= 0.135n + 99.865 100 , R3= n+ 1 2 . (6)
In this setting, the notation[R] is defined as the greatest in-teger less than or equal to the number R, and x(i) is defined as the ith order statistic. The sample percentile estimator for the extension CN p(u, v), therefore, can be easily obtained as the following. We note that the percentile formula developed in the following requires no assumption on the underlying pro-cess distributions to be of the Pearson types, and the computa-tion does not require any tables. Hence, the sample percentile method is more general and convenient to use than Clements’ formula: ˆC N p(u, v) = d∗− u ˆA∗ 3 ˆP 99.865− ˆP0.135 6 2 + v ˆA2 , ˆA∗= max d∗( ˆM− T) du , d∗(T − ˆM) dl , ˆA = max d( ˆM− T) du , d(T − ˆM) dl . (7)
An S-PLUS computer program for calculating the percentile formula was developed and is listed in the Appendix. The S-PLUS program reads the sample data as an input, then out-puts with the estimated values of the indices. Since the per-centile method involves a complicated function of linear com-binations of the order statistics and, given that the underlying process distribution is unknown, the problem of finding an ex-act distribution is analytically intrex-actable. Approximation
ap-proaches or bootstrap resampling methods (an effective simu-lation technique for non-normal distributions) may be applied to establish the lower confidence bound for capability testing purposes.
6 Distribution plot of the percentile estimator
In the following, we apply a simulation approach to investigate the distribution of the sample percentile estimator. We set the asymmetric specification limits to (LSL, T, USL) = (8, 18, 23), and the underlying process distribution to:
1. Normal distribution N(17,1), with probability density func-tion f(x) =(√2π)−1e−(x−17)2/2, for−∞ < x < ∞, 2. Uniform distribution U(14, 20), with probability density
function f(x) = 1/6, for 14 < x < 20,
3. Weibull distribution W(2, 2) shifted for 17 units, with proba-bility density function f(x) = 4xe−2x2, for x> 0,
4. Gamma distribution G(289, 17), with probability density function f(x) = 17289x289−1e−17x/Γ(289), for x > 0, 5. Beta distribution B(17, 1), with probability density function
f(x) = [Γ(18)/Γ(17)] x16, for 0< x < 1,
6. Lognormal distribution LN(0.5, 1) shifted for 17 units, with probability density function of LN(0.5, 1) as f(x) = (√2π)−1x−1e−(ln x−0.5)2/2, for x> 0,
7. Chi-square distribution with degrees of freedom two shifted for 17 units, where the probability density function ofχ22is
f(x) = e−x/2/2, for x > 0, and
8. t distribution, shifted 18 units and with eight degrees of freedom, where the probability density function of t8 is,
f(x) = [Γ(9/2)/Γ(4)] (√8π)−1(1 + x2/8)−9/2 for −∞ <
Fig. 4. Distribution plot of CN pmkfor normal distribution N(17,1), with n= 50, 100, 250, 500, 1000 (bottom to top)
x< ∞. For each distribution, we randomly generate N = 15, 000 samples of sizes n = 50, 100, 250, 500, 1000, and then calculate the estimated capability index CN pmk.
Figures 4–11 plot the distribution of CN pmk for the eight process distributions, normal distribution N(17, 1), uniform dis-tribution U(14, 20), Weibull disdis-tribution W(2, 2) shifted for 17 units, gamma distribution G(289, 17), beta distribution B(17, 1),
Fig. 5. Distribution plot of CN pmkfor uniform distribution U(14,20), with n= 50, 100, 250, 500, 1000 (bottom to top)
Fig. 6. Distribution plot of CN pmkfor Weibull distribution W(2,2), with n= 50, 100, 250, 500, 1000 (bottom to top)
Fig. 7. Distribution plot of CN pmkfor gamma distribution G(289,17), with
n= 50, 100, 250, 500, 1000 (bottom to top)
Fig. 8. Distribution plot of CN pmkfor beta distribution B(17,1), with n= 50, 100, 250, 500, 1000 (bottom to top)
lognormal distribution LN(0.5, 1) shifted for 17 units, chi-square distribution with degrees of freedom two shifted for 17 units, and
t8distribution with degrees of freedom eight shifted for 18 units, respectively. For moderate and large sample size n, the distribu-tions of the estimated capability index all appear to be normal. Therefore, for processes where large sample data may be col-lected (product items may be inspected by automatic controlled machines), normal approximations may be used for capability testing.
Fig. 9. Distribution plot of CN pmkfor lognormal distribution LN(0.5,1), n= 50, 100, 250, 500, 1000 (bottom to top)
Fig. 10. Distribution plot of CN pmkfor chi-square distribution with d f= 2,
n= 50, 100, 250, 500, 1000 (bottom to top)
7 Bootstrap for manufacturing capability testing
In statistical analysis, the researcher is usually interested in ob-taining not only a point estimate of a statistic but also an estimate of the variation of this point estimate, and a confidence interval for the true value of the parameter. For example, a researcher may calculate not only a sample mean but also the standard error of the mean and a confidence interval for the mean.Tra-Fig. 11. Distribution plot of CN pmkfor t distribution with d f= 8, and n = 50, 100, 250, 500, 1000 (bottom to top)
ditionally, researchers have relied on the central limit theorem and normal approximations to obtain standard errors and confi-dence intervals. These techniques are valid only if the statistic, or some known transformation of it, is asymptotically normally distributed. Hence, if the normality assumption does not hold, then the traditional methods should not be used to obtain confi-dence intervals. A major motivation for the traditional reliance on normal-theory methods has been computational tractabil-ity. Now, with the availability of modern computing power, re-searchers need no longer rely on asymptotic theory to estimate the distribution of a statistic. Instead, they may use resampling methods, which return inferential results for either normal or non-normal distributions. This is particularly useful in our case, for small or moderate size of non-normal data.
Bootstrapping, as presented by Efron [24, 25] is a data based estimation method for statistical inference that is effective, non-parametric but also computationally intensive. In this method,
B new samples, each of the same size as the observed data, are
drawn with replacement from the available sample. The statis-tic of interest is then calculated for each new set of resampled data, yielding a bootstrap distribution for the statistic. The funda-mental assumption of bootstrapping is that the observed data are representative of the underlying population. By resampling ob-servations from the observed data, the process of sampling obser-vations from the population is mimicked. The merit, in its sim-plest form, is that the nonparametric bootstrap does not rely on any distributional assumptions about the underlying population. Efron and Tibshirani [26] developed three types of bootstrap confidence interval. Those include the standard bootstrap confi-dence interval (SB), the percentile bootstrap conficonfi-dence interval (PB), and the biased corrected percentile bootstrap confidence interval (BCPB). Franklin and Wasserman [27] investigated the lower confidence bounds for the capability indices, Cp, Cpkand
Cpm using the three bootstrap methods. Some simulations were conducted and a comparison was made among the results from the three bootstrap methods against the known values of the ca-pability indices. Their simulation results indicate that, for normal processes, the bootstrap confidence limits perform equally well. And for non-normal processes the SB method performed signifi-cantly better than other methods. Furthermore, the proportion of coverage of the SB method limits and confidence intervals are closer to the desired value, whatever the underlying process dis-tribution, and the lengths of the SB intervals are much shorter than others. Therefore, the SB method is more useful in deter-mining the value of the process index. In the following, we will describe how to construct the bootstrap lower confidence bound by means of the standard bootstrap method.
The bootstrap lower confidence bound
In our application, B= 10000 bootstrap resamples (each of the same size as the available data) are drawn randomly from the ori-ginal sample. A 100(1 − α)% bootstrap lower confidence limit of the SB method for CN p(u, v) is constructed. If the calculated bootstrap lower confidence limit is found to be smaller than the specified index value, we would judge the process as incapable. Quality improvement activities will be initiated. Otherwise, the process is considered to be capable. From the 10000 bootstrap estimates, the sample average can be calculated as:
ˆC N p(u, v) = 1 10000 10000 i=1 ˆC N p(i)(u, v),
and the sample standard deviation of the ˆCN p(u, v) can be
ob-tained as: SˆC N p(u,v) = 1 10000− 1 10000 i=1 ˆC
N p(i)(u, v) − ˆCN p(i)(u, v) 2
.
where ˆCN p(i)(u, v) is the ith bootstrap estimates and by
set-ting u,v = 0, 1, we obtain ˆCN p(0, 0) = ˆCN p, ˆCN p(1, 0) = ˆCN pk, ˆC
N p(0, 1) = ˆCN pm, and ˆCN p(1, 1) = ˆCN pmk. Thus, the 100(1 − α)% SB lower confidence bound (LCB) for C
N p(u, v) can be constructed as:
LCB= ˆCN p(u, v) − Zα× SˆC
N p(u,v). (8)
In order to make use of the methodology more convenient and accelerate the computation time, an integrated S-PLUS com-puter program has been developed (see Appendix). The practi-tioners only need to input the manufacturing specifications, USL, LSL, target value T, a specified quality level of CN p(u, v) and a collected sample data of size n. The estimated value ˆCN p(u, v)
and the SB lower confidence bound of CN p(u, v) may be eas-ily obtained. Thus, whether or not the process is capable may be determined.
28 28 26 32 28 27 29 25 25 28 28 26 27 31 27 26 32 29 27 26 26 26 30 25 27 29 27 31 30 30 27 31 27 25 30 28 26 27 31 27 34 27 28 32 33 25 29 28 28 29 29 25 29 29 30 31 28 28 30 28 26 28 28 25 29 28 29 31 28 28 27 30 27 25 30 25 29 26 26 27 33 29 26 31 32 27 26 29 26 28 30 26 29 28 29 25 30 27 28 32
Table 3. The 100 sample observations
The collected data (a total of 100 observations) are displayed in Table 3. Proceeding with the calculations by running the inte-grated S-PLUS program with 95% of confidence, we obtain the values of the sample percentile estimators and the corresponding bootstrap lower confidence bound (LCB) as:
ˆC N p= 1.35, with LCB= 1.25, ˆC N pk= 1.20, with LCB= 1.10, ˆC N pm= 1.18, with LCB= 1.08, ˆC N pmk= 1.05, with LCB = 0.94.
We note that the estimated index values for all the four ex-tensions are greater than 1.00. In fact, all 100 observations fall within the specification interval (LSL, USL) with one observa-tion (34) fairly close to the upper specificaobserva-tion limit (35). Check-ing the correspondCheck-ing lower confidence bounds, 1.25, 1.10, 1.08, and 0.94, we may only conclude that the process is marginally capable, with 95% of confidence.
8 Concluding remarks
Process capability indices are practical and powerful tools for measuring process performance. Many quality engineers and statisticians have proposed methodologies for assessing prod-uct/process quality but limited their proposals to the inspection of data that are normally distributed where the manufacturing tolerances are symmetric. In this paper, we considered an exten-sion of the existing capability index, called CN p(u, v), to handle non-normal processes with asymmetric tolerances. The exten-sion takes into account the important property of the asymmetric loss function, which is shown to be more sensitive to process shift and more accurate than the existing ones in measuring pro-cess capability, and hence provides better quality assurance. We also presented a percentile estimator to calculate the asymmetric process capability with non-normal data.
Since the proposed percentile method involves a complicated function of linear combinations on the order statistics and, given that the underlying process distribution is unknown, the prob-lem of finding an exact distribution is analytically intractable. We applied the nonparametric bootstrap method to establish the lower confidence bound for capability testing purposes. We also developed an integrated S-PLUS computer program to calculate the percentile estimator and the corresponding lower confidence bound. The practitioners only need to input the manufacturing specifications, USL, LSL, target value T, a specified quality level
of CN p(u, v) and the collected sample data. The estimated value ˆC
N p(u, v) and the SB lower confidence bound of CN p(u, v) may be applied to determine whether the process is capable or not.
Appendix
Integrated S-PLUS computer program
#---# # Input the specification limits, USL, LSL, # and the target value T
#---# USL<-35
LSL<-20 Target<-29
#---# # Store the input of the original sample data # of size n =100 #---# data0<-c( 28,28,26,32,28,27,29,25,25,28, 28,26,27,31,27,26,32, 29,27,26,26,26,30,25, 27,29,27,31,30,30,27, 31,27,25, 30,28,26,27, 31,27,34,27,28,32,33,25,29,28, 28,29,29,25, 29,29,30,31,28,28,30,28,26,28,28,25,29,28, 29,31, 28,28,27,30,27,25,30,25,29,26,26,27, 33,29, 26,31,32, 27,26,29,26,28,30,26,29,28, 29,25,30,27,28, 32) #---# # The function to calculate the estimated # C’’_Np (u, v) based on the given data
#---# CNp.hat<-function(u,v,data){ data.order<-sort(c(data)) n<-length(data) R1<-(99.865*n+0.135)/100 R2<-(0.135*n+99.865)/100 R3<-(n+1)/2 P99.865hat<-data.order[floor(R1)] +(R1-floor(R1))*(data.order[floor(R1+1)] -data.order[floor(R1)]) P0.135hat<-data.order[floor(R2)] +(R2-floor(R2))*(data.order[floor(R2+1)] -data.order[floor(R2)])
Mhat<-data.order[floor(R3)] +(R3-floor(R3))*(data.order[floor(R3+1)] -data.order[floor(R3)]) du<-USL-Target dl<-Target-LSL d<-(USL-LSL)/2 dstar<-min(du,dl) Ahat.star<-max(dstar*(Mhat-Target) /du,dstar*(Target-Mhat)/dl) Ahat<-max(d*(Mhat-Target)/du, d*(Target-Mhat)/dl) CNp.return<-(dstar-u*Ahat.star) /(3*sqrt(((P99.865hat-P0.135hat)/6)^2 +v*Ahat^2)) } #---# # Calculate the estimated C’’_Np (u, v)
# based on the original data
#---# CNp.est<-CNp.hat(0,0,data0) CNpk.est<-CNp.hat(1,0,data0) CNpm.est<-CNp.hat(0,1,data0) CNpmk.est<-CNp.hat(1,1,data0) #---# # Generate 10000 bootstrap resamples from # the original sample of n = 100
#---# m<-10000 CNp.boot<-rep(0,m) CNpk.boot<-rep(0,m) CNpm.boot<-rep(0,m) CNpmk.boot<-rep(0,m) for (i in 1:m){ dataB<-sample(data0,100,replace=T) CNp.boot[i]<-CNp.hat(0,0,dataB) CNpk.boot[i]<-CNp.hat(1,0,dataB) CNpm.boot[i]<-CNp.hat(0,1,dataB) CNpmk.boot[i]<-CNp.hat(1,1,dataB) } #---# # Calculate the lower confidence bound based # on the bootstrap resampling
#---# CNp.boot.95lowerbound<-mean(CNp.boot) -qnorm(0.95)*var(CNp.boot)^0.5 CNpk.boot.95lowerbound<-mean(CNpk.boot) -qnorm(0.95)*var(CNpk.boot)^0.5 CNpm.boot.95lowerbound<-mean(CNpm.boot) -qnorm(0.95)*var(CNpm.boot)^0.5 CNpmk.boot.95lowerbound<-mean(CNpmk.boot) -qnorm(0.95)*var(CNpmk.boot)^0.5 #---#
The output of sample program based on the above settings:
1. The estimated CN p(u, v) based on the original data are: > CNp.est = 1.353432
> CNpk.est = 1.20305 > CNpm.est = 1.178897 > CNpmk.est = 1.047908
2. The bootstrap lower confidence bound of CN p(u, v) are: > CNp.boot.95lowerbound = 1.250352
> CNpk.boot.95lowerbound = 1.104946 > CNpm.boot.95lowerbound = 1.084890 > CNpmk.boot.95lowerbound = 0.9366828
References
1. Kane VE (1986) Process capability indices. J Qual Technol 18(1): 41–52
2. Chan LK, Cheng SW, Spiring FA (1988) A new measure of process capability Cpm. J Qual Technol 20(3):162–175
3. Pearn WL, Kotz S, Johnson NL (1992) Distributional and inferen-tial properties of process capability indices. J Qual Technol 24(4): 216–231
4. Vännman K (1995) A unified approach to capability indices. Stat Sin 5:805–820
5. Choi BC, Owen DB (1989) A study of a new capability index. Commun Stat Theory Methods 19(4):1231–1245
6. Pearn WL, Lin GH, Chen KS (1998) Distributional and inferential properties of the process accuracy and process precision indices. Com-mun Stat Theory Methods 27(4):985–1000
7. Kotz S, Pearn WL, Johnson NL (1993) Some process capability indices are more reliable than one might think. J R Stat Soc C 42(1):55–62 8. Hoskins J, Stuart B, Taylor J (1988) A Motorola commitment: a six
sigma mandate. The Motorola guide to statistical process control for continuous improvement towards six-sigma quality
9. Rado LG (1989) Enhance product development by using capability in-dexes. Qual Progress 22(4):38–41
10. Hubele NF, Montgomery DC, Chih WH (1991) An application of sta-tistical process control in jet-turbine engine component manufacturing. Qual Eng 4(2):197–210
11. Noguera J, Nielsen T (1992) Implementing six sigma for interconnect technology. ASQC Quality Congress Transactions, Nashville, TN, pp 538–544
12. Pearn WL, Kotz S (1994) Application of Clements’ method for cal-culating second and third generation process capability indices for non-normal Pearsonian populations. Qual Eng 7(1):139–145
13. Lyth DM, Rabiej RJ (1995) Critical variables in wood manufacturing’s process capability: species, structure, and moisture content. Qual Eng 8(2):275–281
14. Pearn WL, Chen KS (1997) Capability indices for non-normal dis-tributions with an application in electrolytic capacitor manufacturing. Microelectron Reliab 37(12):1853–1858
15. Pearn WL, Chen KS (1998) New generalizations of the process capa-bility index Cpk. J Appl Stat 25(6):801–810
16. Pearn WL, Chang CS (1998) An implementation of the precision index for contaminated processes. Qual Eng 11(1):101–110
17. Spiring FA (1989) An application of Cpm to the tool-wear problem.
ASQC Quality Congress Transactions, Toronto, ON, pp 123–128 18. McCoy PF (1991) Using performance indexes to monitor production
processes. Qual Progress February:49–55
19. Kushler RH, Hurley P (1992) Confidence bounds for capability indices. J Qual Technol 24(4):188–195
20. Vännman K (1997) A general class of capability indices in the case of asymmetric tolerances. Commun Stat Theory Methods 26(8):2049–2072
21. Chen KS, Pearn WL, Lin PC (1999) A new generalization of the ca-pability index Cpmfor asymmetric tolerances. Int J Reliab Qual Safety
Eng 6(4):383–398
22. Pearn WL, Chen KS, Lin PC (1999) On the generalizations of the ca-pability index Cpmkfor asymmetric tolerances. Far East J Theor Stat
3(1):47–66
23. Pearn WL, Chen KS, Lin GH (1999) A generalization of Clements‘ method for non-normal Pearsonian processes with asymmetric toler-ances. Int J Qual Reliab Manage 16(5):507–521
24. Efron B (1979) Bootstrap methods: another look at the Jackknife. Ann Stat 7:1–26
25. Efron B (1982) The Jackknife, the bootstrap and other resampling plans. Society for Industrial and Applied Mathematics, Philadelphia, PA 26. Efron B, Tibshirani RJ (1986) Bootstrap methods for standard errors, confidence interval, and other measures of statistical accuracy. Stat Sci 1:54–77
27. Franklin LA, Wasserman GS (1992) Bootstrap lower confidence limits for capability indices. J Qual Technol 24(4):196–210
