A Simulation Comparison Study for Estimating the Process Capability Index C pm with Asymmetric Tolerances

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1 Available online at ijims.ms.tku.edu.tw/list.asp International Journal of Information and Management Sciences 20 (2009), A Simulation Comparison Study for Estimating the Process Capability Index C pm with Asymmetric Tolerances Shu-Fei Wu Department of Statistics Tamkang University R.O.C. Abstract Process capability indices had been widely used to evaluate the process performance. The process capability index C pm proposed by Chan et al. [2] does take into account of the proximity of the process mean to the target value T for asymmetric tolerance. For point estimation of this index, a Jackknife method is presented to reduce bias. Five interval estimation methods for obtaining approximate confidence intervals are presented and compared. One is based on the chi-squared approximation to the distribution of the natural estimator of C pm given in Boyles [1], three are based on the bootstrap including standard bootstrap (SB), percentile bootstrap (PB) and bisaed-corrected percentile bootstrap (BCPB) and the last one is based on the Jackknife technique. A simulation comparison study of the performance of five methods is done under normal process environment and the results show that the Jackknife interval outperforms the other four methods. Boyles s [1] confidence interval and the SB confidence interval are more reliable than the PB and BCPB methods. At last, one real life example is used to demonstrate the use of the confidence interval to decide if the process is capable. Keywords: Process Capability Index, Bootstrap Method, Jackknife Method. 1. Introduction Process capability indices had been widely used to evaluate the capability of the manufacturing process to reach the preset quality requirements (See, for examples, Montgomery [9], Kane [7] and Ryan [11]. Three basic process capability indices are defined as follows (See Kane [7] and Pearn et al. [10]): C p = USL LSL, 6σ Received October 2007; Revised March 2008; Accepted June 2008.

2 244 International Journal of Information and Management Sciences, Vol. 20, No. 2, June, 2009 C pk = min(c pl,c pu ), USL LSL C pm = 6 σ 2 + (µ T) = d 2 3 σ 2 + (µ T) = d 2 3τ, where C pl = USL µ 3σ, C pu = µ LSL 3σ, USL and LSL are the upper and lower specification limits preset by the process engineers, µ is the process mean, σ is the process standard deviation, m = (USL + LSL)/2 is the midpoint of specification limits, d = (USL LSL)/2 is the half length of the specification interval and τ 2 = σ 2 + (µ T) 2. The index C p only measures the process variation without considering the process centering. The index C pk takes the process variation and process centering into account, but not the proximity of the process mean to the target value. The index C pm does take into account of the proximity of the process mean to the preset target value. For asymmetric tolerance (T m ), Chan et al. [2] developed the process capability index Cpm which is a genelization index of C pm and is defined as: C pm = min(d L,D U ) 3 σ 2 + (µ T) 2 = D σ 2 + (µ T) 2 = D τ = 3D d C pm, where D L = T LSL, D U = USL T and D = min(d L,D U )/3. Note that Cpm is reduced to C pm for symmetric tolerance. Clearly Cpm will not only continue to take the proximity of the target value into consideration as C pm does, but also taking into account the asymmetric specification limits. Please see the following figure (Kane [7]) for the reaction of Cpm to the departures from the non-central target value, where USL = 18, LSL = 10, T = 16 and the normal mean is µ = 13(1)17. The expected yield associated with a given value of Cpm USL µ is 1 yield = 1 φ( ) λ 2 (µ T) 2 LSL µ D +φ( ), where λ = λ 2 (µ T) 2 C. It revealed that the larger value of pm C pm results in a smaller expected proportion of nonconforming. 2. Point Estimation and Interval Estimation of C pm 2.1. Point Estimation of C pm Let X 1,...,X n denote a random sample from a normal distributed process with mean µ and standard deviation σ, denoted by N(µ,σ 2 ). The sample mean and the maximum likelihood estimate of the variance are n i=1 X = X n i and S 2 i=1 = (X i X) 2. n n Replacing the parameters µ and σ 2 in the index Cpm by X and S 2, then we have the natural estimator of Cpm given by: where ˆτ 2 = S 2 + ( X T) 2 = P n i=1 (X i T) 2 n. Ĉ pm = D ˆτ, (1)

3 A Simulation Comparison Study for Estimating the Process Capability Index C pm 245 Figure 1. Boyles [1] showed that ˆτ 2 is an unbiased estimator of τ 2 and has smaller variance than the biased estimator τ 2 = n n n 1 S2 + ( X T) 2 i=1 = (X i T) 2. n 1

4 246 International Journal of Information and Management Sciences, Vol. 20, No. 2, June, 2009 Ignoring the fact that the esimator is a random variable with distribution, many process engineers is simply comparing the calculated value of Cpm with a preassigned minimum value to determine whether if the process is capable by a given sample. For point estimation, unbiasedness is a good feature of an estimator should have. Therefore, we proposed a Jackknife estimator of the index. Jackknife method was originally introduced by Quenouille [13] in order to reduce the bias of an estimator of a serial correlation coefficient. We employ his method as follows: Let ˆθ = Ĉ pm denote the natural estimator of θ = Cpm based on the complete sample. Eliminating the first observation, we make use of the remaining n 1 observations to calculate the first natural estimator of Cpm and denoted by ˆθ (1). Similarly, eliminating the second observation, we can have the second natural estimator of Cpm and denoted by ˆθ (2) based on the remaining n 1 observations. Repeat the same procedure, we can have n natural estimators denoted by ˆθ (1), ˆθ (2),..., ˆθ (n) based on the subsample of size n 1. The ith pseudovalue is defined as ˆθ i = nˆθ (n 1)ˆθ (i), i = 1,2,...,n. The Quenoulli s estimator is the mean of the ˆθ i s, and benoted by ˆ θ. The Jackknife estimator of standard error is Sˆ θ = estimator is defined as Pn i=1 (ˆθ i ˆ θ) 2 n(n 1). For point estimation, the MSE of an MSE(ˆθ) = Var (ˆθ) + [Bias(ˆθ)] 2, where Bias(ˆθ) = E(ˆθ θ). A 1000 iteration runs of Monte-Carlo simulation were done to obtain the coverage probabilities of five methods by setting USL = 60, LSL = 40 and m=50. Following the same structure of Franklin and Wasserman [6], the random samples of size n = 10(10)40(20)60 are generated from a normal distribution with mean and variance given by (µ,σ 2 )=(50,4), (50,9), (52,4), (52,9). Two target values T =51 and T =55 are considered for asymmetric tolerance and the corresponding true index values are Cpm =(1.342,.949,1.342,.949) and Cpm =(.309,.286,.462,.393) for four different combinations of means and variances given by (50,4), (50,9), (52,4), (52,9). The average bias and MSE by using estimator defined in equation (1) and Jackknife estimator are listed in Table 1. From Table 1, we can see that the bias and MSE for both methods are smaller for more asymmetric tolerance (T = 55). The Jackknife estimator has almost the same MSE as the natural estimator given in equation (1) and the Jackknife estimator can reduce the bias a lot for small, moderate or large sample size. More over, the bias and MSE of both methods are decreasing and the discrepancy of bias between two methods is decreasing when the sample size is increasing. Overall speaking, Jackknife estimator has better performance than the other method especially for cases Interval Estimation of C pm Considering the sampling error, it is better to use the interval estimation or hypothesis testing to reflect the uncertainty about the true index value. For that reason, Chou et al. [3] provided confidence limits for the index C pk for normal process environment.

5 A Simulation Comparison Study for Estimating the Process Capability Index C pm 247 Table 1. The Bias (upper entry) and the MSE (lower entry) of the estimator given in equation (1) and Jackknife method for Cpm under normal process. T = Eqn. (1) Jackknife Eqn. (1) Jackknife Eqn. (1) Jackknife Eqn. (1) Jackknife Eqn. (1) Jackknife (50,4) C pm= (50,9) Cpm = (52,4) Cpm = (52,9) Cpm = T = 55 (50,4) C pm= (50,9) Cpm = (52,4) Cpm = (52,9) C pm= Franklin and Wasserman [6] make used of the three Bootstrap confidence interval techniques introduced by Efron and Tibshirani [5] to construct the confidence intervals for C pk. Franklin and Wasserman [6] also offered three bootstrap lower confidence limits for C p,c pk and C pm. They compare the three bootstrap methods and a parametric method given in Boyles [1] and claimed that the bootstrap lower confidence limits performed as well as the parametric method. The advantage of bootstrap methods is nonparametric and free from assumptions of the distribution of X. In addition to the previous four method, we also proposed another nonparametric method called Jackknife method. In this paper, these five methods for the interval estimation of Cpm are presented and introduced in more detail as follows: (1) Boyles s method For normal process, Chan et al. [2] had shown that the pdf of C pm is f(x) = exp[ (n 1)C2 pm/x 2 + λ ] 2 (n 1)Cpm/x 2 2 ] (n/2)+j (λ j ) Γ( n 2 + (2 n/2 x) 1, where j)22j j! j=1 0 < x <. Since ˆτ 2 has a noncentral chi-squared distribution, Boyles [1] used the central chi-squared distribution to approximate the noncentral chi-squared distribution and showed that (C pm/ĉ pm) 2 is approximated χ 2 ν/ν, where χ 2 ν is a chi-squared distribution with ν degrees of freedom, where ν = n(1 + ξ2 ) ξ 2 and ξ = µ T σ.

6 248 International Journal of Information and Management Sciences, Vol. 20, No. 2, June, 2009 The natural estimator of ξ is ˆξ = X T S. Thus the (1 α)100% approximate confidence interval for Cpm is χ 2 ν (1 α/2) χ 2 ν (α/2) (Ĉ pm ν,ĉ pm ), ν where χ 2 ν (α/2) is the right tail α/2 percentile of a chi-squared distribution with ν degrees of freedom. (2) The Standard Bootstrap Confidence Interval of C pm (SB) The Bootstrap method was introduced by Efren [4]. Let X 1,...,X n be the original random sample from a process with distribution F. A Bootstrap sample is one of size n drawn ( with replacement )from the original sample and is denoted by X 1,...,X n. There are a total of n n such possible samples. Let B be the number of Bootstrap samples and B is taken to be 1000 throughout this paper. Let X (i) and S 2 (i) be the sample mean and sample variance based on the ith Bootstrap sample. First, calculate the natural estimator of C pm given by Ĉ pm (i) = D S 2 (i)+( X (i) T) 2 based on the ith Bootstrap sample, i = 1,...,B. Then calculate the sample average of the Bootstrap estimates Ĉ pm( ) = 1 B B i=1 Ĉ pm(i) and the sample standard deviation of 1 Bootstrap estimates SĈ = B pm B i=1 [Ĉ pm(i) Ĉ pm( )] 2. Then the (1 α)100% confidence interval for Cpm is (Ĉ pm ± Z α/2sĉ pm ), where Z α/2 is the right tail α/2 percentile of a standard normal random variable Z. If a 95% confidence interval is desired, then Z α/2 =1.96. If a 97.5% lower confidence interval of the index is desired, the lower confidence limit can be easily obtained by simply selecting the lower value of the two-sided confidence interval. (3) Percentile bootstrap confidence interval of C pm ( PB) Let Ĉ pm (1) Ĉ pm (2) Ĉ pm (B) be the sorted Bootstrap estimates. Then Ĉpm ([B α/2] + 1) and Ĉ pm ([B (1 α/2)] + 1) are the α/2 and (1 α/2) percentile points of the distribution of Ĉ pm(i), where [x] denotes the largest integer being less than or equal to x. The (1 α)100% approximate confidence interval for Cpm is given by (Ĉ pm([b α/2] + 1), Ĉ pm([b (1 α/2)] + 1). (4) Biased corrected percentile bootstrap confidence interval of Cpm (BCPB) Since the Bootstrap distribution may be a biased distribution, the third method was developed to correct for this potential bias. For example, if Ĉ pm is 1.63 and in the order values of Ĉ pm (i) we have Ĉ pm (412)=1.61 and Ĉ pm (423)=1.66, then p 0 = P(Ĉ pm 1.63) = 412/1000 =.412. Calculate Z 0 = φ 1 (p 0 ) = φ 1 (.412) =.222, where φ 1 is the inverse of the distribution function standard normal random variable Z. Then calculate P L = φ(2z 0 Z α/2 ) and P U = φ(2z 0 + Z α/2 ), where φ is the cdf of a standard

7 A Simulation Comparison Study for Estimating the Process Capability Index C pm 249 normal random variable Z. Then the (1 α)100% approximate confidence interval for C pm is given by [Ĉ pm ([P L B] + 1), Ĉ pm ([P U B] + 1)]. (5) Jackknife confidence interval of C pm : Tukey [12] suggested that the statistic ˆt = ˆ θ θ should be distributed approximately Sˆ θ as Student s t with n 1 degrees of freedom. Then the (1 α)100% approximate confidence interval for Cpm is given by [ˆ θ ± tα/2 (n 1)Sˆ θ], where t α/2 (n 1) is the right tail α/2 percentile of a Student s t distribution. For symmetric tolerance (T = m), the index C pm reduced to the index C pm. Therefore, all results for the index C pm are applicable for the index C pm. Under the same simulation set up for point estimation, we compare the performance of five methods based on their coverage probabilities and their simulation results are listed in Table 2-3. The frequency of coverage is a Binomial event with p =.95 and n = Thus a 95% confidence interval surrounding the expected coverage frequency.95 would have a bound of ±1.96 (.95)(.05)/1000 = ± Hence, one would be 95% confident that the true coverage percentage would have a proportion of coverage between (.9365,.9635). The frequencies of coverage falling into this interval are marked by an asterisk (*) in Tables 2-3. From Tables 2-3, the coverage probabilities are increasing and the average lengths are decreasing when the sample size n is increasing for most cases. Five methods have higher coverage percentages for the case of (µ,σ 2 )=(50,4) than the other three cases for any given T and for larger T value (more asymmetric tolerance) for any given combination of (µ,σ 2 ). The simulation results also showed that the Jackknife method always has the highest coverage probability and the highest rates of reaching the nominal confidence coefficient.95 among five methods. Therefore, the Jackknife method is recommended for used. The parametric Boyles s method is also better than the other three bootstrap methods. The performance of three Bootstrap methods based on the closest coverage rates to the nominal confidence coefficient are ranked as SB>BCPB>PB. For not normal process environment, the nonparametric method should be more suitable since the nonparametric methods do not need any distributional assumptions on the process. When C pm > 1, the process is capable, and conversely. Therefore, only the normal process with mean and variance given by (50,4) when T = 51 is capable since C pm exceeding 1.

8 250 International Journal of Information and Management Sciences, Vol. 20, No. 2, June, 2009 Table 2. The coverage probability of five confidence intervals for Cpm with T=51 under normal process. (µ, σ 2 )=(50,4) n = 10 n = 20 n = 30 n = 40 n = 60 Coverage Coverage Coverage Coverage Coverage Boyles * 0.939* 0.947* 0.949* SB * 0.93 PB BCPB Jackknife 0.941* * 0.954* (µ, σ 2 )=(50,9) Boyles * 0.941* * SB * * PB BCPB Jackknife 0.951* 0.947* 0.941* 0.946* 0.937* (µ, σ 2 )=(52,4) Boyles * 0.938* 0.945* 0.939* SB * PB BCPB Jackknife * * (µ, σ 2 )=(52,9) Boyles * 0.938* SB * 0.940* PB BCPB Jackknife 0.950* 0.951* 0.949* 0.940* 0.944* 3. Numerical Example The Example 5-1 in Montgomery [9] is used to demonstrate the construction of 90% and 95% confidence interval estimates and the 95% and 97.5% lower confidence limit of C pm and C pm. In that example, the inside diameter measurement data of the 125 Piston rings for an automotive engine produced by a forging process is recorded in Table 5-1 of Montgomery [9]. The sample mean and the sample variance are obtained as and The upper limit, lower limit of the specification interval are assumed to be and respectively and thus the midpoint is m= The target is given by for symmetric tolerance and is given by for asymmetric tolerance. The natural point estimates of the corresponding index are Ĉpm=1.333 for symmetric tolerance and Ĉ pm=0.771 for asymmetric tolerance. Their confidence interval estimates or the lower confidence limits are presented in Table 4. Usually, if a process has C pm > 1 or C pm > 1, then it can be considered to be a capable process. From Table 4, we can conclude that this Piston rings manufacturing process is capable with symmetric tolerance (T= = m) since the lower confidence limit is greater than 1 and is

9 A Simulation Comparison Study for Estimating the Process Capability Index C pm 251 Table 3. The coverage probability of five confidence intervals for C pm with T=55 under normal process. (µ, σ 2 )=(50,4) n = 10 n = 20 n = 30 n = 40 n = 60 Coverage Coverage Coverage Coverage Coverage Boyles * 0.940* 0.941* SB * 0.950* 0.956* PB * BCPB * 0.942* 0.947* Jackknife 0.958* 0.947* 0.956* 0.956* 0.960* (µ, σ 2 )=(50,9) Boyles * * 0.938* SB 0.942* * 0.938* PB * BCPB * Jackknife 0.955* 0.943* * 0.943* (µ, σ 2 )=(52,4) Boyles * 0.938* 0.946* SB * PB * BCPB Jackknife * * 0.941* (µ, σ 2 )=(52,9) Boyles * * 0.951* SB 0.941* * 0.947* 0.955* PB BCPB Jackknife 0.957* 0.943* 0.946* 0.954* 0.960* incapable with asymmetric tolerance (T= m) since the lower confidence limit is less than 1 for confidence coefficient of.95 or Conclusion In estimating any process capability index that confidence interval estimates should be used instead of the simple point estimates. For point estimation, the Jackknife estimator has almost the same MSE as the natural estimator given in equation (1) and the Jackknife estimator can reduced the bias a lot for small, moderate or large sample size. For interval estimation, the Jackknife method and Boyles method are recommended for use for normal process. The nonparametric confidence interval estimates can protect the user from the error of calculating confidence intervals based on an assumed normal process if the process is a distinctly non normal process. In this case, the Jackknife method is recommended. A software program to obtain the point estimation and interval estimates for the index C pm with asymmetric tolerance is written by the use of the IMSL Library of Micrisoft Fortran [8] software package and is available upon request.

10 252 International Journal of Information and Management Sciences, Vol. 20, No. 2, June, 2009 Table 4. The 90% and 95% confidence intervals (length) or the 95% and 97.5% lower confidence bound for C pm with T = (symmetric tolerance) and for Cpm with T = (asymmetric tolerance). T= % confidence intervals (length) T = % confidence intervals (length) Ĉ pm = % lower confidence bound Ĉpm= % lower confidence bound SB (1.188, 1.478) (0.290) SB (0.694, 0.849) (0.154) (1.188, ) (0.694, ) PB (1.202, 1.496) (0.293) PB (0.702, 0.852) (0.150) (1.202, ) (0.702, ) BCPB (1.192, 1.481) (0.289) BCPB (0.703, 0.852) (0.149) (1.192, ) (0.703, ) Jackknife (1.172, 1.485) (0.313) Jackknife (0.689, 0.848) (0.159) (1.172, ) (0.689, ) T = % confidence intervals (length) T= % confidence intervals (length) Ĉ pm = % lower confidence bound Ĉpm= % lower confidence bound SB (1.155, 1.512) (0.364) SB (0.683, 0.860) (0.178) (1.155, ) (0.683, ) PB (1.176, 1.544) (0.370) PB (0.691, 0.869) (0.178) (1.176, ) (0.691, ) BCPB (1.165, 1.525) (0.377) BCPB (0.684, 0.858) (0.174) (1.165, ) (0.684, ) Jackknife (1.142, 1.516) (0.374) Jackknife (0.674, 0.864) (0.190) (1.142, ) (0.674, ) References [1] Boyles, R. A., The Taguchi capability index, Journal of Quality Technology, Vol. 23, No. 1, 17-26, [2] Chan, L. K., Cheng, S. W. and Spiring, F. A., A new measure of process capability:c pm, Journal of Quality Technology, Vol. 20, pp , [3] Chou, Y., Owen, D. B. and Borrego, A., S. A., Lower confidence limits on process capability indices, Journal of Quality Technology, Vol. 22, No. 3, pp , [4] Efron, B., Bootstrap methods: Another look at the Jackknife, The annals of statistics, Vol. 7, No. 1, pp.1-26, [5] Efron, B. and Tibshirani, R. J., Bootstrap Method for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy, Statistical Science, Vol. 1, pp.54-77, [6] Franklin, L. A. and Wasserman, G., Bootstrap confidence interval estimates of C pk : An introduction, Communications in Statistics-Simulation and Computations, Vol. 20, pp , [7] Kane, V. E. (1986). Process capabillity indices, Journal of Quality Technology, Vol. 18, pp.41-52, [8] Microsoft Developer Studio Fortran Powerstage 4.0 and IMSL, 1995, Microsoft Corporation. [9] Montgomery, D. C., Introduction to Statistical Quality Control. John Wiley & Sons, New York, NY, [10] Pearn, W. L., Lin, G. H. and Chen, K. S. Distributional and Inferential Properties of the process accuracy and process precision indices, Communications in Statistics-Theory and Methods, Vol. 27, pp , [11] Ryan, T. P., Statistical Methods for Quality Improvement, John Wiley & Sons, New York, NY, 1989.

11 A Simulation Comparison Study for Estimating the Process Capability Index C pm 253 [12] Tukey, J. W., Bias and Confidence in not quite large samples., ANnals of Mathematical Statistics, Vol. 29, p.614, [13] Quenouille, M. H., Approximate tests for the correlation in time series, Journal of the Royal Statistical Society, B, Vol. 11, pp.68-84, Author s Information Shu-Fei Wu is a Professor of Department of Statistics at Tamkang University. Her research interests are in the areas of screening, multiple comparisons with the average, subset selection and statistical inferences. Her work has appeared in IIE Transaction, CSDA, Communication in Statistics, JSPI, etc. Department of Statistics, Tamkang University, Tamsui, Taipei, Taiwan 251, R.O.C @mail.tku.edu.tw TEL: ext.2876