Lower Confidence Bound for Process-Yield Index S pk with Autocorrelated Process Data

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1 Quality Technology & Quantitative Management Vol. 1, No., , 15 QTQM IAQM 15 Lower onfidence Bound for Process-Yield Index with Autocorrelated Process Data Fu-Kwun Wang * and Yeneneh Tamirat Deartment of Industrial Management National Taiwan University of cience and Technology, Taiei, Taiwan (Received October 13, acceted May 14) Abstract: In this aer, the lower confidence bound for the rocess-yield index is develoed for autocorrelated rocess data. A simulation study is conducted to assess the erformance of the roosed index and two existing indices and for various combinations of samle size, autoregressive arameter, and true index value. The simulation results confirm that the estimated erforms better than the other two indices and regarding bias and standard deviation. Additionally, the coverage rates of in most cases are greater than a 95% lower limit of the stated nominal. Two real examles are used to demonstrate the alication of the roosed aroach. Keywords: Autocorrelated rocess, lower confidence bound, rocess-yield index. 1. Introduction rocess yield is a measure of degree of conformance to the design secification. It is a P common ractice in the manufacturing sector to use rocess yield as a criteria to measure rocess erformance. It is closely related to the roduction cost as well as customers satisfaction. Process caability indices, including UL LL, 6 UL LL min,, and 3 3 m 6 UL LL T, where LL and UL are the lower and uer secification limits, resectively, and are the rocess mean and standard deviation, resectively, and T is the target value, have been widely used in many industries. However, the index only reflects rocess otential (or rocess recision) and the index only rovides an aroximate measure rather than an exact measure of the rocess yield. That is, for a normal rocess, we have (3 ) 1 yield (3 ), where () is the cumulative distribution function of the standard normal distribution. To obtain an exact measure of the rocess yield, Boyles [1] considered a yield index, referred to as, for a rocess with normal distribution. The index is defined as where UL 1 LL 3, (1) 1 () is the quantile function of the standard normal distribution. * orresonding author. fukwun@mail.ntust.edu.tw

2 5 Wang and Tamirat Process data in continuous manufacturing rocesses such as chemical rocess data and Internet traffic data are correlated or self-deendent [5, 7]. Thus, the autocorrelated data may affect the erformance of rocess caability analysis. cagilarini [9] investigated the effects of the index for autocorrealted data and measurement errors. hore [1] conducted a simulation study for rocess caability indices and when data follow a third order autoregressive model AR(3). Zhang [15] rovided the interval estimation rocedures for and when data follow an AR(1) model. Wallgren [13-14] alied the Taylor linearization aroach to derive aroximate lower confidence limit for when data follow an AR(1) model or a first order moving average model MA(1). Noorossana [6] resented the estimated values of and when data follow an ARMA(1,1) model. Vännman and Kulchci [1] offered a model-free aroach based on the iterative skiing strategy to estimate the rocess caability indices and when data follow an AR(1) model. Lovelace et al. [3] develoed through simulation 1(1 )% lower confidence limits for and when data follow an AR(1) model. un et al. [11] analyzed five estimation schemes for,, and m when data follow an AR(1) model. Lundkvist et al. [4] rovided a comarison study of four decision methods for when data follow an AR(1) model. The results showed that no single method works well in a general situation, but the method by Wallgren [14] erforms better than the other methods when the rocess mean is close to the target value. As shown above caability studies on autocorrelated rocess is based on the indices,, and m. However,,, and m are aroximations rather than an exact measure and subjected to various limitations; see Montgomery [5] and Lee et al. []. The urose of this article is to investigate the erformance of when data follow an AR(1) model. The aroximate lower confidence bound (LB) of the true rocess yield is shown in the Aendix. The LB cannot only be used in the rocess yield assurance, but also be used in rocess caability testing for decision-making. A simulation study is conducted to evaluate the erformance of the roosed method and two existing methods in the following section. Two real examles are used to demonstrate the alication of our roosed method. Finally, we offer a conclusion and suggestions for future studies.. Process-yield Index with Autocorrelated Process Data For a normally distributed rocess based on individual observations, we have the samle size is n. Let xi, i 1,,, n be the characteristic of the n samles with mean and standard deviation. The index roosed by Boyles [1] can be rewritten as ( ) ( ), () 3 d d where m( UL LL)/, d ( UL LL)/, ( m)/ d, d / d. The above index rovides an exact measure of rocess yield. There exists a one-to-one relationshi between and the rocess yield. If c, the rocess yield is then derived by P (3 c) 1. (3) ome various values with their corresonding rocess yield and non-conformities are shown in Table 1. The Lithograhic industry suggests that (a) the rocess has serious roblems if 1., (b) the rocess needs imrovement if , (c) the rocess is satisfactory if and (d) the rocess is world class if..

3 Lower onfidence Bound for Process-Yield Index with Autocorrelated Process Data 53 To estimate the rocess-yield index, we consider the following estimator, where x and are the samle mean and the samle standard deviation, which may be obtained from a stable rocess. 1 1 UL x 1 x LL (4) 1 1 ( ) ( ) ( ) ( ) 3 3 d d Table 1. Various values with their corresonding rocess yield and non-conformities. Process yield Non-conformities (m) Note: m reresents arts er million. To indicate its reliability we are often interested in constructing the confidence interval of the rocess yield. Lee et al. [] rovided the asymtotic distribution of the estimator of. Thus, an aroximate 1(1 )% confidence interval and lower confidence limit for are derived by a b a b Z1 /, Z1 /, (5) 6 n (3 ) 6 n (3 ) and a b Z1, (6) 6 n(3 ) where 1 1 a d/ (1 ) ( ) (1 ) ( ), 1 1 b ( ) ( ), d d d d and Z1 / is the uer 1 (1 /) % oint of the standard normal distribution, is the robability density function of the standard normal distribution, and is the estimated rocess standard deviation. An aroximate 1 (1 ) % lower confidence bound for can be derived by 1Z 1 9n 1 1 ( n 1) 1/ or Z 1 1 9n ( n1) 1/. (7)

4 54 Wang and Tamirat In the resence of autocorrelation the standard error of the estimated index is affected. Hence tests and confidence intervals, based on the assumtion of indeendence, are not valid when autocorrelation is resent. We considered an examle from hore [1] where the secification limits are UL = 61 and LL = 19, resectively, the target rocess mean (T ) is 4, and the actual rocess means ( ) are 4, 4, and 5, resectively. Based on the AR (1) model, the true values of,, and for different rocesses are given in Table. In Table, we found that the higher the autocorrelation the lower the caability index value. In addition the higher the target mean shift the lower the caability index value. For examle, if 5, the values for d, and 5 are 1.4, 1.35 and , resectively Note: Table. The true values of three different indices for AR(1) model. d d d d T. When data follow an AR (1) rocess, Zhang [15] roosed an aroximate 1(1- )% lower confidence bound for which is given by k, (8) where z z g F V ( ) z f n n f 9 ( 1) n1, f ( n, i) 1 ( ni) i, nn ( 1) i 1 n n n n i i i i j F( n, i ) n ( n i) 1 n ( n i) 1 ( n i j), i n i n i j n 1 ri g( n, i) 1 ( ni) i, i, n i 1 r i is the rocess autocorrelations (from a lag of 1 to n), and k z is a constant to be chosen by the user. Zhang [15] did not rovide suggestion how to choose this constant. However, o

5 Lower onfidence Bound for Process-Yield Index with Autocorrelated Process Data 55 Zhang [15] used k z = and 3 for an AR (1) rocess in a simulation study, the average coverage rates for these two cases are aroximately 95% and 99.7%, resectively. Wallgren [13] alied the Taylor linearization aroach to derive the aroximate lower confidence bound for which is given by Z1, (9) n 1 9n 1 where the estimated is obtained by the equation: xi ( xi 1 ) i, i 1,,, n, where i ~ N (, ). The exectation and variance of for an AR(1) rocess are derived in Aendix. We have that 1 1 n E ( ) ( (3 )) a( f 1), (1) 6 n and 1 na F V( ) b g. (11) 36 n( (3 )) n 1 By the entral Limit Theorem, an aroximate 1(1 )% confidence interval for under an AR(1) rocess is derived by na F na F b g b g n1 n1. (1) Z1, Z 1 6 n(3 ) 6 n(3 ) imilarly, an aroximate 1( 1 )% lower confidence bound for under an AR(1) rocess is derived by 3. imulation tudy na F b g n 1 Z1. (13) 6 n(3 ) We considered an AR(1) rocess where UL = 61, LL = 19, and the target value = 4. The simulation was conducted with three samle sizes (n = 5, 1, and ) and four different autocorrelation coefficients ( =,.5,.5, and.) for three different rocesses: centered rocess (d = ), a small shifted rocess (d = ), and a large shifted rocess (). Furthermore, we considered an incaable rocess that the true values of and are less than 1. (see Table 3) and a satisfactory rocess that the true values of and are greater than 1.33 (see Table 4). Three measures including average, standard deviation, and bias of three different indices are summarized in Tables 3 and 4. The simulation rogram was written in R language [8] and all results are based on 1, simulation runs. 1/

6 56 Wang and Tamirat In Tables 3-4, we found that the standard deviation and the magnitude of bias of are the smallest in most cases. The samle size affects the standard deviation and the bias of all three indices; that is, when the samle size increases, the estimates erform better. Table 3. omarison of, n Measures 5 1 Note: , and for incaable rocesses ( 4 and 7 ). d d d 5 d d d 5 Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias d T ; td=standard deviation; Bias=estimated value-true value.

7 Lower onfidence Bound for Process-Yield Index with Autocorrelated Process Data 57 Table 4. omarison of, n Measures 5 1 Note: , and for incaable rocesses ( 4 and 5 ). d d d 5 d d d 5 Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias Average td Bias d T ; td=standard deviation; Bias=estimated value-true value. In Figure 1, the standard deviation decreases as the samle size increases. In addition, when autocorrelation value increases, the standard deviation increases. For examle, the standard deviations of a small shifted rocess for =,.5,.5, and. under samle size 1 are.698,.731,.798 and.897, resectively. We also found that no or week autocorrelation for a centered rocess has the highest standard deviation followed by a small shifted rocess. However, as shown in Figure 1(d) an increase in autocorrelation

8 58 Wang and Tamirat makes the reverse true; that is, a rocess with the largest shift has the highest standard deviation. Figure illustrates the relationshi between the bias and the samle size. In no autocorrelation case, the bias decreases raidly as the samle size increases. A larger shifted rocess has the larger bias. a) b) d = d =.1 d =.1 d = tandard Deviation tandard Deviation amle size amle size c).5 d) d = d =.13 d = d = tandard Deviation tandard Deviation amle size amle size Figure 1. Relationshi between samle size and standard deviation under different correlations. a) b) d = d =.7 d = d = Bias Bias amle size amle size c).5 d) d = d =.7 d = d = Bias.4 Bias amle size amle size Figure. Relationshi between bias and samle size under different correlations.

9 Lower onfidence Bound for Process-Yield Index with Autocorrelated Process Data 59 Figure 3 shows that a larger shifted rocess under has the lowest standard deviation. When increases, the standard deviation increases. When the samle size increases, the standard deviation decreases. For examle, when the samle size n 5, the standard deviation is.135 for d. When the samle size n 15, the standard deviation is reduced to.55 for d. Finally, each run relicated 1, times to yield the average, confidence interval, coverage rate, and LB of at a 95% confidence level (see Tables 5 and 6). A 95% lower limit of the stated nominal value for the coverage rate is obtained as /1 1% 94.57%. We found that the coverage rates in most cases are greater than 94.57%. The samle size affects the interval length and the LB at a 95% confidence level; that is, when the samle size increases, the estimates erform better. As simulated case for d =,.5, 4, and 5 with the samle size n 5, an aroximate 95% confidence interval and 95% lower confidence bound for are obtained as (1.773, 1.658) and , resectively. We concluded that the LB value for is within the range of (1.33, 1.5) at a 95% confidence level. It should be noted that this rocess is satisfactory. Additionally, the coverage rate ranges between 94.61% and 97.69% excet in one case where the samle size is 5 and the autocorrelation coefficient is.. a) n 5 b) n 1.13 d = d =.95.9 d = d = tandard Deviation tandard Deviation Autocorrelation oefficient Autocorrelation oefficient.7.8 c) n d = d = tandard Deviation Autocorrelation oefficient.7.8 Figure 3. Relationshi between and standard deviation under different samle sizes. 4. Illustrative Examles Examle 1: We considered the emirical data given by Lundkvist et al. [4]. wedish steel AB wanted to find out whether it is ossible to claim that the true value exceeds 1. with significance level of 5%. Here we used the rocess-yield index instead of. The rocess data is normally distributed and reasonably modeled by AR (1) model with ositive autocorrelation. The roduction secification limits are (LL, T, UL) = (4.4,

10 6 Wang and Tamirat 4., 5.1). ummary statistics of the data set are shown in Table 7. The 95% lower confidence bounds for and are also listed in Table 7. We tested the null hyothesis Ho : 1 against the alternative H 1 : 1 with a 95% confidence level. In Table 7, we found that the 95% confidence lower bound of for samle sizes of and 4 are and 1.11, resectively. These two values are greater than the minimum value 1.. Therefore, we can conclude that the null hyothesis is rejected for both data sets. wedish steel AB can claim that the rocess is caable with 1 at a 95% confidence level. Table 5. onfidence intervals of for incaable rocesses ( 4 and 7 ). n d True value of Average confidence overage Average LB interval rate 1. (.871, 1.5) % (.768, ) % (.6477, ) % (.4113,.9685) %.964 (.7814, 1.164) % (.7434, ) % (.631, 1.884) %..65 (.439,.9568) %.8451 (.691, 1.16) % (.6549, 1.3) % (.5597,.9866) % (.3694,.8998) % 1. (.863, 1.144) % (.866, ) % (.715, 1.37) % (.488,.863) %.964 (.834, 1.17) % (.7987, 1.83) % (.6916, 1.9) %..65 (.4783,.8518) %.8451 (.7346,.9666) % (.74,.958) % (.618,.914) % (.435,.7985) % 1. (.935, 1.13) % (.8687, 1.74) % (.83,.9816) % (.548,.7978) %.964 (.8719, 1.61) % (.838, 1.386) % (.7351,.9578) %..65 (.596,.7868) %.8451 (.7664,.995) % (.7366,.9163) % (.651,.863) % (.4819,.735) %

11 Lower onfidence Bound for Process-Yield Index with Autocorrelated Process Data 61 Table 6. onfidence intervals of for satisfactory rocesses ( 4 and 5 ). n d True value of Average confidence overage Average LB interval rate 1.4 (1.137, 1.688) % (1.773, 1.658) % (.9117, ) %..96 (.5868, 1.368) % 1.35 (1.8, ) % (1.65, 1.583) % (.85, 1.546) % (.5718, 1.333) % (.956, 1.373) % (.885, 1.363) % (.7, ) %..794 (.583, 1.98) % 1.4 (1.87, ) % (1.158, ) % (.999, 1.446) %..96 (.6876, 1.91) % 1.35 (1.146, 1.515) % (1.989, ) % (.9547, 1.396) % (.6668, ) % (.9844, 1.95) % (.9436, 1.785) % (.835, 1.81) %..794 (.5873, 1.639) % 1.4 (1.649, 1.545) % (1.163, 1.515) % (1.6, ) %..96 (.86, ) % 1.35 (1.1964, ) % (1.151, 1.444) % (1.116, 1.317) % (.7335, 1.89) % (1.66, 1.451) % (.988, 1.31) % (.87, 1.141) %..794 (.6444,.95) % Table 7. The summary results of Examle 1. ase n x Estimated Estimated LB LB value value Note: LB is obtained at a 95% confidence level. Examle : We considered the viscosity measurements from a chemical rocess taken every hour resented in examle 1. from Montgomery [5]. This rocess is stable and in statistical control. Based on the behavior of the samle autocorrelation function, the

12 6 Wang and Tamirat rocess is modeled using the first-order autoregressive AR(1) model. The fitted value of the viscosity data is xt xt 1 t, where t ' s are identically and indeendently distributed normal random variables with mean zero and standard deviation of. Taking value of.5, the simulated data are shown in Table 8. The mean and standard deviation of the AR(1) data from Table 8 became and.8983, resectively. The roduction secification limits are given as (LL, T, UL)=(8, 85, 9). ummary statistics of the data set are shown in Table 8. The 95% lower confidence bounds for and are also listed in Table 8. We test the null hyothesis H : 1.33 against the alternative H1 : 1.33 at a 5% significance level. We found that the 95% confidence lower bound of for samle sizes of is The value is greater than the minimum value Therefore, we can conclude that the null hyothesis is rejected and the rocess is satisfactory since its value is between 1.33 and.. Table 8. imulated data for the viscosity measurements. t x t t x t t x t t x t t x t

13 Lower onfidence Bound for Process-Yield Index with Autocorrelated Process Data 63 Table 9. The summary results of Examle. n x Estimated value LB Estimated value LB Note: LB is obtained at a 95% confidence level. 5. onclusion Process yield lays an imortant role for a manufacturing rocess. We resent a rocess-yield index to evaluate the rocess yield when data follow an AR(1) model. The aroximate lower confidence bound (LB) for the true rocess yield is also derived. This methodology is easy to understand and use. The simulation results showed that the roosed method outerforms two existing indices and. The roosed method can be used to determine whether the rocess erformance meets the resent yield requirement leading to a reliable decision. Future studies may include more comlex models such as MA and ARMA models. References 1. Boyles, R. A. (1994). Process caability with asymmetric tolerances. ommunications in tatistics imulation and omutation, 3(3), Lee, J.., Hung, H. N., Pearn, W. L. and Kueng, T. L. (). On the distribution of the estimated rocess yield index. Quality and Reliability Engineering International, 18(), Lovelace,. R., wain, J. J., Zeinelabdin, H. and Guta, J. D. (9). Lower confidence limits for rocess caability indices and when data are autocorrelated. Quality and Reliability Engineering International, 5(6), Lundkvist, P., Vännman, K. and Kulchci, M. (1). A comarison of decision methods for when data are autocorrelated. Quality Engineering, 4(4), Montgomery, D.. (13). tatistical Quality ontrol - A Modern Introduction. Wiley, ingaore. 6. Noorossana, R. (). hort communication rocess caability analysis in the resence of autocorrelation. Quality and Reliability Engineering International, 18(1), Psarakis,. and Paaleonida, G. E. A. (7). P rocedures for monitoring autocorrelated rocesses. Quality Technology and Quantitative Management, 4(4), R Develoment ore Team (14). R: A Language and Environment for tatistical omuting. Vienna, Austria: R Foundation for tatistical omuting. 9. cagliarni, M. (). Estimation of for autocorrelated data and measurement error. ommunications in tatistics Theory and Methods, 31(9), hore, H. (1997). Process caability indices when data are autocorrelated. Quality Engineering, 9(4), un, J., Wang,. and Fu, Z. (1). Process caability analysis and estimation scheme for autocorrelated data. Journal of ystems cience and ystems Engineering, 19(1), Vännman, K. and Kulchci, M. (8). A model-free aroach to eliminate autocorrelation when testing for rocess caability. Quality and Reliability Engineering International, 4(), 13-8.

14 64 Wang and Tamirat 13. Wallgren, E. (1). onfidence limits for the rocess caability index for autocorrelated quality characteristics. Frontiers in tatistical Quality ontrol, eds., Lenz, H. J. and Wilrich, P. T. Heidelberg: Physica-Verlag. 14. Wallgren, E. (1). A generalization of the Taguchi caability index for data generated by a first order moving average rocess. tatistical Methods, 3(1), Zhang, N. (1998). Estimating rocess caability indexes for autocorrelated data. Journal of Alied tatistics, 5(4), Aendix The mean and variance of with the autocorrelation function are derived as follows: Lee et al. [] rovided the asymtotic distribution of the estimator of. Let Z n( X ), and Y n( ), then Z and Y are indeendent and since the first two moments of X and exist, by the entral Limit Theorem they converge to N (, ) and N (, 4 ), resectively, as n goes to infinity. onsequently, can be exressed as 1 1 ( (3 )) W 6 n, (A1) where d W Y (1 ) (1 ) 3 r Z, d d dd d d and W ~ N(, a b ). Let W AY BZ, where d 1 1 A (1 ) (1 ) 3 r, d d B. dd d d We have that E( W) AE( Y) BE( Z) and VW ( ) AVY ( ) BV( Z). The exectation and variance of Z are obtained as E( Z) and V( Z) nv( X) n g g. n Thus, we have Z n( X ) N(, g), where g is given in Equation (8). The exectation and variance of Y are obtained as 4 4 F EY ( ) n( E ( ) ) n ( f 1) and VY ( ) nv ( ) n 3 n. n 1 ( n 1) Thus, we have 4 F Y n( ) N n ( f 1), n, ( n 1) where f and F are given in Equation (8). Thus, the exectation and variance of are derived as follows: n E ( ) ( (3 )) A n ( f 1) ( (3 )) a( f1), (A) 6 n 6 n and

15 Lower onfidence Bound for Process-Yield Index with Autocorrelated Process Data An F 1 naf V( ) B g b g. (A3) 36 n( (3 )) n1 36 n( (3 )) n1 An aroximate 1(1 )% confidence interval for is derived by na F na F b g b g n1 n1 Z1, Z 1. (A4) 6 n(3 ) 6 n(3 ) An aroximate 1( 1 )% lower confidence limit for in the resence of autocorrelation is obtained as na F b g n 1 Z1. (A5) 6 n(3 ) Authors Biograhies: Fu-Kwun Wang is a Distinguished Professor in the Deartment of Industrial Management at the National Taiwan University of cience and Technology, Taiwan. His rimary research interests are in reliability, quality and roduction management. Yeneneh Tamirat is a Ph.D. candidate in the Deartment of Industrial Management at the National Taiwan University of cience and Technology, Taiwan. His research areas are statistical quality control and rocess caability analysis.

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