UPPER CONFIDENCE LIMIT FOR NON-NORMAL INCAPABILITY INDEX: A CASE STUDY

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1 مو تمر الا زهر الهندسي الدولي العاشر AL-AZHAR ENGINEERING TENTH INTERNATIONAL ONFERENE December 4-6, 008 ode: M 01 UPPER ONFIDENE LIMIT FOR NON-NORMAL INAPABILITY INDEX: A ASE STUDY A. Rotondo 1 and Amir A. Mahdy 1 Dipartimento di Ingegneria Meccanica e Gestionale, Politecnico di Bari, viale Japigia 18, 7016, Bari, Italy Mining and Petroleum Eng. Dept., Faculty of Engineering, Al-Azhar University, (Post Doc. Fellowship, School Mechanical and Manufacturing Eng., Dublin ity University, Dublin 9, Ireland) ABSTRAT: Process capability indices (PIs) are widely adopted in industrial environment since they are a simple tool to assess the level of conformation of products to customers specifications and, hence, to monitor process performances. Among PIs, the process incapability index, pp, has progressively obtained large consideration since it is easy to apply and analytically convenient. Moreover, it keeps uncontaminated information about process accuracy and process precision. As for other PIs, the applicability of pp can also be extended to non-normal process by means of lements s method [1]; however, with this reformulation, the non-normal pp distribution becomes mathematically intractable. Another question with the non-normal PIs concerns the reliability and the accuracy of the Pearson distributions system percentiles. In this study, a non-normal pp, pp(q), based on Burr XII distribution is proposed and its upper confidence interval is evaluated by means of the bootstrap resampling method. The upper confidence bound of pp(q) can be considered a reliable indicator of product quality and process performance, since it allows avoiding problems caused by sampling errors. The procedure proposed in this paper can be easily implemented and very poor statistical background is sufficient for handling its results. Finally, a numerical example is presented. 008 Faculty of Engineering, Al-Azhar University, airo, Egypt. All rights reserved. KEYWORDS: Process Incapability Index; Burr Xii Distribution; Bootstrap Methodology. 1.INTRODUTION Process capability analysis is used to evaluate the level of conformation of products to their specifications. It enables both the monitoring and the reduction of the variability of industrial processes []. The simplest tools to assess process capability are process capability indices (PIs). They provide a single number and unitless measure of process potential and performance which reflects the ability of the process to meet specifications limits [3]. Their simplicity makes them usable as a communication medium between both quality engineers and shop floor controllers, interested in improving process performance. They can be also used during contract stipulation [4]. Al-Azhar University Engineering Journal, JAUES Vol. 3, No. 11, Dec

2 In the last two decades, several PIs have been developed involving even more criteria in capability measurement: process variation, process departure, process yield and process loss [5]. The first indices were defined by Juran and Kane as follows: p USL LSL = (1) 6σ {, } USL-µ µ LSL pk = min pu pl = min ; () 3σ 3σ where µ and σ are the process mean and process standard deviation and USL and LSL are the upper and the lower specification respectively. When process parameters are unknown, the sample mean, x, and the sample standard deviation, S, can replace them. With respect to p, pk presents the advantage to consider, along with process variability, the degree of process mean shift from the center of the specification interval. Moreover, the two sub-indices that appear in pk definition, pu and pl, can be used in case of unilateral specifications. However, this first category of indices does not take the target value, T, into any account. That means, process loss caused by mean departure from the target is completely neglected [6]. A new class of indices has been introduced to remedy this disadvantage: pm USL LSL = 6 σ ( µ T ) + (3) pmk = min 3 USL-µ ; µ LSL ( µ T ) + σ 3 ( µ T ) + σ (4) The statistical properties of pm natural estimator are mathematically intractable. Accordingly, Greenwich and Jahr-Schaffrath [7] proposed an incapability index pp, which is a simple transformation of pm, but it results easier to use and analytically convenient [8]. An overview on the pp properties will be presented in the next section. The definition of such PIs has been generally based on three assumptions: (1) the system determining which data are collected is under control; () the collected data are independent and identically distributed; (3) the process data are normally distributed, that is the process must be normal []. However, not always these conditions and above all the last one are respected in an industrial environment. An uncritical use of these indices may lead to erroneous considerations. Many studies have been conducted for handling non-normal processes and two simple approaches have emerged. The first one consists of transforming original data into normally distributed data: models suitable for any type of data are available. See, for instance [9-11]. The second one suggests the use of non-normal percentile in place of sample mean and standard deviation to calculate PIs. lements used the last approach when proposing the method of non-normal percentiles to calculate p and pk for a distribution of any shape, using the Pearson family of curves []. Based on lements method, Pearn and Kotz [1] further modified PIs as follows: Vol. 3, No. 11, Dec. 008

3 p( q ) = USL LSL x x (5) pk( q ) USL x = min x x ; x x LSL x (6) pm( q ) ( USL T; T LSL) min = (7) x x ( x ) T 6 = min 3 USL x x 3 LSL pmk ( q ) ; (8) x x x ( ) 0. x + ( ) T 3 x T where p is the p*100 percentile value of non-normal data; in particular, in lements method, it represents the p*100 percentile of the Pearson distribution family and can be easily obtained by evaluating sample skewness and kurtosis coefficients. Lin and hen [] demonstrated that the Burr type XII distribution, if used for evaluating nonnormal percentiles, in place of the Pearson curves, generally provides slightly better results and works especially well under distributions which moderately depart from normality. In spite of noticeable efforts have addressed non-normal process capability analysis, the problem of the mathematical intractability of the non-normal PIs does not allow to neither investigate their statistical properties nor analytically make statistical inference on them. However, due to sampling errors, which frequently occur during data collection, the comparison between different entities under investigation, for instance suppliers or processes, based on an estimation of the chosen PI cannot be considered reliable. That s why confidence interval evaluation is favourable in making decisions. At the authors knowledge, a confidence interval of pp(q) based on Pearson distribution family has never been developed. Moreover, the calculation of pp(q) using the Burr distribution system has never been proposed in the open literature. In this study, the upper confidence bound of the non-normal incapability index, pp(q), built by means of the bootstrap resampling simulation method, is proposed as an effective indicator usable for both selecting suppliers and monitoring process or guiding quality engineers in improving process performance. The Burr-based approach has been used. About the index proposed, a case study reveals its relevant capability of discerning between processes of different quality level.. INAPABILITY INDEX PP The process incapability index, pp, was introduced by Greenwich and Jahr-Schaffrath [7] as a simple transformation of pm 3 x Vol. 3, No. 11, Dec

4 pp 1 = pm µ T = D σ + D (9) where D ( USL LSL) / 6 =. This index is easy to apply and enjoys a large consideration in studies on capability analysis since it provides more information than other commonly used PIs. Indeed, first, it inherits the own properties of pm, which primarily are the consideration of process location and process centering (that is the ability to distinguish between on-target and off-target processes). Second, it can be easily considered as the sum of two sub-indices, ( T ) / D µ, also called inaccuracy index, ia, and σ / D denoted by ip, or imprecision index. ia takes into account process loss caused by mean departure from the target, while ip reflects the extent of process variation [3]. The uncontamination of information concerning accuracy and precision represents a useful tool for both individuating the reasons of an inadequate process capability (too high pp values) and distinguishing between entities eventually characterized by the same pp, or pm, value. Some commonly used values of pp with the related process denomination are reported in table 1. Table lists some commonly used values of ip and the corresponding quality condition. The pp distribution was studied by hen [13] under normality assumption; hence confidence interval evaluation is possible. Even when process parameters are unknown, an interval estimate of pp can be obtained, since Greenwich and Jahr-Schaffrath [7] found that Ĉ pp / pp is approximately distributed as χ v / v, Ĉ pp being the natural estimator of pp, χ v a chi-square distribution with v degrees of freedom and v n + [( T )/ σ ] { 1 µ } /{ 1+ [ ( µ T )/ σ ] } =. Due to its features, the pp index has been widely used in process capability analysis. hen et al. used pp for contract manufacturers selection by both developing a score index based on pp confidence bounds [14], and constructing a supplier capability and price analysis chart, where considerations about product price were also included [3]. Pearn et al. proposed a multiprocess performance analysis chart based on pp, which displays multiple processes with departure and process variability relative to the specification tolerances on one single chart [15]. Vol. 3, No. 11, Dec

5 Table 1 Some commonly used pp and the corresponding pm values Table Some commonly used ip and the corresponding quality condition pp pm Quality condition Precision requirements apable 0.56 ip 1.00 Satisfactory 0.44 ip 0.56 Good 0.36 ip 0.44 Excellent 0.5 ip 0.36 Super ip 0.5 When the process characteristic is not normally distributed, the relation between pp and pm can be used to define a non-normal pp, pp(q), as follows: pp ( q ) 1 = pm( q ) x = x 6D* x T + D* (10) being D* min (( USL T ); ( T LSL) )/ 3 =. Note that, unlike pp, due to pm(q) formulation, pp(q) can also be adopted in case of asymmetric tolerances. However, the distribution of pp(q) results mathematically intractable. The percentiles in Eq. 10 are usually evaluated using Pearson distribution family of curves. Basing on Lin and hen s suggestion [], in this study, pp non-normal percentiles are evaluated using the Burr system of distribution, which proved to be more accurate in PIs estimates than lements s method. 3. THE BURR XII DISTRIBUTION The Burr type XII distribution was introduced by Burr in 194 [16]. Its cumulative distribution function and the corresponding probability density function respectively are: F f c 1 ( ) ( 1+ x ) x = 0 k c ( ) 1 c ( 1 ) ( k + x = kcx + x 1) for x 0 for x < 0 (11) (1) Vol. 3, No. 11, Dec

6 The region of coverage of standardized moments a 3 and a 4 was still further extended, toward low a 4 for given a 3, by means of the reciprocal transformation x=1/y which yields the following cumulative distribution function: G c ( ) ( 1+ y ) y = 0 k for y 0. for y < 0 (13) With this extension, Burr showed that an appropriate choice of the distribution parameters, c and k, allows coverage of a large proportion of curve shape characteristics of types I, II, II, IV and VI in the Pearson family [17]; that means that the well known normal, Weibull, logistic, lognormal, gamma, beta and extreme value type I distributions are all covered by the Burr XII system. Burr [18] provided tables where several combinations of c and k are related with the expected value, the standard deviation, the standardized skewness coefficient and the standardized kurtosis coefficient of the Burr XII distribution. These two last parameters represent the link between a random variate (X) and a Burr variate (U). When a sample is drawn from X, the sample skewness coefficient (a 3 ) and the sample kurtosis coefficient (a 4 ) allow the fitted parameters of the Burr XII distribution to be obtained using the tables. Then, a standardized transformation relates X with U: X x S x U = µ σ where x and S x are the sample mean and standard deviation and µ and σ are the mean and the standard deviation of the corresponding Burr variate. Besides the extended coverage area, the Burr XII distribution has the property that its cumulative distribution function exists in closed form [19]; hence, an algebraic transformation allows distribution percentiles to be obtained, useful in non-normal PIs evaluation. Its extreme flexibility makes it suitable for describing data from the real world. Many authors have used it as a life time model in reliability analysis. Wang, based on Burr XII distribution, presented a model for modelling failure rate of mechanical and electronic components [0]. It was also used by Abdel-Ghaly et al. for developing a reliability growth model [1]. Good fits are also obtained when biological, clinical and other experimental data have to be described. Ali Mousa et al. provided maximum likelihood and Bayes estimates of the two parameters of the Burr XII distribution for progressive type II censored data []. Zimmer reported the maximum likelihood method for fitting Burr XII parameters to data under any type of conditions [19]. 4. THE BOOTSTRAP METHOD The normality assumption, often used in statistical research and generally made possible by the central limit theorem, presents several advantages among which the possibility of evaluating statistical errors and confidence intervals. However, it provides accurate results only if statistics or statistics transformations are asymptotically normally distributed. In an industrial environment, such a property is not always verified and a normality approximation may yield misleading conclusions. In order to avoid these problems, thanks to computation Vol. 3, No. 11, Dec

7 advances, more realistic models can be adopted by using methods which often are too complicated for practical analytical treatment. Among these, the bootstrap methodology has obtained great success and diffusion. It is used to solve problems generally characterized by mathematical intractability and works well when the population under examination is either normally or not normally distributed. It was initiated by Efron [3] in 1979 and it relies on the idea that the sampling distribution for a statistic, θ, is estimated by the relative frequency distribution of the θ values calculated for each one of the samples of size n randomly drawn from a random population sample. The unique requirements are that the random sample must be representative of the population and that the observations are independent and identically distributed. The original sample, of size n, {x 1, x,,x n }, is treated as the population X; then, a Monte arlo style procedure draws, with replacement, from it a large number, B, of resamples, also called bootstrap samples, * * * { x 1,x, K,xB }. Theoretically, the replacement allows to create n n different samples from the original one, but 1000 resamples are generally considered enough for obtaining an accurate confidence interval estimate [4]. If θ is the parameter by which process performance is * controlled, the bootstrap estimate of θ, ˆθ, can be evaluated for each bootstrap sample. Then, the relative frequency histogram of the B bootstrap estimates, { ˆ * * θ, ˆ θ,, ˆ θ } * 1 K B, can be built; the distribution obtained is the bootstrap estimate of the sampling distribution of θ. This distribution can be used to make statistical inference on it. Different types of bootstrap confidence interval have been developed; the most commonly used are the standard bootstrap (SB) confidence interval, the percentile bootstrap (PB) confidence interval and the biased corrected percentile bootstrap (BPB) confidence interval [6]. onstruction of a two-sided (1- α )100% BPB confidence limits will be described, so that a lower (1-α )100% confidence interval can be obtained by using only the lower limit [6]. The bootstrap distribution built on only a sample of the complete bootstrap distribution may be biased, that is, shifted higher or lower than expected. In order to correct the possible bias, a third procedure was developed, which consists of the following steps. First, the probability p P[ ˆ θ * ˆ 0 = θ ] is calculated using the ordered distribution of { ˆθ } * i, i=1,, B; θˆ is the value of θ estimated from the original sample. Secondly, the inverse of the cumulative distribution function of a standard normal based upon p 0 is computed as z 0 = Φ -1 (p 0 ), p L = Φ( z 0 z α ), p U = Φ( z 0 + z α ), where Φ () is the cumulative standard normal distribution function. Finally, the BPB confidence interval is obtained as follows [ ˆ * ( ) ˆ * plb, θ ( pu B) ] θ. In the literature, confidence intervals of some PIs have been computed by means of bootstrap method. hen and Jang constructed the BPB confidence interval of the difference between pk s of two suppliers and proposed it as a supplier selection criterion. They proved the robustness of BPB confidence interval with varying of distribution parameters and sample size [4]. hen and Pearn derived the confidence interval for S pk index [5]. Pearn et al. evaluated the upper confidence limit for the Y q index and found that the bootstrap method provides more reliable statistical inferences than those obtained by means of normality approximation of the Q-yield distribution when large samples are not available [6]. 5. BPB UPPER ONFIDENE BOUND FOR pp(q) Unlike the most common PIs, pp(q) is a the-smaller-the-better type index. That implies that when it is chosen as a critical indicator, a reliable analysis on process or suppliers Vol. 3, No. 11, Dec

8 performances should be based on its upper confidence limit, that is, on the worst conditions the process or the supplier could perform. If the processes under investigation are independently distributed, without any other requirement on neither specifications or distributions, the BPB upper confidence limit for pp(q), for each process, can be computed as follows: Step 1. onsider a single process. Step. Randomly select a sample of size n from the process population. This is called the original sample. Step 3. Use the bootstrap resampling method to draw, with replacement, a bootstrap sample of size n from the original one. Step 4. From the i-th (i=1,, B) bootstrap sample calculate pp(q). This estimate is denoted by * Ĉ pp( q )i. Step 5. Repeat steps 3 and 4 B times (B=1000). Step 6. Sort in ascending order the B values of Ĉ * pp( q ) i. They are referred to as Ĉ * () 1,Ĉ ( ),..., ( B) * pp( q ) pp( q ) Ĉ * pp ( q ). Step 7. Use the BPB confidence interval procedure to obtain the upper confidence bound of pp(q). Step 8. Repeat from steps 1 to 7 for each process. In terms of capability, the best process should be the one characterized by the lowest upper confidence limit. The entire procedure is implemented in a MATLAB 7.0 code. 6. AN APPLIATION In order to demonstrate how to use the upper confidence bound of pp(q) in making decisions, the proposed procedure is applied to data available in [4]. These data refer to two suppliers, who provided aluminium foil materials to an electronics company. The capability analysis is conducted on the voltage of aluminium foil which is considered one of the most important characteristic of the foils when they are used as capacitors components. The production specifications, expressed in [WV], have been set to USL= 530, T=, LSL=510. If the voltage exceeds specification limit, the aluminium foil must be rejected. Statistical information about the random samples drawn from the aluminium foil production process are listed in table 3. The 95% unilateral confidence interval of pp(q) has been derived for both the suppliers. For the first one the upper confidence bound (UB) of pp(q) resulted equal to 0.701, so that the process can be nearly considered super, for the second one the simulation produced a pp(q) upper confidence bound of , so the process is no more capable (Table 3). Evidently, with a 95% confidence, the first supplier presents better performances than the second one. This result confirms the consideration made in [4]. Table 3. Statistical information about the two samples pp(q) and pp(q) UB values. x [WV] S [WV] skewness kurtosis a3 a4 pp(q) pp(q) UB Supplier Supplier Vol. 3, No. 11, Dec

9 7. onclusion The process incapability index, pp, differs from the other commonly used PIs since it is easy to apply, analytically convenient and, above all, because of its property of keeping uncontaminated information about process accuracy and process precision. But as many other PIs, in its original formulation, can only be applied on normal processes. This limitation can be easily overcome if the non-normal pm formulation is used to derive a non-normal pp index, pp(q). Basing on lements s method, the required percentile values for non-normal PIs have usually been estimated by means of Pearson frequency distribution family. However, recent studies have proved that the percentiles of the Pearson distribution system can t be considered very reliable when used in capability analysis. The use of the Burr XII distribution is suggested. On the other hand, the Burr XII distribution appears particularly convenient since it is expressed in closed form and its percentiles can be calculated by algebraic transformations. In this study, the pp(q) based on Burr XII distribution is proposed. Because of its mathematical intractability, its unilateral BPB confidence interval has been calculated by the bootstrap resampling method. The upper confidence limit of pp(q) can be considered an efficient indicator when suppliers and processes performances have to be compared or monitored; moreover, the procedure followed to calculate it can be easily implemented and very poor statistical background is sufficient for handling its results. A numerical example presented confirmed the considerations obtained using other approaches. REFERENES: [1] lements, J.A. Process capability calculations for non-normal distributions. Quality Progress, Vol, pp 95-98, [] Liu, Pei-His; hen, Fei-Long Process capability analysis of non-normal process data using the Burr XII distribution. International Journal of Advanced Manufacturing Technology, Vol 7, pp , 006. [3] hen, K.L.; hen, K.S.; Li, R.K. Suppliers capability and price analysis chart. International journal of production economics, Vol 98, pp , 005. [4] Huang, M.L.; hen, K.S. apability analysis for a Multi-Process Product with Bilateral Specifications. International Journal of Advanced Manufacturing Technology, Vol 1, pp , 003. [5] Pearn, W.L.; Shu, Ming-Hung Measuring manufacturing capability based on lower confidence bound of pmk applied to current transmitter process. International Journal of Advanced Manufacturing Technology, Vol 3, pp , 004. [6] Pearn, W.L.; hang, Y..; Wu, hien-wei Bootstrap approach for estimating process quality yield with application to light emitting diodes. International Journal of Advanced Manufacturing Technology, Vol 5, pp , 005. [7] Greenwich, M.; Jahr-Schaffrath, B.L. A process incapability index. International Journal of Quality and Reliability Management, Vol 1(4), pp 58-71, [8] Wu,..; Kuo, H.L.; hen, K.S. Implementing process capability indices for a complete product. International Journal of Advanced Manufacturing Technology, Vol 4, pp , 004. [9] Johnson, N.L. System of frequency curves generated by methods of translation. Biometrika Vol 36, pp , [10] Box, G.E.P.; ox, D.R. An analysis of transformation. Journal of the Royal Statistical Society: Series B, Vol 96, pp 11-43, Vol. 3, No. 11, Dec

10 [11] Sommerville, S.; Montgomery, D. Process capability indices and non-normal distributions. Quality Engineering, Vol 19(), pp , [1] Pern, W.L.; Kotz, S. Application of lements method for calculating second and third generation process capability indices for non-normal Pearsonian populations Quality Engineering, Vol 7(1), pp , [13] hen, K.S. Estimation of the process incapability index ommunications in statistics: Theory and Methods, Vol 7(4), pp , [14] hen, K.S.; hen, K.L.; Li, R.K. ontract manufacturer selection by using the process incapability index pp International Journal of Advanced Manufacturing Technology, Vol 6, pp , 005. [15] Pearn, W.L.; Ko,.H.; Wang, K.H. A multiprocess performance analysis chart based on the incapability index pp: an application to the chip resistors Microelectronics Reliability, Vol 4, pp , 00. [16] Burr, I.W. umulative frequency functions Annals of Mathematical Statistic, Vol 13, pp 15-3, 194. [17] Burr, I.W.; islak, P.J. On a general system of distribution, I. Its curve characteristics, II. The sample median. Journal of American Statistical Association, Vol 63, pp , [18] Burr, I.W. Parameters for a general system of distributions to match a grid of 3 and 4. ommunications in Statistics, Vol (1), pp 1-1, [19] Wang, F.K.; Keats, J.B.; Zimmer, W.J. Maximum likelihood estimation of the Burr XII parameters with censored and uncensored data. Microelectronics Reliability, Vol 36(3), pp , 1996 [0] Wang, F.K. A new model with bathtub-shaped failure rate using an additive Burr XII distribution. Reliability Engineering and system safety, Vol 70, pp , 000. [1] Abdel-Ghaly, A.A.; Al-Dayian, G.R.; Al-Kashkari, F.H. The use of Burr type 1 distribution on software reliability growth modelling Microelectronics Reliability, Vol 37(), pp , [] Ali Mousa, M.A.M.; Jaheen, Z.F. Statistical inference for the Burr model based on progressively censored data. omputers and Mathematics with Applications, Vol 43, pp , 00. [3] Efron, B. Bootstrap methods: another look at jackknife. Annals of Statistics, Vol 7, pp 1-6, [4] hen, J.P.; Tong, L.I. Bootstrap onfidence Interval of the Difference between Two Process apability Indices. International Journal of Advanced Manufacturing Technology, Vol 1, pp 49-56, 003. [5] hen, J.P.; Pearn, W.L. Testing process performance based on the yield: an application to the liquid-crystal display module. Microelectronics Reliability, Vol 4, pp , 00. Vol. 3, No. 11, Dec