Contributions to Multivariate Process Capability Indices

Transcription

1 LICENTIATE T H E S I S Contributions to Multivariate Process Caability Indices Ingrid Tano

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3 Contributions to Multivariate Process Caability Indices Ingrid Tano Luleå University of Technology Deartment of Engineering Sciences and Mathematics Division of Mathematical Sciences

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5 Abstract The work resented in this thesis considers multivariate rocess caability indices (MPCIs) with focus on confidence intervals and tests for MPCIs. It also includes a case study, where multivariate statistical analysis and MPCIs are alied to data from a thermal sraying rocess at Volvo Aero Cororation. A rocess caability index measures the ability of a rocess to satisfy customers demands. Since the knowledge about the rocess erformance is based on a random samle, it is imortant to be able to handle the uncertainty this imlies. If the uncertainty is not dealt with and the estimated index is used directly without considering, e.g. a confidence interval, an erroneous conclusion may be drawn about the rocess caability. The thesis consists of a summary and of three aers, one of which has been resented at an international conference, one has been acceted for ublication in an international journal, and the third has been submitted for ublication. In Paer I, multivariate statistical analysis is used for screening and tentative model building to describe the relationshi between the orosity and the heat conductivity in the thermal sraying rocess. Object-oriented finite element analysis (OOF) is subsequently used for verification of the statistical model. The new aroach functions well and confirms the findings from the statistical model. Paer II reviews and comares four different MPCIs with confidence intervals, requiring multivariate normal distribution. MPCIs are needed when the quality characteristic of interest is multivariate, i.e. when the quality characteristic consists of several correlated variables. The review shows that more research is needed to obtain an MPCI with a confidence interval or a test that works roerly. In articular, it is required that a stated value of the MPCI at least limits the robability of nonconformance in a known way. A drawback is also elucidated with existing MPCIs based on rincial comonent analysis. Paer III resents some method develoment with the urose of meeting the deficiencies identified in Paer II. A new index together with a confidence interval and a decision rocedure is develoed that converts a multivariate situation into a famil-

6 iar univariate situation. Well-known statistical theory for univariate rocess caability indices can then be used. Proerties, like significance level and ower, of the roosed decision rocedure is evaluated through a simulation study in the two-dimensional case. A comarative simulation study between our new MPCI and an MPCI reviously suggested in the literature is also done. These studies show that our roosed MPCI with accomanying decision rocedure has desirable roerties and is worth to study further.

7 Acknowledgements Starting my doctoral studies at the age of forty with three children aged, 7 and 9 was robably not a very wise move. But here I am, halfway, and I consider it to have been very beneficial to send all my working hours develoing myself. During this time many colleagues, friends and family members have encouraged and suorted me, which I really areciate. Therefore many thanks to: -My managers, Liselott Lycke & Jonas Tosteby, who let me have this great oortunity for caacity building. -My assisstant suervisors Jan Wigren, Volvo Aero Cororation, and Per Nylen, University West, for having faith in me. -My suervisor Kerstin Vännman: The journey makes the effort worthwhile and not the destination, and we have certainly made this journey together. - Members of the Thermal Sray Grou, who always make me feel welcome even if I am a statistician, joining you very seldom. -My dear, dear colleagues in mellanrummet, who have to listen to my angst and agony, endlessly suorting me. -My female friends Ak, Camilla, Eva and Ulrika, dearly loved, who follow me thru life sharing hainess and sorrows, roviding me with laughter, energy and love. -My mother, who once a week throughout these years has icked u the children from daycare and school, giving me and my husband oortunity to catch u with work. -My beloved children: Hektor, Rakel and Torkel, the most imortant ersons in my life who make me see things in a different ersective and force me to be resent and not miss out of life. -My husband Robert, always standing by me, working art time to make the familyuzzle work; never, ever comlaining of all the work and travelling I have carried out. You truly want my best and that is real and ure love, and I am honoured to return the same to you. Finally, I do wish my father could have been here. He would have been the roudest of all. This thesis is dedicated to him, Thor Andersson, who assed away in November 00. Ingrid Tano, Trollhättan, December 0 3

8 Publications The following aers are included in the thesis: Paer I- Tano, I., Guta, M., Curry, N., Nylén, P. and Wigren, J. (00), Relationshis between coating microstructure and thermal conductivity in thermal barrier coatings -A modeling aroach. In: ITSC00, International thermal sray confernce & exosition 00, Thermal sray: Global solutions for future alication. Singaore: DVS-ASM, Paer II- Tano, I. & Vännman, K. (0), Comaring confidence intervals for multivariate rocess caability Indices. Quality and Reliability Engineering International. DOI:0.00/qre.50 Paer III- Tano, I. & Vännman, K. (0). A multivariate rocess caability index based on the first rincial comonent only. Submitted for ublication 4th of January 0. 4

9 Contents.Introduction Relationshis in the thermal sraying rocess An evaluation of MPCIs with confidence interval A new caability index, C,TV Concluding remarks and future research References Paers Paer I Paer II Paer III

10 .Introduction Caability analysis is a suitable tool when there is a need to understand the ability of a rocess to erform against customers demands. Juran s definition of a rocess erformance is what a rocess actually does. Comaring it with customers demands comares it to what the rocess should do. Caability analysis was initially erformed by lotting rocess data, e.g. using histograms, control charts or robability lots (see Montgomery ), and comaring these visually with the uer and lower secification limits, USL and LSL (Figure ). If the rocess erformance is outside the secification limits, there is a high risk of roducing oor roducts with soilage and/or rework as a result. Percent 99, , LSL -3 Probability Plot of X Normal X USL 3 Frequency Histogram of X LSL X USL 3 Figure. Examles of a robability lot and a histogram with uer and lower secification limits using the same dataset. Since the evaluation was erformed visually and conclusions could differ from erson to erson there was a need for a more standardized way to evaluate rocess caabilty. In 974, Juran 3 resented the caability index C, which made it ossible to determine whether a rocess is caable or not by calculating a unitless value. The index is exressed by USL LSL C () 6 6

11 where is the rocess standard deviation. In the exression for C, stands for rocess (see Bergman & Klefsjö 4, age 64). When rocess caability indices started to gain accetance in the 980s, a requirement for a caable rocess was that C, which when C, means that the length of the tolerance interval is equal to six standard deviations of the rocess. The index could also be exressed roughly as customers demand C () rocess erformance and when C, the customers demands and the rocess erformance are equal. When this is the case and the rocess is normally distributed, 99.73% of the rocess erformance lies within 3. The robability of non-conformance, PNC, ( ) i.e. that the rocess erforms outside the secification limits, is then The index C catures only the rocess variation and does not take the rocess location or the closeness to target into consideration, i.e. it catures only the otential caability. In 986, Kane 5 develoed C k in order to take the rocess location also into consideration. However, this index does not consider whether the rocess location,, diverges from the target, T, or not. The Cm index overcomes this roblem, but assumes that the target is in the middle of the secification interval. In 99, Pearn et al. 6 develoed C, which is a combination of C and C. Figure resents a mk summary and visualization of the above indices. Several caability measures have been subsequently resented after the late 980s, see, e.g. Kotz & Johnson 7, Siring et al. 8, Yum & Kim 9 and Wu et al. 0 The indices resented in Figure are based on the assumtion that the rocess is normally distributed. This assumtion is essential and should not be neglected, since a deviation from normality will mislead the interretation of the rocess caability with resect to PNC. ( ) Other distributions, such as the t distribution and skew distributions, do not have the same interretation ossibility, see Somerville & Montgomery. If a rocess arameter is not normally distributed, caability indices can be k m 7

12 calculated by, e.g., transforming the data, see Wu et al. or by using indices that do not require normality, see, e.g. Pearn & Kotz 3. Juran 3,974: Hsiang 4, 985 and Chan et al. 5, 988: C USL LSL 6 C m 6 USL LSL T Kane 5,986: Pearn et al. 6, 99 C k USL LSL min, 3 3 C mk min USL LSL, 3 T 3 T Figure. Visualizations and descritions of imortant, early caability indices. Another essential assumtion when calculating a caability index is that the rocess is stable or under statistical control. This means that it has only natural variability, which is the variability that arises by chance. If this is the case, the rocess is redictable. It is common to use a control chart to monitor whether a rocess is stable or not, see Montgomery. The author s interest in rocess caability comes from Volvo Aero Cororation, Sweden, who desires to secure the rocess caability of a thermal sraying rocess. 8

13 In ractice a random samle is normally required when evaluating rocess erformance. Then every samle is a snashot of the rocess erformance and it is very unlikely that it shows the whole, true, icture. Since the knowledge about the rocess erformance is based on a random samle, it is imortant to be able to handle the uncertainty this imlies. For the univariate indices resented above, statistical theory stiulates how to treat the uncertainty. By resenting, e.g., a confidence interval, it is ossible to draw conclusions with a high confidence level, see, e.g. Pearn & Kotz 3. If the uncertainty is not dealt with and the estimated index is used directly without considering the confidence interval, an erroneous conclusion may be drawn about the rocess caability. The thermal sraying rocess at Volvo Aero Cororation requires knowledge of the rocess caability with many, correlated variables simultaneously. A first naïve aroach might be to ignore the multivariate situation and consider a univariate caability index for each variable searately. And then define the rocess as caable if each univariate caability index shows a caable rocess. However, then the robability of non-conformance for the rocess can be very large even if each univariate index shows a caable rocess. This is illustrated in below. To simlify, consider the case when the quality characteristic is two-dimensional and the i:th variable has secification interval [ LSL, USL ], i,. Let the tolerance region R be defined by a rectangular region with the lengths of the sides equal to the lengths of the univariate secification interval. Then i i X P NC P R P LSL X USL LSL X USL. (3) If X and X are indeendent stochastic variables, equation (3) corresonds to PNC PNC PNC ( ) PLSL XUSL PLSL X USL ( ) ( ). (4) For the more general case when the dimension of the quality characteristic is v we get similarly 9

14 P NC P NC P NC P NC v ( ) ( ) ( )... ( ) (5) This means that and if PNC ( ) for each univariate variable, i,,..., v, then i PNC ( ) ( 0.007) v. For v = 3 we then get PNC 3 ( ) (0.9973) which indicate a non-caable rocess although PNC ( ) for each univariate i variable. The roblem with using the naïve aroach is even more evident when, e.g. v = 5, since then 5 PNC ( ) ( 0.007) 0.034, which clearly shows a noncaable rocess. If the univariate variables are indeendent Equation (5) can be used to define the smallest value of the univariate caability index that would give PNC ( ) 0.007for the multivariate situation. As an examle, consider the case P( NC )... P( NC ), 5 when v = 5. We get PNC, from Equation (5). This /5 ( ) (0.9973) corresonds to C.53. Hence, if the rocess is on target and each univariate C.53 the robability of non-conformance for the multivariate situation will be But if the variables are correlated we cannot use Equation (5) and it is not ossible to easily obtain what value of the univariate index is required to give P( NC) for the multivariate situation. This is one reason for the need of MPCIs. An MPCI is able to manage information from many correlated variables simultaneously, resulting in a single value that can be used to determine whether a multivariate rocess is caable or not. Many MPCIs have been described in the literature, and they manage the correlations resent in the multivariate situation in many different ways. See, e.g. Goethals & Cho 6 and Shinde & Khadse 7. Some calculate the ratio between should do/actually do, but now base this measure on an area or a volume, instead of an interval. Others use, e.g. rincial comonent analysis, PCA. In Jolliffe 8 PCA is described further. Not many of the MPCIs described in the literature manage the estimation and uncertainty due to random samle, i.e. have confidence 0

15 intervals or tests nor do they secify the PNC ( ) related to the MPCI. The area of MPCIs is an area under develoment. The overall aim of the work resented here is to contribute to the area of MPCIs with focus on confidence intervals and tests for MPCIs. More secific aims are to review the existing MPCIs with confidence intervals or tests and to evaluate their alicability on data from a thermal sraying rocess. Volvo Aero Cororation wanted to determine the caability of its thermal sraying rocess as early as ossible in their rocess, and therefore the relationshis between the variables in the rocess were established first and are resented in Paer I. The review and evaluation of MPCIs with confidence intervals are subsequently resented in Paer II. The results in Paer II show that existing MPCIs do not work as well as desired. Hence, a new MPCI together with a confidence interval and a decision rocedure are resented in Paer III. Section describes the thermal sraying rocess while Section 3 resents the results from Paer I, relationshis in the thermal sraying rocess, among other material. Sections 4 and 5 resent results from Paer II, and start with a short review of MPCIs found rior to the start of Paer II. These sections evaluate the MPCIs found with confidence intervals or tests. Here the concentration is on the simlest form of an MPCI, MC, since it is simle and at the same time sufficient to identify the essential ros and cons. Section 6 resents a new index C TV,. Finally, conclusions and future work are resented in Section 7.. The case studied: A thermal sraying rocess The author came into to contact with Volvo Aero Cororation through the thermal sraying research grou at University West, Trollhättan, Sweden. The thermal sraying rocess is studied in the work resented in Paers I and II, and can be characterized as a high temerature sray ainting to rotect the srayed surface against, heat, corrosion, erosion, and other effects. It is a surface-coating rocess in which

16 melted or heated materials are srayed onto a surface. A rough outline of the thermal sraying rocess is shown in Figure 3. Volvo Aero Cororation wants to evaluate the rocess caability as early as ossible in the rocess, referably already at the flame stage, using the correlated variables Temerature, Light intensity and Velocity, where the variables Light intensity and the Velocity are the light intensity and velocity of the articles, resectively, and are measured in the flame. To be able to do this more knowledge is needed about the relationshi between the variables in the rocess. Today, e.g., customers demand focuses on orosity, while heat conductivity is actually a better variable. The relationshi between orosity and heat conductivity, however, is not comletely defined. A fundamental understanding of the relationshis between coating microstructure and heat conductivity is imortant in order to understand the influence of coating defects, such as delaminations and ores, on the heat insulation of thermal barrier coatings. Figure 3. Overview of the thermal sraying rocess. The sraying material, in owder or wire form, is fed into a hot gas jet flame (at C ), where it melts. The orosity desired for the final coating deends on the desired roerties of the coating, and whether it is rimarily intended to rotect against heat, corrosion, erosion, or another effect. Rocket nozzle extensions are among the objects coated at Volvo Aero Cororation, Sweden, see Figure 4, while others are gas turbines, which must be rotected from heat.

17 Figure 4. Thermal sraying of a rocket nozzle. Porosity is the fraction of hollow sace throughout the total volume. Different amounts of hollow sace corresond to different orosities, and thus different heat conductivities. Saces arise when a melted material is srayed onto a surface. The attern and shae of the hollow saces differ. Figure 5 shows an image of a srayed surface taken in an otical microscoe. The dark sots corresond to hollow saces, and software is able to differentiate between cracks and ores by using the height to length ratio. Analyzing the arrangement of ixels makes it ossible to distinguish between ores in contact with cracks, free ores and free cracks, and to define the crack angle. Figure 5. An image showing different sizes and forms of ores and cracks (dark areas). Section 3 will establish the relationshi between orosity and heat conductivity, and then the relationshi between orosity and the correlated, in-flame variables. 3

18 3. Relationshis in the thermal sraying rocess The object-oriented finite element model, OOF, is a relatively new finite element model aroach that can be used to determine macroscoic roerties from images of real microstructures. OOF can be used to determine various roerties of a structure, and it is imortant to know what it is that is desired. The work resented in Paer I, therefore, involves multivariate statistical analysis to screen and to build tentative relationshis between orosity and heat conductivity. OOF is then used to verify the obtained statistical model. An exeriment with 0 different coatings was erformed in which the heat conductivity was measured for each, as were the shaes and amounts of hollow saces (orosity). The orosity was characterised by 7 shae variables (Table ), and measured by the roortion of ixels included in the secified shae in each image. That is e.g., X 5 shows the roortion of ixels included in cracks having the crack angel between 5 and 30 in the images. Table. Porosity variables showing the roortion of ixels included in the secified shae variable. X only the cracks in contact with ores X 0 75 X 90, crack angle 0 X free cracks X 90 X 05, crack angle X only the ores in contact with cracks X 3 05 X 0, crack angle X free ores 0 X 35, crack angle 4 X 5 X 5 X 6 X 6 X 7 X 7 X 8 X 8 X 9 X 9 0 5, crack angle , crack angle , crack angle , crack angle X crack angle X 3 3 X X 4 X X 5 X X , crack angle 50 65, crack angle 65 80, crack angle 0 and 80, crack angle The orosity variables are correlated, and thus the situation have multicollinearity, see, e.g. Afifi 9. In this case, the estimates of the regression coefficients are unstable and have large standard errors. Therefore, rincial comonent regression, PCR, is carried out first, see, e.g. Jolliffe 8 in order to establish the relationshis between the 4

19 heat conductivity and the orosity variables, and thereafter derived the following regression model: heat conductivity 0.Z 0.09Z 0.07Z 0.06Z 0.05Z 0.05Z Z +0.03Z 0.03Z 0.03Z (6) where Zi equals the Z scores of X i. The orosity variables describing cracks and ores in contact, X 3 and X, have the largest regression coefficients, imlying that they have largest imact on heat conductivity. The analysis by OOF confirmed the findings of the statistical model, and both methods indicate that cracks and ores in contact affect the heat conductivity more than crack angle, free cracks and free ores. The statistical model is based on ten observations only and should thus be considered tentative, but it does allow the relationshi between the orosity and heat conductivity to be suggested, and this suggestion is confirmed by OOF. Another dataset from a thermal sraying rocess at Volvo Aero Cororation was used to determine the relationshi between orosity and the in-flame variables intensity, temerature and velocity (Figure 4). The dataset consisted of 70 observations from three different sray guns, denoted by Gun_, Gun_ and Gun_3. The relationshi between the in-flame variables and orosity was established through rincial comonent regression, PCR, and the regression equation is exressed by: Porosity 6.0.4velocity.4intensity 0.0tem 6.7Gun_ 7.33Gun_3 (7) The Gun variables are dummy variables, such that Gun_ equals when data come from sray Gun_ and 0 otherwise and similarly for Gun_3. The regression equation for Gun_ is obtained by setting the coefficients for Gun_ and Gun_3 to zero. The variables velocity and intensity have equal regression coefficients but with oosite signs, while temerature has the smallest regression coefficient. The regression model has a high ability to make redictions, with R ( red) 87%. 5

20 Equations (6) and (7) define tentative relationshis that enable us to determine heat conductivity from the in-flame variables, using orosity as an intermediate stage. Now there is a need to find a suitable MPCI for the in-flame variables that has the ability to handle the uncertainty that arises due to samling. 4. A review of multivariate rocess caability indices It is often necessary to evaluate the caability of a rocess having many correlated variables, but this is comlex to comrehend and calculate. Research on multivariate rocess caability indices, MPCIs, started to be ublished in the late 980s, but it was not until 005 that the number of ublications increased. Table resents a summary of MPCIs found in the literature ublished from 000 and onward and some MPCIs ublished earlier as well, since they are commonly occurring references. Many of the indices resented in Table utilize the univariate aroach by using a threshold value of to determine whether a rocess is caable or not. Assumtions such as multivariate normal distributions, multivariate stable rocesses, and the use of a confidence interval or a test are also imortant, in order to interret the results roerly and in order to handle the uncertainty that samling introduces. MPCIs can roughly be divided into four different grous, as suggested by Shinde & Khadse 7. The grous are MPCIs that are a) based on the ratio of a tolerance region to a rocess region b) based on the robability of the non-conforming roduct c) based on rincial comonent analysis (PCA) d) other. Table shows that there are many MPCIs described in the literature, but confidence intervals have been derived for only 4 of the 6 resented. Furthermore, six of 6 indices do not require multivariate normal distributions, and half of them are based on the ratio of a tolerance region to a rocess region. It can also be noted that there is an abundance of different names for the MPCIs. 6

21 Table. Summary of indices found in the literature mainly from the eriod In the table MND denotes that a multivariate normal distribution is required and Test is used to show that a test or confidence interval exists. Year Author Index MND Test Grou 00 Goethals & Cho 6 yes no d 00 Pan &Lee 0 NMC m yes yes a 009 Shinde & Khadse 7 M and M yes no b, c 009 González & Sánchez s yes and C n no 009 Ahmad et al. no no b 009 Shahriari et al. 3 NMPCV yes no a 008 Castagliola et al. 4 BC and BC no no b 007 Pearn et al. 5 MC yes yes a 006 Wang 6 no yes d 005 Wang 7 MC and MC k yes no a, c 005 Castagliola et al. 8 BCP and BCPK yes no b 00 Yeh & Chen 9 MC f no no b 000 Wang & Du 30 yes and MC and MC c no yes a, c MC, MC, MC k m 998 Wang & Chen 3 and MC yes no a, c MC mc PNC Total 995 Shahriari et al. 3 MPCV Yes no d 994 Chen 33 MC yes no a 993 Taam et al. 34 MC m yes no a P MC c Paer II reviews and comares the three different methods available for calculating confidence intervals for MPCIs that require a multivariate normal distribution. The first two are based on the ratio of a tolerance region to a rocess region, Pearn et al. 5 and Pan & Lee 0. The third is based on PCA, Wang & Du 30. Furthermore, the index by Wang 7, which is based on PCA, was reviewed and comared. Wang 7 did not resent any confidence interval for the index but since it is based on PCA the mk PK no b, c 7

22 confidence interval can be calculated following the method resented by Wang & Du 30. For more details about PCA, see, e.g., Johnson and Wichern 35. The four methods to be comared are described in more detail below. First, the assumtions and notations are stated. Let X ( X, X,..., X )' v denote the v-dimensional quality characteristic that is studied. It is assumed that X is distributed according to a multivariate normal distribution N (, ) v, where is the mean vector and the variance-covariance matrix. Furthermore, let S denote the samle variancecovariance matrix. It is exected that each univariate quality characteristic X i has a given secification interval [ LSL, USL ], i,,... v. Since it is of interest to find out i i if the MPCI exceeds, only the lower confidence bound of one-sided confidence intervals is considered. Method : Pearn et al. The index is exressed as MC Vol( R ) Vol(modified tolerance region), (8) () v/ / Vol( R ) ( ), v / v where R is the largest ellisoid centred at the target value comletely within the original rectangular tolerance region, and R is the ellisoid that contains 99.73% of the multivariate normal distribution. The index MC is estimated by MC () Vol( R ) v/ / ( ) S, v / v and the lower 00 % confidence bound is exressed as () F ( ) Y MC v. ( n ) (9) (0) 8

23 In Equation (0), F ( ) denotes the quantile of the distribution of the random Y variable Y, where Y is defined as a roduct of v indeendent -distributed random variables with n, n,..., n v degrees of freedom, resectively. Method : Pan & Lee They exressed their index as MC Vol( R ) A * Vol R () 3 /, () where the elements of the matrix * A are given by * USL LSL USL LSL i i j j A ij ij v, v, () and ij is the correlation coefficient between the ith and jth univariate quality characteristic. The index () MC in () is estimated by MC A S () * / (3) and the lower 00(- )% confidence bound is MC () w, (4) v v where w is the quantile of the distribution of /( n ) and i n i -distributed random variable with n i degrees of freedom. n denotes a i Method 3: Wang & Du Wang & Du 30 resented a confidence interval based on the index (3) MC that had been introduced by Wang & Chen 3. (3) MC is based on rincial comonent analysis, 9

24 PCA. The first rincial comonent, PC, of a rincial comonent analysis accounts for as much of the variability in the data as ossible, and each succeeding PC accounts for as much of the remaining variability as ossible. Furthermore the PCs are uncorrelated. For more details about PCA, see, e.g. Johnson and Wichern 35. The index resented by Wang & Du 30 is exressed as MC (3) v C i, PCi / v. (5) In (5), CPC, is the univariate caability index for the ith rincial comonent, PC i i, and is given by C USL LSL (6) 6 PCi PCi, PC, i i where i is the PC s i eigenvalue and also its variance. The uer- and lower- secification limits of PC i in (3) are defined as, see Wang & Du 30, USL uusl and PCi i X LSL ulsl, (7) PCi i X where u i is the eigenvector corresonding to PC, USL, USL,..., i USL USL, X v and LSL LSL, LSL,..., X LSL. Furthermore, the index v 3 MC is estimated by MC (3) v C i, PCi / v, (8) where USL LSL PCi PCi C PC, i. (9) 6 i The aroximate lower confidence bound, 00 %, given by Wang and Du 30, is 0

25 m C i n, PC, i n m, (0) where -distribution with n degrees of freedom. denotes the quantile of a n, Method 4: Wang Wang 7 roosed an MPCI also based on PCA, based on the use of a weighted geometric mean of C PC, in Equation (6), with the eigenvalues, i,,..., m i i as weights. The objective was to give the largest weight to the index for PC, the second largest weight to the index for PC, and so on. The MPCI Wang 7 suggested is MC (4) m C i i, PCi / i, () where CPC, is given in Equation (6) and m is the number of PCs selected. The in- i dex is estimated by MC (4) m C i i, PCi / i () and C PC, is given by Equation (9). Paer II uses the idea of Wang & Du 30, and i suggests the following aroximate lower confidence bound for MC, (4) m n, C PC, i i n i / i. (3) The indices with confidence interval resented above will now be comared using the data from Volvo Aero Cororation resented in Section 3.

26 5. An evaluation of MPCIs with confidence interval Paer II resents an evaluation of the four MPCIs described in detail in the revious section, for the in-flame variables from Volvo Aero Cororation that were studied. Table 3 shows the secification intervals of the variables. Figure 6 shows that all of the observations fall within the tolerance region for this samle. Table 3. The in-flame variables studied and their secification intervals. Variable Light intensity (unitless) Temerature (ºC) Notation X X X 3 Velocity (m/s) LSL, USL 394,603 95,668 98,8 i i 3D Scatterlot Temerature Intensity Velocity Figure 6. A 3D scatterlot of the in-flame variables where the box shows the tolerance region. It is assumed that the in-flame variables are multivariate normally distributed. Normal robability lots of the individual X variables do not contradict this assumtion, as can be seen in Figures 7 a)-c). The observations follow the line rather well.

27 a) b) c) Figure 7. Normal robability lots of the in-flame variables. a) X, b) X, and c) X 3. All of the MPCIs evaluated use a threshold value of to deem a multivariate rocess caable or not. Results are shown in Table 4, and it can be seen that all estimated indices are fairly similar, ranging from 0.89 to.05. Method 3 has the smallest estimated index and Method 4 the largest. Table 4. Summary of the results obtained when alying the methods to the case studied Method Estimated index Lower confidence bound, Pearn et al. 5 MC Pan & Lee 0 MC Wang & Du 30 MC Wang 7 MC Since the indices are based on a samle, it is not aroriate to use only the estimated indices to deem a rocess caable or not: the lower confidence bound must be considered instead. Table 4 shows that the lower confidence bounds vary more than the estimated index, and that Method has the lowest bound (0.67) and Method 4 the highest (0.90). All lower confidence bounds lie below the threshold value, and thus none of the methods deem the rocess to be caable. The investigation resented in Paer II, however, shows the need of more research. Method does not cature the correlation structure roerly when the correlations between the univariate quality 3

28 characteristics are high, and hence misjudges the rocess caability with resect to the robability of non-conformance. However, when the correlations among the univariate quality characteristics are small or moderate, as in our case, Methods and are fairly similar. The MPCIs based on PCA (Methods 3 and 4) transform the tolerance region in an inaroriate way, which strongly affects the result. Shinde & Khadse 7 resent an aroriate way of calculating the transformed tolerance region for the PCs, which, however, becomes comlicated for v. Paer II shows by a simle examle that a threshold value of does not always corresond to PNC ( ) in the multivariate case. Of the four indices with confidence intervals for MPCI considered, the index by Pan & Lee 0 is to be referred. The work resented in Paer II shows that more research is needed to obtain an MPCI with a confidence interval or a test that works roerly. In articular, it is required that a stated value of the MPCI at least limits the robability of nonconformance in a known way. 6. A new caability index, C,TV This section resents results from Paer III, where a new index denoted C TV,, is resented. In C TV,, " TV " stands for Tano & Vännman and/or transformed variables. The essential idea behind our suggested MPCI is illustrated in Figure 8. Figure 8. An illustration of the leading idea behind our roosed index. If PCA is alied on the variance-covariance matrix of X the first PC will contain as much variability in the original data as ossible. But if a univariate variable, X i, has a large variance, and dominate the first PC, it does not necessarily imly that this variable is non-caable. Its caability deends on the relation to its secification interval. Furthermore, when doing a PCA most books recommend standardization before 4

29 the eigenvalue decomosition, when variables are measured on different scales to obtain variables that arise on an equal footing. Based on these asects a transformation of the original variables X, i,,..., v was suggested according to i X M i i X, i,,.., v. (4) TVi d i In Equation (4), M i denotes the midoint of i:th secification interval and d i denotes half the length of the of i:th secification interval. The effect of the transformation is illustrated in Figure 9, where v 3. It can be seen that the X -variable, before the transformation, with the smallest variance but most sread out relative its secification limits, i.e. X, becomes the transformed variable X TV with largest variance. X X TV LSL X USL X - 0 LSL X USL X 0 LSL X3 USL X3 0 Figure 9. An illustration of how the transformation affects the distribution of X TVi. The urose of the transformation was to obtain new variables on a common scale, such that those with largest variability relative to its secification interval contribute substantially in the first PC. Paer II shows that the secification limits for PC deend on the value of PC and vice versa. When the dimension,v, is greater than, this roblem becomes even more comlicated than when v. As a result of the roosed transformation the 5

30 new index, C TV,, is based on the first rincial comonent only where the PCA is based on the transformed variables, X TVi. According to the reasoning above, the variables with largest variability relative to its secification interval, i.e. the variables that are least caable, will contribute substantially in the first PC. If PC indicates a caable rocess all other PCs will also indicate a caable rocess, since their variances are smaller. On the other hand if PC indicates a non-caable rocess it does not matter if any of the other PCs indicate a caable rocess. It is still not ossible to claim the rocess to be caable. A consequence of this is that a multivariate situation has turn into a univariate situation and well-known statistical theory from univariate caability indices can be used to form confidence intervals or tests. Paer III describes a method of determining the uer and lower secification limits of PC. The roosed limits are USL PC max u LSLPC i,, (5) max u i resectively, where max u i is the maximum absolute value of the comonents in the first eigenvector corresonding to PC. Then the new index is defined in accordance with Equation () as C TV, USL LSL 6 max u 3 PC PC, (6) PC i where is the largest eigenvalue and the variance of PC. Furthermore, it is essential to know that when the rocess caability index exceeds a stated value this imlies a certain P(NC). The rocess is defined as caable if the roosed rocess caability index exceeds a given threshold value, k. As mentioned 0 before, it has been common to let k 0 in revious work concerning MPCIs. However, this will not be the case here. In Paer III it is shown, for v, that k deends 0 on the correlation coefficient,, the ratio between the variances, c / and the 6

31 magnitude of, where is defined to be the largest variance. It is also shown that needs to be varied with c and to obtain PNC values of k 0 for c 0.,0.,..., and 0.,0.,...0.9,0.95. Table 5 gives the corresonding to PNC ( ) The largest values of k 0 are obtained when c, i.e. when is small and. Table 5. Value of the threshold value k 0 corresonding to borderline cases where PNC ( ) for different,c. c In most situations, the arameters of the distribution for the quality characteristic studied are unknown, and a random samle is needed to estimate the arameters. Then an estimated index together with a decision rule neded to be formed to be able to decide whether a rocess can be claimed to be caable or not at a stated significance level,. A ossible estimator of C, TV is C TV, (7) max u i 3 where is the first eigenvalue of S. In aer III the hyotheses TV 7

32 H : C k H : C k 0 TV, 0 TV, 0 (8) is used with the following decision rule to deem a rocess caable or not n, reject H if C 0 TV, k. 0 n (9) The estimated value of k, k 0 0, is obtained from Table 5 by using c S / S and the samle correlation coefficient,. It is of interest to find out if the significance level is at most. The significance level is the robability to reject the null hyothesis when it is true or, equivalently, the robability to deem the rocess caable when it is not. Note that, since both the secification limits of PC and k 0 are estimates the actual significance level is not known. It is also of interest to study the ower of the test, i.e. the robability to deem the rocess caable when it is caable. In order to estimate the actual significance level for the decision rule in Equation (9) a simulation study is resented in aer III. The results when n50, n 00 is shown in Table 6. Result of more n-values is resented in aer III. It is seen that the estimated significance level is close to or less than 0.05 for both samle sizes, imlicating that the decision rule in Equation (9) works well with regard to the significance level of the test. The lowest values of the estimated significance levels are obtained for combination of large values of c and small values of. In aer III it is concluded that, although both k 0 and the secification limits for PC are estimated, our decision rule seems to have an actual significance level of aroximate 0.05 or smaller. This means that the nominal significance level can be trusted. However, the low actual significance levels for some combinations of c and induces lower ower than necessary for these cases. 8

33 Table 6. Estimated significance level for different combinations of c, and giving PNC ( ) 0.007, when n 50,00 v and n = 50 n = 00 c = 0. c = 0.3 c = 0.5 c = 0.7 c = 0.9 c =.0 c = 0. c = 0.3 c = 0.5 c = 0.7 c = 0.9 c = The ower, i.e. the robability to deem the rocess caable when it is caable, of the decision rule in Equation (9) is investigated for three cases when the robability of non-conformance is much smaller than 0.007, in aer III. One of the cases corresonding to PNC ( ) , i.e. C 4/3 is shown in Table 7. As exected the ower is lowest when c is close to and is small. But the lowest estimated ower value, is larger than 0.55, according to Table 7. Table 7: Estimated ower corresonding to C 4/3with the PNC ( ) n 50 n 00 c = 0. c = 0.3 c = 0.5 c = 0.7 c = 0.9 c =.0 c = 0. c = 0.3 c = 0.5 c = 0.7 c = 0.9 c = In aer III a small comarative simulation study, when having a two-dimensional quality variable, between our new decision rule and the decision rule based on the index, MC, PL, suggested by Pan & Lee 0 was erformed. Their confidence interval corresonds to the result obtained by our rocedure. 9

34 7. Concluding remarks and future research The overall aim of the work resented in this thesis has been to contribute to the area of MPCIs with focus on confidence intervals and tests for MPCIs. This has been accomlished by reviewing the existing MPCIs with confidence intervals or tests, evaluating their alicability on data from a thermal sraying rocess at Volvo Aero Cororation and also roosing a new MPCI with an aroriate decision rocedure. Concluding remarks Volvo Aero Cororation wanted to determine the caability of the thermal sraying rocess as early as ossible in the rocess. Therefore the relationshis between the variables the rocess was established first and is described in Paer I. This work took a new aroach by combining a statistical modelling aroach with a finite element model aroach to create what is known as an object-oriented finite element model, OOF. Multivariate statistical analysis was initially used for screening and tentative relationshi building, followed by OOF for verification of the statistical model. The findings in the statistical model were confirmed by OOF. Even if the result is based on ten observations only and should be considered tentative, a otential method to identify relationshis in the thermal sraying rocess has been develoed. The review and evaluation of existing multivariate rocess caability indices with confidence intervals is resented in Paer II. The review shows that this is an area under develoment and that research about multivariate rocess caability indices started to be ublished during the late 980s, but it was not until 005 that the number of ublications increased. Only a handful of the MPCIs found have derived confidence intervals or tests. The threshold value equal to is used almost automatically, by analogy with the univariate situation, without reflecting whether the interretations may have changed. Paer II shows by a simle examle that a threshold value equal to does not always corresond to PNC ( ) in the multivariate case. 30

35 Paer III shows, when v, how PNC ( ) deends on the arameters, c /. Hence, needs to be varied with c and to obtain and P NC The method, used by, e.g. Wang 3 and Wang 30, of transforming the tolerance region using MPCIs based on PCA aeared to be inaroriate, and the use of the inaroriate method strongly affects the result. Shinde & Khadse 7 resented the aroriate method to calculate the transformed tolerance region which, however, becomes comlicated for v. Paer II shows that more research is needed to obtain an MPCI with a confidence interval or a test that works roerly and at a stated value of the MPCI infer a known P(NC). Such a new method should at least limit the robability of non-conformance in a known way. Paer III resents a new index, C, TV, with a confidence interval and a test, based on PCA. The index limits the robability of non-conformance in a known way. The new index is obtained by transforming the original variables relative to their secification interval. Next, a PCA is erformed using the transformed variables, in which the variables that are least caable, i.e. the transformed variables with largest variability, will contribute substantially to the first PC. If PC indicates a caable rocess, all other PCs will also indicate a caable rocess, since their variances are smaller. On the other hand, if PC indicates a non-caable rocess, it does not matter if any of the other PCs indicate a caable rocess. It is still not ossible to state that the rocess is caable. Hence, a multivariate situation has been converted into a familiar univariate situation. Finally, well-known statistical theory for univariate rocess caability indices can be used for drawing conclusions about rocess caability in the multivariate situation. Suggestions for future research Although the simulation studies erformed are quite limited they indicate that our roosed MPCI with accomanying decision rocedure is worth to study further. Esecially in the view of the lack of MPCIs, with confidence intervals and tests, such 3

36 that the knowledge that the rocess caability index exceeds a stated number imlies that the robability of non-conformance, or an uer bound of it, is known. The two-dimensional case, i.e. v, is studied in detail and more work is needed to obtain threshold values to define a rocess caable based on our index when v and simulation studies need to be done when v, which is more comlex. The threshold value, k 0, limits the robability of non-conformance in a known way and PNC ( ) deends on three different arameters when v the correlation, the ratio between the variances and the magnitude of the largest variance. When v, there is only one correlation, one ratio and one maximum level of the variance to consider. When v, the PNC ( ) will deends on more than these three arameters. There is still only one maximum level of the variance, but there are several correlations and ratios to handle. Future research could consider whether it is ossible to find out if one of the ratio or the correlation is more dominant than the other. Furthermore, the attention has been restricted to the simlest form of a MPCI, MC, measuring the otential caability only. Future work needs to include MPCIs taking location as well as closeness to target into consideration, i.e. generalizations of the univariate indices C k and C m. If a rocess cannot be deemed caable it would be nice to be able to trace which one of the original variables is least caable. This issue is of interest to investigate in the future. The new index requires several rather advanced stes in its calculation. From a ractitioner s oint of view there is a need to develo tools to make the calculations easier to do. 3

37 8. References. Juran JM. Juran on lanning for Quality, The Free ress: New York, Montgomery D. Statistical quality control A modern introduction, 6 edn Wiley, Juran JM. Quality control handbook, McGraw-Hill: New York, Bergman B, Klefsjö B. Quality from customer needs to customer satisfaction, Studentlitteratur, Kane VE. Process caability indices. Journal of Quality Technology 986; 8:. 6. Pearn. WL KS, Johnson. NL,. Distributional and inferential roerties of rocess caability indices. Journal of Quality Technology 99; 4(4): Kotz S, Johnson NL. Process caability indices - a review, Journal of Quality Technology 00; 34(): Siring F, Leung B, Cheng S, Yeung A. A Bibliograhy of rocess caability aers. Quality and Reliability Engineering International 003; 9(5): Yum B-J, Kim K-W. A bibliograhy of the literature on rocess caability indices: Quality and Reliability Engineering International 0; 7(3): Wu C-W, Pearn WL, Kotz S. An overview of theory and ractice on rocess caability indices for quality assurance. International Journal of Production Economics 009; 7(): Somerville SE, Montgomery DC. Process caability indices and non-normal distributions. Quality Engineering 996; 9():

38 . Wu C-W, Pearn WL, Chang CS, Chen HC. Accuracy analysis of the ercentile method for estimating non normal manufacturing quality. Communications in Statistics - Simulation and Comutation 007; 36(3): Pearn WL, Kotz S. Encycloedia and handbook of rocess caability indices : a comrehensive exosition of quality control measures, World Scientific: Singaore ; Hsiang TC, Taguchi, G. Tutorial on quality control and assurance - the taguchi methods. In: Joint Meetings of the American Statistical Association, Las Vegas, Nevada, Chan LCSSF. A new measure of rocess caability: Cm. Journal of Quality Technology 988; 0 (3 ). 6. Goethals PL, Cho BR. The develoment of a target-focused rocess caability index with multile characteristics. Quality and Reliability Engineering International 0; 7(3): Shinde RL, Khadse KG. Multivariate rocess caability using rincial comonent analysis. Quality and Reliability Engineering International 009; 5(): Jolliffe IT. Princial Comonent Analysis, Sringer-Verlag New York, Incororated: Secaucus, NJ, USA, Afifi AA, Clark C. Comuteraided multivariate analysis, vol. 3, Chaman & Hall, Pan JN, Lee CY. New caability indices for evaluating the erformance of multivariate manufacturing rocesses. Quality and Reliability Engineering International 00; 6(): González I, Sánchez I. Caability indices and nonconforming roortion in univariate and multivariate rocesses. International Journal of Advanced Manufacturing Technology 009; 44(9-0):

39 . Ahmad S, Abdollahian M, Zeehongsekul P, Abbasi B. Multivariate nonnormal rocess caability analysis. International Journal of Advanced Manufacturing Technology 009; 44(7-8): Shahriari H, Abdollahzadeh M. A new multivariate rocess caability vector. Quality Engineering 009; (3): Castagliola P, Castellanos JVG. Process caability indices dedicated to bivariate non normal distributions. Journal of Quality in Maintenance Engineering 008; 4(): Pearn WL, Wang FK, Yen CH. Multivariate caability indices: Distributional and inferential roerties. Journal of Alied Statistics 007; 34(8): Wang FK. Quality Evaluation of a Manufactured Product with Multile Characteristics. Quality and Reliability Engineering International 006; (): Wang CH. Constructing multivariate rocess caability indices for short-run roduction. International Journal of Advanced Manufacturing Technology 005; 6(-): Castagliola P, Castellanos. J-VG. Caability Indices Dedicated to the Two Quality Characteristics Case. Quality Technology and Quantitative Management, 005; (): Yeh AB, Chen H. A nonarametric multivariate rocess caability index. International Journal of Modelling and Simulation 00; (3): Wang FK, Du TCT. Using rincial comonent analysis in rocess erformance for multivariate data. Omega 000; 8(): Wang FK, Chen JC. Caability index using rincial comonents analysis. Quality Engineering 998; (): Shahriari H, Lawrence FP. A multivariate rocess caability vector. In: 4th Industrial Engineering Research Conference,

40 33. Chen H. A multivariate rocess caability index over a rectangular solid tolerance zone. Statistica Sinica 994; 4: Taam W SP, Liddy JW. A note on multivariate caability indices. Journal of Alied Statistics 993; 0(3):. 35. Johnson RA, Wichern DW. Alied multivariate statistical analysis, Pearson Prentice Hall: Uer Saddle River, N.J.,

41 Paer I

42

43 Relationshis between Coating Microstructure and Thermal Conductivity in Thermal Barrier Coatings A modelling Aroach I. Tano, M. Guta, N. Curry, P. Nylén and J. Wigren, Trollhättan/S Fundamental understanding of relationshis between coating microstructure and thermal conductivity is imortant to be able to understand the influence of coating defects, such as delaminations and ores, on heat insulation in thermal barrier coatings. Object-Oriented Finite element analysis (OOF) has recently been shown as an effective tool for evaluating thermo-mechanical material behaviour, because of this method s caability to incororate the inherent material microstructure as an inut to the model. In this work, this method was combined with multi-variate statistical modelling. The statistical model was used for screening and tentative relationshi building and the finite element model was thereafter used for verification of the statistical modelling results. Characterisation of the coatings included microstructure, orosity and crack content and thermal conductivity measurements. A range of coating architectures was investigated including High urity Yttria stabilised Zirconia, Dysrosia stabilised Zirconia and Dysrosia stabilised Zirconia with orosity former. Evaluation of the thermal conductivity was conducted using the Laser Flash Technique. The microstructures were examined both on as-srayed samles as well as on heat treated samles. The feasibility of the combined two modelling aroaches, including their caability to establish relationshis between coating microstructure and thermal conductivity, is discussed.. Introduction Plasma srayed ceramic coatings are commonly used for thermal rotection of comonents as thermal barrier coatings (TBCs) in gas turbine engine alications []. The coating microstructures in TBC alications are highly heterogeneous, consisting of defects such as ores and cracks of different sizes. The density, size and morhology of these defects determine the coating s final thermal and mechanical roerties, and the service lives of the coatings. The usual aroach to ensure that a coating has desired roerties is to develo an exerimental recie, i.e. to find an adequate arameter setting. The drawbacks in this aroach are that it is time consuming, exensive, secifically if new coating designs are to be develoed and that it does not rovide any fundamental understanding of microstructure-roerty relationshis. Quantitative microstructure roerty correlations is thus of interest. A number of analytical aroaches have been roosed for this urose []. Examles are: ) Homogenizing for volume averaging in eriodic orous media [3]. The disadvantage in this aroach is that the real in-homogeneity in microstructure might be difficult to catch in the model. ) Statistical model building, tyically using different regression techniques [4]. The major disadvantage with this aroach is that no hysical understanding is gained. 3) Finite element models based on true microstructures [5], which have the major advantage over other classical modeling techniques, that it is a true microstructure-based technique. One major drawback with this aroach is that it is time consuming. The objective of this work was to evaluate if the two last aroaches could be combined i.e. to utilize the statistical aroach for screening and tentative microstructure roerty relationshi building and then the finite element model aroach for verification and hysical understanding. The aer is divided into three searate sections. In the first section, the evaluated coating systems together with the exerimental results are resented. In the second section, the statistical model aroach and its results are resented. The last section resents the finite element model aroach and the verification of the statistical modeling results.. Exerimental Procedure.. Samle Production The samles in this roject were srayed using a combination of owders and sray equiment to roduce 6 different coatings. Powders were chosen with modified chemistry including; high urity owders for sintering resistance, Dysrosia stabilised Zirconia owders for lower thermal conductivity and a owder containing a orosity former in order to achieve a range of microstructure. Agglomerated & Sintered (A&S) and Hollow Oxide Sherical Powder (HOSP) morhologies were used. Thick dual layer coatings were finally roduced using both owder morhologies. Sraying was carried out by Atmosheric lasma Sraying (APS) using either the Suzer Metco F4 or Trilex gun. The substrate material was the nickel based sueralloy Hastelloy X. A standard NiCoCrAlY (Sulzer Metco AMDRY 365-) Bondcoat of 00 ±0μm was alied for all samles using the Sulzer Metco F4 gun. The coatings are summarised below in Table. Two different samle geometries were srayed in the investigation, cylindrical Ø 5.4 x 5mm couons, and 30 x 30 x.54mm lates for microstructure analysis and thermal conductivity measurement. Coating thickness was aroximately 300μm for the standard coatings and aroximately 800μm for the dual layer coating. DVS 64

44 Since sintering resistance is of high imortance for the lifetime of a TBC coating; a short term heat treatment was carried out on a samle of each coating. The heat treatment was carried out in standard atmoshere at 50 C for 00 hours. Table. Descrition of the coatings Coating System Material Process Comments A&S YPSZ APS Reference (F4) Coating High Purity HOSP YPSZ APS (F4) (Metco 04C-NS) 3 High Purity HOSP YPSZ Trilex (Metco 04C-NS) High Purity A&S YPSZ (AE 9538) High Purity A&S DyPSZ (AE 958) + High Purity HOSP DyPSZ (AE 9556) High Purity A&S YPSZ (AE 9538) + High Purity HOSP YPSZ (Metco 04C-NS).. Thermal Conductivity APS (F4) APS (F4) APS (F4) Double Layer Double Layer Laser flash analysis has become the most acceted method for measurement of thermal diffusivity of TBC s at varying temeratures. In this study a Netzsch Laser Flash Analysis 47 (Netzsch Gerätebau GmbH Germany) was used to assess the coatings thermal diffusivity at room temerature in a dynamic argon atmoshere. During the test the samle hit with a laser ulse that increases the temerature on the back face. This temerature rise is detected using an InSb infra-red detector. The resonse is normalised and thermal diffusivity calculated from the following formula [6]: where L is the thickness of the samle and t is the time taken for the total temerature increase. Corrections are taken on the data to account for radiative losses in heat. Differential Scanning Calorimeter 404C (Netzsch Gerätebau GmbH Germany) was used to establish the secific heat caacity of the 6 coatings, bond coat and substrate. The combination of measured secific heat caacity and thermal diffusivity was used to calculate thermal conductivity using the following equation [6]: where is thermal conductivity (W m - K - ), is thermal diffusivity (m s - ), is density (kg m -3 ) and C is secific heat caacity (J kg - K - ). Density for the coatings was calculated using Archimedes method. Fig.. Thermal Conductivity Results Thermal conductivity results are resented above in Fig.. Calculated conductivity for all six coating systems is shown at room temerature for as-srayed samles and samles heat treated for 00 hours at 50 C..3. Image Analysis and Microstructure A samle from each set of coatings was mounted, cut and olished in cross-section in the standard way. Samles were then insected using Otical microscoy and Scanning Electron Microscoy (SEM). The orosity measurement was carried out using an image analysis routine develoed in the HITS- Brite Euram roject [7]. By this rocedure, 5 images were taken across the coating cross section to reresent the coating microstructure. The images were then rocessed using the Ahelion image analysis software (ADCIS, France). In the first analysis case all images used were obtained using otical microscoy at 50x magnification. Fig. shows examles of images taken by the otical microscoe for coating system 4 and 6 resectively. The image (originally in grey scale) is converted into binary format by the use of thresholding; with the matrix reresented by the black and the cracks and ores by the white. This is normally oerator deendent why an auto-thresholding algorithm was used to remove errors caused by oerator variations. The Ahelion software creates a grey level histogram of every ixel for each row and then takes the average of two neighbouring rows. From this, the software can establish the critical threshold level for the image and so what should be used for the conversion. DVS 64

45 The software is able to differentiate between cracks and ores by using the height to length ratio and searate them into images of only ores and only cracks. By analysing the arrangement of ixels it is ossible to differentiate ores in contact with cracks and vice versa and so select either as required. Again, this rocess is automated to remove oerator deendent errors. (a) (b) Fig.. Image taken by an otical microscoe for (a) the coating system 4 (as-srayed) and (b) the coating system 6 (as-srayed). The information on crack angles is found from the image of only cracks. Firstly, the software searates the cracks. It then creates a tie line (a straight line from the start of the crack to the end), one ixel in width, for each crack. The lengths of the tie lines can be measured in terms of number of ixels. The tie lines are groued into secified angle ranges (based on the angle they make to the horizontal reference). All the tie line lengths in each grou are added together, resulting in the total number of ixels which create the tie lines in each angle range. The images used in the finite element model building were roduced using SEM imaging. All images used were backscattered electron images taken at 500x magnification. Again the autothresholding algorithm was used to roduce a binary image for inut to the model. 3. Statistical model of the thermal conductivity 3.. Princile of PCA and PLS modelling Since the variables, see Table, were highly correlated they were not used searately as redictor variables in a multivariate regression analysis. Two alternative statistical methods were therefore evaluated. One was rincial comonent regression (PCR) and the other artial least square (PLS). PCR is a regression analysis where the redictor variables are rincial comonents derived through rincial comonent analysis (PCA) on the correlated original redictor variables. PCA transforms a set of correlated variables into a smaller set of uncorrelated variables called rincial comonents. The rincial comonents (PCs) are linear combinations of the original variables derived such that they are uncorrelated. The first PC (PC) has the largest variance, the second PC (PC) the next largest variance and so on. Since the variables, see Table, are not measured in comarable units and do not have similar variances, the variables should be standardized before the PCA [8]. Table. Predictor variables (all crack angles include both free and connected cracks) only the cracks which are in contact with ores free cracks only the ores which are in contact with cracks free ores, crack angle, crack angle, crack angle, crack angle, crack angle, crack angle, crack angle, crack angle, crack angle, crack angle, crack angle, crack angle 0 and 80, crack angle PLS is similar to PCR. PCR start with PCA, with the urose to transform the redictor variables into a smaller uncorrelated set and then the regression analysis is erformed. PLS roject to latent structures in the redictor variables and the resonse variable simultaneously, with the urose to connect the information and align these structures to each other [9]. 3.. Modelling of thermal conductivity The aim is to obtain a statistical model, based on the redictor variables in Table, useful to redict the resonse variable thermal conductivity. The data available to estimate the model is the data obtained DVS 64

46 from the image rocessing for all the twelve coating systems using Ahelion as described in section.3. First a PCA on the redictor variables in Table was erformed. The software Minitab was used for this urose. As it can be seen in Table 3 below, the 3 first PCs exlain 93% of the total samle variance and they all have variances (eigenvalues) greater than one. The remaining PCs have variance much smaller than one and can be considered as noise. Hence, the first 3 PCs were selected as redictor variables in the regressions analysis for modelling the thermal conductivity. The weights or loadings in the first 5 PCs are given in Table 4. Table 3. Eigenvalues in the correlation matrix showing the first 5 PCs *samle variance exlained by the PC individually Table 4. The weights (loadings) for the first 3 PCs In PC the weights (loadings) are largest, and aroximately equal, for the variables, different crack angles. Furthermore, the variables (free cracks) and (free ores) have almost as large weights as the crack angles but (only the cracks which are in contact with ores) and (only the ores which are in contact with cracks) have very small weights. All weights in PC have ositive signs. PC are dominated by, aroximately around 0.4, where two of the variables have ositive signs and one negative. PC3 are dominated by, both having negative signs. The multile regression analysis with thermal conductivity as resonse variable and PC-PC3 as redictor variables showed that PC was nonsignificant at 5% significance level (-value = 0.30). Hence PC was excluded from the model and a new regression analysis was erformed. The final regression model is given in Equation (). () Combining the information from Table 4 and the estimated regression model in (), it can be seen that the thermal conductivity will become large if are close to zero and all other variables are ket constant, e.g. there are no cracks and ores in contact. This is due to the dominance of and in and at the same time their small weights in. The effect of the different variables in Table is more easily seen if the exressions for the PCs from Table 4 are relaced in the estimated model in (), Table 5. By using PLS together with cross-validation two PLS comonents was selected and. The ten largest estimated regression coefficients from the PLS analysis, when using standardized variables, are resented in Table 5. In both the PCR model and PLS model the variables and have the largest coefficient. The third largest regression coefficient in the PCR model is but in the PLS model the got much less weight. On the third lace in the PLS model is. Table 5. The ten largest estimated regression coefficients, using standardized variables, from the PCR and the PLS model, resectively. Regression coefficient PCR model Regression coefficient PLS model Both the PCR and PLS methods have roughly the same and are dominated by the variables. Hence both models indicate that cracks and ores in contact affect the thermal conductivity more than crack angle, free cracks and free ores. Since these models are based on twelve observations only, the models should be considered tentative and more observations are needed to confirm the results. The reason to that crack angles are not among the two most dominated variables maybe that there are interactions between some of the crack angle DVS 64

47 variables and the variables. This interaction was not considered in this analysis. The results from the statistical analysis were thereafter evaluated by finite element modelling in the next section, secifically the individual influence of ores and cracks and ores and cracks in contact on thermal conductivity. This analysis was made by finite element modelling using OOF. zirconia [, ]. The value of thermal conductivity taken in this study for YSZ was. W/(m.K) and for ores/cracks 0.06 W/(m.K), assuming it to be same as that of air. (a) 4. Modelling using OOF 4.. Procedure OOF is a rather new finite element model aroach which can be used to determine the macroscoic roerties from images of real microstructures [5]. The current version of OOF, which is OOF, is limited to two-dimensional images and solving linear roblems. OOF has been successfully imlemented in many alications in the ast, aart from comuting the thermal conductivity of thermal srayed coatings [, 0]. For this study, the inuts to the software are the microstructure image and the thermal conductivities of ores and bulk thermal srayed material. Two binary images (A and A ) taken from the samle with heat treated reference coating were taken into consideration in the models. Images reresenting free ores (B i ), free cracks (C i ) and ores and cracks in contact (D i ) were generated using Ahelion for i =,. These binary images were used as inut in OOF. First, two ixel grous were generated in OOF on the basis of colour and assigned as YSZ and orosity. Then a finite element mesh was generated on the basis of features and colours resent in the image by using an adative meshing rocedure. During the meshing rocedure, the elements were refined and the nodes were moved so that the material interfaces are well defined. Finer elements were generated near the interfaces to account for higher thermal gradients. The two constituents were then assigned material roerties, i.e. thermal conductivity values, according to the ixel grous defined in the binary image. The mesh generated by OOF from image A had aroximately elements and nodes, Fig. 3. The thermal conductivity value was determined by solving the heat equation in the D domain. A temerature difference of 0ºC (T) was set-u as boundary conditions between the to and bottom surfaces, while the sides were assumed to be comletely insulated in the model. In steady state, the resultant heat flux across the cross-section in vertical direction will be constant. Thus, this value can be used in the following heat equation to determine the thermal conductivity (K) of the coating microstructure. Several studies have been erformed to determine the value of thermal conductivity for bulk (b) Fig. 3. (a) The image taken using SEM which was used to roduce binary image A. The red box denotes the area reresented in Fig. 4(b). (b) A art of the finite element mesh generated by OOF from image A (red area denotes YSZ and white area denotes orosity ) 4.. Results and discussions The redicted thermal conductivity values are resented in Table 6. The numbers in the second column reresent normalised values which have been scaled by dividing the numbers in the first column by the first value in the first column (done searately for each image). The table also shows, for comarative urose, thermal conductivity values calculated when the microstructure is considered as a homogenous mixture of YSZ and orosity, the fractions (f YSZ, f orosity ) calculated on the basis of the ixels of each grou resent in the image (using the equation K total = f YSZ.K YSZ + f orosity.k orosity ). Normalised values are also resented in the next column which have been scaled in the same fashion as earlier described. Comaring the values for each image in the columns S and S, in Table 6, it can be seen that DVS 64

48 the value of thermal conductivity when they are simly calculated from a volume fraction becomes fairly constant, while the values change significantly when comuted by the finite element model. The effect is most significant for the image reresenting ores and cracks in contact, which seem to confirm the findings in the statistical model i.e. deicting the fact that ores and cracks in contact have much more significance on the thermal conductivity comared to free ores or free cracks. Table 6. Thermal conductivity values obtained from the analysis Thermal conductivity Image OOF (W/(m.K)) Scaled (S) Calculated (W/(m.K)) Scaled (S) A B C D A B C D The differences in the values redicted by the model and the exerimental values can be attributed to several factors. The major factor is most likely the limitation of the model to D. If one tries to extraolate the D model to 3D, it is quite obvious that the cracks and ores resent in the images are considered to be infinitely long in the model, while in reality, they are of finite length. So the image, and hence the thermal conductivity value, deends uon the cross-section of the coating considered. Also, the area reresenting the image is only a small fraction of the total crosssectional area of the coating. So the microstructure features might vary to a great extent from one art of the coating to the other. Another factor which affects the comuted value is the limitation of the resolution of the image, as all small cracks and ores (and secifically SEM invisible cracks) cannot be included in the analysis. Desite these limitations the results indicate that the ores and cracks when in contact will have a significant effect on the thermal conductivity comared to when they are freely distributed in the matrix. This result is consistent in the two modelling aroaches. 5. Summary and Conclusions The aroach undertaken in this study can be summarised as follows. First, statistical modelling was used as a screening method to determine the imortant microstructure arameters influencing the thermal conductivity. The most interesting finding in this analysis was that ores and cracks in contact seem to have significant influence on thermal conductivity comared to free ores and free cracks. The major limitation with this modelling aroach is that it does not consider any hysical interretation of the results. The results from this analysis were then validated by finite element modelling using microstructure images obtained by SEM which rovided hysical verification of the statistical results. The results, although tentative in nature, indicate that the combination of the two modelling aroaches seems to be a feasible aroach. Through this combination, relationshis between microstructure defects and thermal conductivity can be evaluated and verified. The interesting result that the interaction between ores and cracks i.e. ores and cracks in contact, is the most imortant arameter to achieve a low thermal conductivity TBC. This finding will be further investigated in future work. References: [] Markocsan, N., Nylén, P., Wigren, J., Li, X.-H. and Tricoire, A.: Effect of Thermal Aging on Microstructure and Functional Proerties of Zirconia- Base Thermal Barrier Coatings. Journal of Thermal Sray Technology (June 009), Vol. 8(),. 0/08. [] Wang, Z., Kulkarni, A., Deshande, S., Nakamura, T. and Herman, H.: Effects of ores and interfaces on effective roerties of lasma srayed zirconia coatings. Acta Materialia 5 (003),. 539/34. [3] Kaviany, M.: Princiles of heat transfer in orous media. nd ed. Sringer-verlag New York Inc (995). [4] Friis, M.: A Methodology to Control the Microstructure of Plasma Srayed Coatings. Ph.D. Thesis, Lund University, Lund, Sweden (00). [5] Langer, S.A., Fuller Jr, E.R. and Carter, W.C.: OOF: an image-based finite-element analysis of material microstructures. Comut Sci Eng (00), Vol. 3(3),. 5/3. [6] Taylor, R.E.: Thermal conductivity determinations of thermal barrier coatings. Materials Science and Engineering A (May 998), Vol. 45,. 60/67. [7] Wigren, J.: High Insulation Thermal Barrier Systems HITS Brite Euram Project BE (00). [8] Wichern, D.W. and Johnson, R.A.: Alied multivariate statistical analysis. Pearson Prentice Hall (007),. 430/63. [9] Eriksson, L., Johansson, E., Kettaneh-Wold, N., Trygg, J., Wikström, C. and Wold, S.: Multi- and Megavariate Data Analysis, Part I: Basic Princiles DVS 64

49 and Alications. Umetrics Academy, Umea, Sweden (006),. 63/99. [0] Jadhav, A.D., Padture, N.P., Jordan, E.H., Gell, M., Miranzo, P. and Fuller Jr., E.R.: Low-thermalconductivity lasma-srayed thermal barrier coatings with engineered microstructures. Acta Materialia 54 (006),. 3343/49. [] Hasselman, D.P.H., Johnson, L.F., Bentsen, L.D., Syed, R., Lee, H.L. and Swain, M.V.: Thermal diffusivity and conductivity of dense olycrystalline ZrO ceramics: a survey. J Am Ceramic Soc (987), Vol. 66(5),. 799/806. [] Bansal, N.P. and Zhu, D.: Thermal conductivity of zirconia alumina comosites. Ceramics International 3 (005),. 9/6. DVS 64

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53 Case Study (wileyonlinelibrary.com) DOI: 0.00/qre.50 Published online in Wiley Online Library Comaring Confidence Intervals for Multivariate Process Caability Indices Ingrid Tano, a,b * and Kerstin Vännman, a,b,c Multivariate rocess caability indices (MPCIs) are needed for rocess caability analysis when the quality of a rocess is determined by several univariate quality characteristics that are correlated. There are several different MPCIs described in the literature, but confidence intervals have been derived for only a handful of these. In ractice, the conclusion about rocess caability must be drawn from a random samle. Hence, confidence intervals or tests for MPCIs are imortant. With a case study as a start and under the assumtion of multivariate normality, we review and comare four different available methods for calculating confidence intervals of MPCIs that generalize the univariate index C. Two of the methods are based on the ratio of a tolerance region to a rocess region, and two are based on the rincial comonent analysis. For two of the methods, we derive aroximate confidence intervals, which are easy to calculate and can be used for moderate samle sizes. We discuss issues that need to be solved before the studied methods can be alied more generally in ractice. For instance, three of the methods have aroximate confidence levels only, but no investigation has been carried out on how good these aroximations are. Furthermore, we highlight the roblem with the corresondence between the index value and the robability of nonconformance. We also elucidate a major drawback with the existing MPCIs on the basis of the rincial comonent analysis. Our investigation shows the need for more research to obtain an MPCI with confidence interval such that conclusions about the rocess caability can be drawn at a known confidence level and that a stated value of the MPCI limits the robability of nonconformance in a known way. Coyright 0 John Wiley & Sons, Ltd. Keywords: multivariate rocess caability index; lower confidence bound; multivariate normal distribution; rincial comonent analysis. Introduction O ur interest in multivariate caability indices is based on a case from a thermal sraying rocess at the Volvo Aero Cororation, Sweden, who wanted to determine the rocess caability of a three-dimensional variable. Hence, a multivariate rocess caability index (MPCI) was needed. The uroses of this article are to investigate how four of the MPCIs and their confidence intervals, described in the scientific literature, erform in our case and to comare the results and discuss the exeriences of the MPCIs and their confidence intervals. We concentrate on the simlest form of an MPCI, MC, because it is simle and at the same time sufficient to identify the essential ros and cons. The corresonding caability index in the univariate case is (Kane ) USL LSL C ¼ () 6s where [LSL,USL] is the secification interval and s is the rocess standard deviation of the in-control rocess. This index catures only the rocess variation and comares it with the rocess sread. It does not take the rocess location or the closeness to target into consideration, so other univariate indices, like C k and C m, have been develoed. Multivariate counterarts to C k and C m have been suggested but will not be dealt with in this article. For a review of univariate caability indices as well as some multivariate indices, see, for examle, Kotz and Johnson. For more recent reviews of MPCIs see, for examle, Gonzáles and Sánchez 3 and Pan and Lee 4. Yum and Kim 5 resented a bibliograhy of aroximately 530 articles and books on rocess caability indices, including MPCIs, for the eriod a Deartment of Engineering Sciences and Mathematics, Luleå University of Technology, Luleå, Sweden b Deartment of Engineering Science, University West, Trollhättan, Sweden c Deartment of Statistics, Umeå University, Umeå, Sweden *Corresondence to: Ingrid Tano, Deartment of Engineering Science, University West, SE Trollhättan, Sweden. Ingrid.tano@hv.se Coyright 0 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 0,

54 I. TANO AND K. VÄNNMAN During our literature review, we discovered that there are only a few articles that deal with confidence intervals or tests for MPCIs. The ones found are those of Wang and Du, 6 Wang, 7 Pearn et al., 8 and Pan and Lee. 4 The methods we found of interest to aly to our case study are those given by Wang and Du, 6 Pearn et al., 8 and Pan and Lee. 4 The method by Wang 7 deals with a situation on the basis of a geometric distance aroach not suitable for our case. Then we also consider an MPCI suggested by Wang 9 and roose a confidence interval to be used for that index. Furthermore, we restrict our interest to the situation where the studied quality characteristic follows a multivariate normal distribution. In the next section, a brief review of MPCIs is made. This section is followed by a more detailed descrition of the four studied methods. In Section 4, we describe the thermal sraying rocess and resent the result obtained when alying the studied methods to our case. In the last section, we discuss some issues found of interest to emhasize after we have alied the considered methods.. A brief review of MPCIs The develoment of MPCIs started in the early 990s with indices suggested by Chan et al. 0 and Hubele et al. Since then, several different multivariate indices have been resented. For references, see the revious section. MPCIs can roughly be divided into four different grous, as suggested by Shinde and Khadse. They are as follows: grou, based on the ratio of a tolerance region to a rocess region (see, e.g. Taam et al. 3 ); grou, based on the robability of the nonconforming roduct (see, e.g. Castagliola and Castellanos 4 ); grou 3, based on the rincial comonent analysis (PCA; see, e.g. Wang and Chen 5 ); and grou 4, others (see, e.g. Shahriari et al. 6 ). In our comarison, we will consider two indices from grou and two from grou 3. In the beginning of the develoment of MPCIs, the focus was on the theoretical indices, with suggestions of corresonding oint estimators. However, in ractice, the conclusions drawn about the rocess caability are based on a random samle, and oint estimators are random variables. Hence, confidence intervals or tests for MPCIs need to be develoed to make it ossible to draw conclusions with high confidence or low robability for the tye I error. The first article we found that dealt with this issue was that of Wang and Du. 6 They resented a confidence interval on the basis of the index by Wang and Chen, 5 which belongs to grou 3. They also suggested a confidence interval for an MPCI on the basis of a univariate index roosed by Luceño 7, which belongs to grou 3. Confidence intervals for the indices roosed by Taam et al. 3, belonging to grou, were derived by Pearn et al. 8 Pan and Lee 4 modified the index by Taam et al. 3 and resented a confidence interval for this modified index, which also belongs to grou. Wang 9 suggested a modification of the index by Wang and Chen 5 but did not roose any confidence interval. We considered the index belonging to grou 3 and also roosed a confidence interval for this index on the basis of the idea from Wang and Du 6. The methods we alied in our case study are those given by Pearn et al., 8 Pan and Lee, 4 Wang and Du, 6 and Wang 9. It can be noted that Shinde and Khadse have discovered a roblem with how to obtain the transformed tolerance region when using MPCIs on the basis of PCA. They showed that the way to transform the tolerance region, as suggested by Wang and Chen 5 and also used by Wang and Du 6, is not aroriate. However, they did not suggest how to convert this knowledge to use the index suggested by Wang and Chen. 5 Instead, they roosed a new index without a confidence interval, which they called a robability-based index, to handle the roblem. We will not consider their new index but aly the methods by Wang and Du 6 and Wang, 9 including their suggested transformation of the tolerance region. This issue is discussed further in Section Four MPCIs with confidence intervals In this section, we will describe in more detail the four methods we have chosen to aly to our case. First, we state the assumtions and notations used throughout the entire article. Let X =(X,X,...,X v ) denote the studied v-dimensional quality characteristic. We assume that X is distributed according to a multivariate normal distribution N v (m,σ), where m is the mean vector and Σ is the variance covariance matrix. Furthermore, let S denote the samle variance covariance matrix. We will assume that each univariate quality characteristic X i has a given secification interval [LSL i,usl i ], i =,,...v. Hence, the tolerance region for X is a hyerrectangular region. The indices we will consider are multivariate counterarts to the univariate caability index C in Equation () and will be denoted MC ðþ j ; j ¼ ; ; 3; 4. In accordance with the univariate case, a rocess is defined as caable if MC ðþ j exceeds a given threshold value k 0. In the literature, it is common to let k 0 = when dealing with MPCIs. This threshold value is also used here. Because it is of interest to find out if the MPCI exceeds, we only consider the lower confidence bound of one-sided confidence intervals. 3.. Method : Pearn et al. 8 Under the assumtion of multivariate normality, Taam et al. 3 resented a multivariate exansion of C by comaring the volume of a modified tolerance region, R, with the volume of the region, R, containing 99.73% ercent of the multivariate normal distribution. This index reduces to C in the univariate case. It can be exressed as Coyright 0 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 0

55 I. TANO AND K. VÄNNMAN ¼ Vol ð R Þ Volðmodified tolerance regionþ ¼ VolðR Þ v=jσj ; () = ½Γðv= þ ÞŠ MC ðþ w v;0:9973 where R equals the largest ellisoid centered at the target value comletely within the original rectangular tolerance region. In Equation (), w v;0:9973 denotes the quantile of a w -distribution with v degrees of freedom. The index MC ðþ is estimated by ^MC ð Þ ¼ VolðR Þ v=jsj : (3) = ½Γðv= þ ÞŠ w v;0:9973 Pearn et al. 8 derived confidence intervals for MC ðþ, where the lower 00(a)% confidence bound is exressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðþ FY ðaþ ^MC ðn Þ v : (4) In Equation (4), FY ðaþdenotes the a quantile of the distribution of the random variable Y, where Y is defined as a roduct of v indeendent w -distributed random variables with n,n,...,nv degrees of freedom, resectively. For v=, Pearn et al. 8 showed that For v=3, they showed that FY ðaþ ¼ w n4;a 4 : (5) F Y ðþ¼ y Z y Z 0 0 x = z ðn5 Þ= ex x = z= ðxþ dxdz: (6) ð ÞΓððn 3Þ=Þ ðnþ= Γ n Because the quantile FY ðþmay a be difficult to calculate for v >, we derive a confidence interval for MC ðþ with an aroximate confidence level of a. According to Anderson 8 ffiffiffi (68), n ððjsj=jσjþþ is asymtotically normally distributed with mean 0 and variance v. On the basis of this result, we show in Aendix B that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðþ ^MC l a v ffiffiffi n is a lower confidence bound for MC ðþ with an aroximate confidence level of a. In Equation (7), l a denotes the (a) quantile of the N(0,) distribution. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi As a comarison, the values of FY ðþ= a ðn Þ v ffiffiffi and l a v = n are given in Table I, when v =,3 and a = 0.05, for some values of n. From Table I, we can see that the aroximate value works fairly well for as small values as n=00. It should also be noted that q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi l a v = n ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < FY ðþ= a ðn Þ v. This means that if the aroximate value is used to calculate the lower confidence bound, the confidence level will be at least A numerical method is needed to calculate FY ðþwhen a v >. Such a method usually needs some starting values and then the aroximate value is useful to find suitable starting values. (7) Table I. A comarison of the square root exressions in Equations (4) and (7), when v =,3,a = 0.05, and n = 50, 70, 00, 00, 500, 000 v = v =3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi n F ðaþ= ðn Þ v l a v = n ðþ= a ðn Þ v l a v = n Y FY Coyright 0 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 0

56 I. TANO AND K. VÄNNMAN 3.. Method : Pan and Lee 4 Pan and Lee 4 claim that the index MC ðþ in Equation () may overestimate the true rocess erformance in certain situations, when the univariate quality characteristics are not indeendent. They revised the modified engineering tolerance region by Taam et al. 3 to overcome this roblem and roosed n o R 3 ¼ X X mþ ða Þ ðx mþ w v;0:9973 (8) to be used in Equation () instead of R. The elements of the matrix A is given by 0 0 A BUSL i LSL i CB ij ¼ r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia USL j LSL j qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia; (9) w v;0:9973 w v;0:9973 where r ij is the correlation coefficient between the ith and the jth univariate quality characteristic. See Figure, where R is the modified tolerance region according to Taam et al. 3 and R 3 is the slanted ellisoid centered at the target value defined in Equation (8). Pan and Lee 4 then exressed their index as ¼ Vol R ð 3Þ VolðR Þ ¼ ja j = (0) jσj MC ðþ and roosed the index MC ðþ in Equation (0) to be estimated by ð ^MC Þ ¼ ja j = () jsj Note that they do not suggest r ij to be estimated. Hence, they must consider r ij to be known. Pan and Lee 4 derived a confidence interval for MC ðþ, where the lower 00( a)% confidence bound is ðþffiffiffiffiffi ^MC w a where w a is the a quantile of the distribution of Π v i¼ w ni = ðn Þv and w ni denotes a w -distributed random variable with ni degrees of freedom. Note that w a is the same as FY ðaþ= ðn Þ v in Equation (4). Hence, a corresonding aroximate confidence bound as in Equation ð (7) is obtained for this method by substituting^mc Þ ð in Equation (7) with^mc Þ. Pan and Lee 4 in their derivation considered the correlation coefficient r ij to be known. However, in ractice, usually r ij needs to be estimated, and this will affect the derivation of the confidence interval. Hence, the confidence interval in Equation () has only an aroximate confidence level of (a) when an estimate of r ij used. It can be noted that Pan and Lee 4 in their examles used the estimated correlation coefficients. () 3.3. MPCI based on rincial comonent Wang and Chen 5 were the first to use PCA when defining an MPCI. PCA reduces the dimensionality by extracting a few new latent, uncorrelated variables, called rincial comonents (PCs), which together exlain the main variability in the data (see, e.g. Johnson and Wichern 9 ). The PCA is briefly outlined in the next aragrah. Figure. R is the modified tolerance region roosed by Taam et al. 3, and R 3 is the revised tolerance region roosed by Pan and Lee 4 Coyright 0 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 0

57 I. TANO AND K. VÄNNMAN Let l ; u ; l ; u ;...; lv ; u v be the eigenvalue eigenvector airs of the variance covariance matrix Σ. The v PCs are formed as linear combinations of the original quality variables X,X,...,X v : PC ¼ u X ¼ u X þ u X þ...þ u v X v PC ¼ u X ¼ u X þ u X þ...þ u v X v : : PC v ¼ u v X ¼ u v X þ u v X þ...þ u vv X v (3) The PCs are orthogonal to one another and ordered according to their variances. The first PC, PC, has the largest variance; the second PC, PC, has the second-largest variance; and so on. The eigenvalue, l i ; is the variance of the ith PC, i =,,...,v. The variance covariance matrix is unknown in ractice and estimated by the samle variance covariance matrix S. Sometimes the correlation matrix r is used instead of the variance covariance matrix Σ and the corresonding samle correlation matrix instead of S. When the variables are correlated, the main variability is catured by the first m, m < v, PCs. The remaining vm PCs are considered to contain mostly random noise Method 3: Wang and Du 6 Wang and Du 6 resented a confidence interval on the basis of the index MC ð3þ by Wang and Chen 5, where MC ð3þ ¼ Π v =v C (5) ;PCi i¼ In Equation (5), C ;PCi is the univariate caability index for the ith PC, that is, in accordance with Equation () C ;PCi ¼ USL LSL PCi PCi 6 ffiffiffiffi (6) l i The uer- and lower-secification limits of PC i in Equation (6) are defined as (see Wang and Du 6 ) USL PCi ¼ u iusl X and LSL PCi ¼ u ilsl X (7) where u i is the eigenvector corresonding to PC i, USL X ¼ ðusl ; USL ;...; USL v Þ, and LSL X ¼ ðlsl ; LSL ;...; LSL v Þ. Furthermore, the index MC ð3þ is estimated by where ^MC 3 ð Þ ¼ Π v =v ^C ;PCi ; (8) i¼ ^C ;PCi ¼ USL LSL PCi PCi 6 ffiffiffiffi : (9) Note that, in Equation (9), l i, i =,,...,v, are the eigenvalues of the samle variance covariance matrix S and USL PCi and LSL PCi are calculated using the eigenvectors of S. Both Wang and Chen 5 and Wang and Du 6 used all v PCs in Equations (6) and (8). However, in their examles, they used only the first m PCs, where m is determined so that the main variability is catured by the first m, m < v, PCs. This is reasonable and we will do so henceforth. The lower aroximate 00(a)% confidence bound given by Wang and Du 6 is Π m ^C ;PCi i¼ l i sffiffiffiffiffiffiffiffiffiffiffi w n;aa n =m ; (0) where w n;a denotes the a quantile of a w -distribution with n degrees of freedom. The interval in Equation (0) is based on the confidence interval for the univariate index C in Equation (6) (for the confidence interval for C, see, e.g. Pearn and Kotz 0 ) Method 4: Wang 9 Wang 9 roosed the use of a weighted geometric mean of C ;PCi in Equation (6), with the eigenvalues l i, i =,,...,m as weights. The objective was to give the largest weight to the index for PC, the second largest weight to the index for PC, and so on. The MPCI Wang 9 suggested is Coyright 0 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 0

58 I. TANO AND K. VÄNNMAN where C ;PCi is given in Equation (6) and m is the number of selected PCs. The index MC ð4þ MC ð4þ ¼ Π m =Σli; i¼ Cli ; PCi () ^MC 4 ð Þ ¼ Π m ^C li ;PCi i¼ is estimated by =Σli () where ^C ;PCi is given in Equation (9). Wang 9 did not resent any confidence interval for the index MC ð4þ 3. Because MC ð Þ and MC ð4þ both are based on C ; PCi, we follow the idea from Wang and Du 6 and suggest the following aroximate lower confidence bound for MC ð4þ 0 Π m i¼ ^C ;PCi sffiffiffiffiffiffiffiffiffiffiffil i w n;aa C A n =Σli (3) 4. The studied case: a thermal sraying rocess The thermal sraying rocess can be characterized as a high temerature sray ainting to rotect the srayed surface against (e.g. heat, corrosion, erosion, etc.). In other words, it is a surface coating rocess in which melted or heated materials are srayed onto a surface. The sraying material, in owder or wire form, is fed into a hot gas jet flame (u to 5,000 C 0,000 C) where it is melted. For a rough outline, see Figure. Deending on what a thermally srayed coating should rotect against (e.g. heat, corrosion, erosion, etc.), different orosities are desirable. At the Volvo Aero Cororation, thermal sraying is erformed, among others, on combustion chambers for rocket nozzles and gas turbines to rotect from heat. The caacity is one art at a time, and the orosity is measured metallograhically after sraying. There is a relation between the orosity and the in-flame variables, e.g. higher temerature and higher velocity often mean lower orosity in the resulting coating. It would be referable to secure the orosity during sraying instead of afterward. One way to do this could be to calculate an MPCI of the in-flame variables, light intensity, temerature, and article s velocity in an effort to secure the final orosity. The studied variables and their secification intervals are given in Table II. The target value is equal to the center of each secification interval. Our data set consists of 70 observations from the thermal sraying rocess (see Aendix A). We assume that the in-flame variables are multivariate normally distributed. The normal robability lots of the individual X-variables do not contradict this assumtion, as can be seen in Figure 3 (a c). Figure 4 shows that all observations fall within the tolerance region. The in-flame variables are slightly correlated with the estimated correlation coefficients ^r ; ¼ 0:; ^r ;3 ¼ 0:3; ^r ;3 ¼ 0:9: (4) The correlation coefficients are significantly different from 0 at a 5% significance level. The samle variance covariance matrix equals 66:5 395:59 3 5:9 S ¼ 4 395:59 86:48 5:84 5 (5) 5:9 5:84 : and we obtain jsj = ¼ 5630:75: (6) Figure. A rough outline of the thermal sraying rocess Coyright 0 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 0

59 I. TANO AND K. VÄNNMAN Table II. The studied in-flame variables and their secification intervals Variable Light intensity (unitless) Temerature ( C) Velocity (m/s) Notation X X X 3 [LSL i,usl i ] [394,603] [95,668] [98,8] The PCA, based on the samle variance covariance matrix, gives that PC and PC together exlain 99.8% of the total variance. The first two eigenvalues are l = 95.3 and l = 079.0, and the corresonding eigenvectors are u ¼ ½0:7 0:976 0:09 Š and u ¼ ½0:976 0:7 0:03 Š. The interretation of the eigenvectors is that PC mainly includes information from X, the a) Percent Probability Plot of Intensity Normal Mean 54 StDev 34,5 N 70 AD 0,500 P-Value 0, intensity b) Percent Probability Plot of Temerature Normal Mean 488 StDev 53,50 N 70 AD 0,535 P-Value 0, Temerature c) Percent Probability Plot of Velocity Normal Mean 0,9 StDev 3,349 N 70 AD 0,53 P-Value 0, Velocity 5 0 Figure 3. (a c) Normal robability lots of the in-flame variables: (a) X, (b) X, and (c)x 3 Coyright 0 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 0

60 I. TANO AND K. VÄNNMAN 3D Scatterlot of Intensity, Temerature and Velocity Velocity temerature Intensity 400 Figure 4. A three-dimensional scatterlot of the in-flame variables where the box equals the tolerance region temerature variable, and PC mainly includes information from X, the light intensity variable. The methods chosen to determine the rocess caability estimate the otential caability only, for examle, it does not take the rocess location or the closeness to target into consideration. We now calculate the lower confidence bound for each studied MPCI using our data set. 4.. Method To calculate the lower confidence bound according to method, we first obtain VolðR Þ ¼ 4 USL i LSL i 3 ¼ Π3 ¼ 407:4: (7) By inserting the results from Equations (6) and (7), together with the chi-square value w 3;0:9973 ¼ 4:6 into Equation (3), we ð obtain^mc Þ ¼ 0:98. To derive the 95% lower confidence bound for MC ðþ, we then multilied the estimated index, according to Equation (4), with the factor obtained from Table I for n = 70 and v = 3 and get the result Method The calculation of the lower confidence bound ursuant to method is carried out in the following manner. Because the correlation coefficients are unknown, we will use the estimated correlation coefficients in Equation (4) to obtain an estimate, ^A, of the matrix A in Equation (9). In the following calculations, we will use ^A instead of A. As noted in Section 3, this will affect the confidence level. Then we get 77:7 30:84 3 5:49 ^A ¼ 4 30:84 457:9 57:36 5 (8) 5:49 57:36 5:9 and jaj ^ ¼ : (9) In Equation (), we will now use jaj ^ ð in Equation (9) together with the result in Equation (6). This results in^mc ¼ 0:90. Subsequently, we obtain the aroximate 95% lower confidence bound for MC, similar to method, by multilying the estimated index with the factor according to Equation () and get the result Method 3 To calculate the lower confidence bound according to method 3, we first calculated the uer and the lower secification limits of the PCs according to Equation (7) by using the results from the PCA earlier and get LSL PC ¼ 37:3; USL PC ¼ 737:3; LSL PC ¼:; USL PC ¼ : (30) Furthermore, to estimate the univariate index for each PC according to Equation (9), we use the results in Equation (30) together with the eigenvalues l = 95.3 and l = and obtain Coyright 0 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 0

61 I. TANO AND K. VÄNNMAN ^C ;PC ¼ :6 and ^C ;PC ¼ 0:63: (3) ð3 By inserting the results in Equation (3) into Equation (8), we obtained^mc Þ ¼ 0:89. To find the 95% lower confidence bound for MC ð3þ, we use the results in Equation (3) together with the chi-square value w 69;0:05 ¼ 50:88 and m = in Equation (0) and get the result Method 4 ð4 To derive^mc Þ, we use the result in Equation (3) and the eigenvalues l = 95.3 and l = 079.0, as weights, in Equation () and ð4 obtained^mc Þ ¼ :05. To obtain the 95% lower confidence bound, we use Equation (3) and insert the eigenvalues and the results from Equation (3) together with the chi-square value w 69;0:05 ¼ 50:88 and m = and obtain the result Table III resents the summary of the results for all four methods. It can be noted that the lower confidence bounds for all four methods are less than the threshold value. Therefore, none of the methods will deem the rocess as caable at the significance level 5%. Comaring the results of the four methods, it can be seen that the estimated indices are fairly similar, varying between 0.89 and.05. Method 3 has the smallest value and method 4 the largest. The lower confidence bounds vary more than the estimates, and here method gives the lowest bound (0.67) and method 4 the highest (0.90). Note that the indices studied here do not take the rocess location or the closeness to target into consideration. Hence, the interretation of the result only concerns the otential rocess caability, which takes the rocess variation only into consideration. 5. Discussion The uroses of this article are to investigate how some of the MPCIs and their confidence intervals, described in the scientific literature, erform in our case, comare the results and discuss the exeriences of MPCIs and their confidence intervals. In the following sections, we discuss some issues we have found of interest and want to emhasize. 5.. Aroximate confidence level Pearn et al. 8 derived the one-sided confidence intervals for MC ðþ in Equation (4) with the exact confidence level a. Because the quantile FY ðaþin Equation (4) may be difficult to calculate for large values of v, we derive, in Aendix B, a confidence interval with an aroximate confidence level of a (see Equation (7)). Pan and Lee 4 in their derivation considered the correlation coefficient r ij to be known. Because in ractice usually r ij needs to be estimated, this will affect the derivation of the confidence interval. Hence, the interval by Pan and Lee 4 has an aroximate confidence level of a only. The confidence interval for MC ð3þ in Equation (8) has an aroximate confidence level of a, according to by Wang and Du 6. They did not investigate how good their aroximation is. The confidence interval for MC ð4þ in Equation (3) is based on the same idea as of Wang and Du 6 and hence has only an aroximate confidence level of a. For methods 4, it is of imortance to know how good the aroximations of the confidence levels are to trust the conclusions drawn from them. It would be of interest to erform a simulation study to investigate this, esecially for method. Methods 3 and 4 have other drawbacks that need to be rectified first (see Section 5.4). For method, the results in Table I indicate that the aroximate confidence interval in Equation (7) is fairly close to the exact interval. However, this needs to be investigated for v Different index grous describe varying amount of information In our comarison, we consider two indices on the basis of the ratio of a tolerance region to a rocess region (grou ) and two indices on the basis of PCA (grou 3). The definitions of the indices in the two grous diverge substantially. The indices in grou take all variables into consideration, whereas the indices in grou 3 may include information from few variables only. For examle, when there are large differences between the variances of the variables and the variables are not highly correlated, each PC includes information from mainly one variable. Because only a few numbers of PCs are selected, information from some variables might be left out in the analysis. In our case, this is seen by the fact that PC includes information from mainly the temerature variable, X, and PC Table III. Summary of the results obtained when alying the methods to the studied case Method Estimated index Lower confidence bound (a = 0.05). Pearn et al. 8 ð ^MC Þ ¼ 0: Pan and Lee 4 ð ^MC Þ ¼ 0: Wang and Du 6 ð3 ^MC Þ ¼ 0: Wang 9 ð4 ^MC Þ ¼ : Coyright 0 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 0

62 I. TANO AND K. VÄNNMAN includes information from mainly the light intensity variable, X. This imlies that methods 3 and 4 include information from mainly two variables, X and X, comared with methods and that include information from all three variables. One could question if it matters that information on some variables are left out. The variables that do not have effect on the selected PCs may or may not be of interest when estimating the rocess caability. This is an issue that would be investigated further. However, there are also differences between the two indices within each grou. According to the result by Pan and Lee, 4 the index in method will always be smaller than the index in method when the variables are highly correlated. When the correlations between the variables are small, the indices will be aroximately equal. The indices in both methods 3 and 4 are based on the roduct of the univariate indices for the PCs. However, the first PCs obtain higher weight in method 4 than that in method 3. Hence, if the univariate indices for the first PCs are the largest ones, the index in method 4 will always be larger than that in method 3. If it is only of interest to follow the develoment of the rocess caability, to see if there is an imrovement in the caability, either of the MPCIs can be used. However, if the index also should reflect the robability of nonconformance for the quality characteristic studied, it is of imortance to relate the actual index value to the robability of nonconformance. This will be discussed in the next section Probability of nonconformance The robability of nonconformance, P(NC), is the robability that the quality characteristic falls outside the tolerance region. For a rocess to be caable, P(NC) should be small. What is considered to be small may differ in different situations. When rocess caability indices started to gain accetance in the 980s, a requirement for a caable rocess was that P(NC)< In the univariate case, with a rocess on target, this corresonds to C >. Using a simle examle, we now demonstrate that this corresondence not necessarily holds for the MPCIs. Consider the very simle situation when the studied variable has a known two-dimensional multivariate normal distribution with the arameters m 0:95 0:76 ¼ ½0 0Š; Σ ¼ (3) 0:76 0:95 This means that the variables X and X are highly correlated with r = 0.9. Furthermore, let the secification interval be [,] for each variable. In this situation, we obtain MC ðþ ¼ :0 using Equation (). However, the robability that the variable (X,X ) falls outside the tolerance region is P(NC) = This means that, for this examle, we do not have the required corresondence between MC ðþ ¼ :0 and P(NC). Furthermore, P(NC) is much larger than If we calculate MC ðþ according to Equation (0) for this examle, we obtain MC ðþ ¼ 0:4, which reflects the large robability of nonconformance better than MC ðþ ¼ :0. The variance covariance matrix Σ in Equation (3) has eigenvalues and eigenvectors l = and l = and u ¼ ½0:707 0:707 Šand u ¼ ½0:707 0:707 Š, resectively. Because the first PC exlains 95% of the total variance, we use m=, which imlies that MC ð3þ ¼ MC ð4þ. According to Equations (5) and (), we obtain MC ð3þ ¼ MC ð4þ ¼ 0:8, which is in between the values of MC ðþ and MC ðþ and also reflects the large robability of nonconformance better than MC ðþ ¼ :0: For a summary of the results, see Table IV. From Table IV, we see that when r =0.9, the values of the indices vary quite a lot although the P(NC) = is constant. Let us now change the variance covariance matrix to 0:089 0:07 Σ ¼ (33) 0:07 0:089 without changing m. The variables X and X are now moderately correlated with r = 0.3, and we obtain P(NC) = 0.006, which is smaller than In this situation, we have MC ðþ ¼ 0:99 and MC ðþ ¼ 0:95. Comaring the result of MC ðþ when r = 0.3 and r = 0.9 shows what Pan and Lee 4 discovered about the index MC ðþ. The modified tolerance region in MC ðþ by Taam et al. 3 does not cature the correlation structure roerly when there is high correlation between the quality characteristics and hence misjudge the rocess caability. The variance covariance matrix Σ in Equation (33) has eigenvalues and eigenvectors l = 0.57 and l = and u ¼ ½0:707 0:707 Šand u ¼ ½0:707 0:707 Š, resectively. Only 65% of the variation is exlained by PC, which is quite low and imlies that we should choose m =. However, roblems aear when calculating the uer and the lower secification limits for PC according to Equation (7), when m =. It turns out that both secification limits become zero. As a consequence, we have C ;PC ¼ 0, and hence MC ðþ ¼ MC ðþ ¼ 0. If we instead use only the first PC we find, according to Equations (5) and (), MC ð3þ ¼ MC ð4þ ¼ :4. The values of these indices are much larger than the values of MC ðþ and MC ðþ. Furthermore, because PC catures only 65% of the Table IV. Summary of results by the four methods for the simle theoretical examle r P(NC) MC ðþ ð Þ MC MC ð3þ MC ð4þ Coyright 0 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 0

63 I. TANO AND K. VÄNNMAN variability, methods 3 and 4 seem less reliable in this case. From Table IV, we see that also when r = 0.3, the values of the indices vary quite a lot although the difference between MC ðþ and MC ðþ is small. It may not be ossible to define an MPCI such that if its value equals, the robability P(NC) = However, is seems as a reasonable requirement that if the value of an MPCI equals, then P(NC) It is clear from the revious examle that the index MC ðþ by Taam et al. 3 does not meet this requirement. Further studies are needed to find out if this is the case for the other indices Secification limits As mentioned in the Section, Shinde and Khadse discovered a major roblem with how Wang and Chen 5 and Wang and Du 6 obtained the transformed tolerance region when using MPCIs based on PCA. Shinde and Khadse ointed out that the tolerance region for the PCs has to be calculated as follows. From Equation (3) and the roerty that the eigenvectors are orthogonal, we obtain X ¼ u PC þ u PC þ...þ u v PC v X ¼ u PC þ u PC þ...þ u v PC v : : X v ¼ u v PC þ u v PC þ...þ u vv PC v (34) Because LSL i < X i < USL i,i =,,...,v, the tolerance region for the PCs should be obtained from the following inequalities: LSL < u PC þ u PC þ...þ u v PC v < USL LSL < u PC þ u PC þ...þ u v PC v < USL : : LSL v < u v PC þ u v PC þ...þ u vv PC v < USL v (35) As an examle, if we insert the values of the eigenvectors from the examle in Section 5.3 into Equation (35), we obtain 0:707 PC þ 0:707 PC 0:707 PC 0:707 PC : (36) It is easily seen that the tolerance region that satisfies the inequalities in Equation (36) is the arallelogram in Figure 5. The correct tolerance region for (PC,PC ) in Figure 5 can be comared with the tolerance region roosed by Wang and Chen, 5 which degenerated because LSL PC ¼ USL PC ¼ 0 (see Section 5.3). This simle examle clearly shows that the definition of the tolerance region for the PCs by Wang and Chen 5 does not work well. This is a major drawback of the indices MC ð3þ and MC ð4þ. If the correct tolerance region in Figure 5 is to be used in the index MC ð3þ or MC ð4þ, a new roblem occurs because it is not obvious how to define the secification limits for the selected PCs when the tolerance region no longer is a rectangle. The secification limits for PC deend on the value of PC and the other way around. When the dimension is v >, this roblem becomes even more comlicated than when v =. This issue has to be solved before an index similar to MC ð3þ and MC ð4þ can be recommended for use PCA comuted on the variance covariance or the correlation matrix In the articles we have found that use PCA to form MPCIs, the eigenvalues and the eigenvectors in the examles are always calculated using the variance covariance matrix. This is the case even if the variances of the individual variables vary a lot or the variables are Figure 5. The correct tolerance region for PC and PC obtained from (35) Coyright 0 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 0

64 measured on scales with widely different ranges. In such situation, the general recommendation when doing PCA is to use the correlation matrix instead of the variance covariance matrix. However, methods 3 and 4 give quite different results, deending on which matrix is used in the PCA. This is illustrated in Table V, where our studied case is used. These differences can be exlained by the way the secification limits for the PCs are calculated according to Wang and Chen 5 and Wang and Du. 6 Figure 6 shows the tolerance regions for the PCs when the correlations and the variance covariance, resectively, are used. If the correct way, described in Section 5.4, is used to obtain the tolerance region for the PCs, it does not matter which matrix is used in the PCA. However, the roblem of finding the secification limits for each selected PC remains Concluding remarks Among the four considered indices with confidence intervals for MPCI, the index MC ðþ in Equation (0) with confidence bound in Equation () by Pan and Lee 4 seems to be the one to refer, when the correlations among the univariate quality characteristics are high. Its drawback is that confidence level is aroximate, and we do not know how good this aroximation is. The index MC ðþ in Equation () by Taam et al. 3 does not cature the correlation structure roerly when the correlations are high between the univariate quality characteristics and hence misjudge the rocess caability with resect to the robability of nonconformance. We show by an examle in Section 5.3 when r = 0.9 and MC ðþ ¼ the robability of nonconformance P(NC)=0.034, which is much I. TANO AND K. VÄNNMAN Table V. The estimated MPCIs and the lower confidence bounds for our case using methods 3 and 4, when the correlation and variance covariance matrix, resectively, are used in the PCA (a = 0.05) Method Correlation matrix Variance covariance matrix Method 3 (Wang and Du 6 ) ð3 ^MC ¼ 3: ð3 ^MC ¼ 0:89 Method 4 (Wang 9 ) Lower confidence bound:.54 Lower confidence bound: 0.77 ð4 ^MC ð4 ¼ :79 ^MC ¼ :05 Lower confidence bound:.53 Lower confidence bound: 0.90 a) b) 5 00 Score Plot based on the variance-covaraince matrix Second Comonent First Comonent Figure 6. The PC scores and the tolerance region for the PCs according to Wang and Du, 6 when the PCA is based on (a) the correlation matrix and (b) the variance covariance matrix Coyright 0 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 0