Diskussionsbeiträge des Fachgebietes Unternehmensforschung

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1 Diskussionsbeiträge des Fachgebietes Unternehmensforschung New Methods in Multivariate Statistical Process Control MSPC) Lorenz Braun November 2001 Universität Hohenheim 510 B) Institut für Betriebswirtschaftslehre Fachgebiet für Unternehmensforschung D Stuttgart

2 New Methods in Multivariate Statistical Process Control MSPC) Lorenz Braun 14th November 2001 Abstract In this article a new multivariate capability index and a multivariate acceptance chart will be introduced. These methods are an integral part of a global approach viewing multivariate quality control. In this approach both, the multivariate process region and the multivariate tolerance region are of elliptical shape. Beside the new methods also the χ 2 and the T 2 -charts are parts of this approach. In addition some methods for interpreting the exceeding of multivariate tolerance or control limits are introduced. Keywords: Multivariate Quality Control, Process Capability, Acceptance Control Charts Introduction In recent years Statistical Process Control SPC) has been established for controlling and judging processes in many industrial enterprises. Above all some new standards in the car industry like QS 9000 in the United States and VDA 6.1 in Germany have contributed to the spreading of the methods of SPC. The most important methods in SPC are the analysis of capability and the techniques of control charts. The process capabilities C p and C pk are calculated to verify whether a quality characteristic lies within its specification limits or not. On the other hand the goal of a control chart is to verify, if a process and therefore its quality characteristics are stable over time Shewhart charts) or if a quality characteristic lies within its specification limits acceptance charts). During an univariate view of all relevant quality characteristics of a product, the process capabilities and the limits of the control cards are calculated independently. Usually the quality characteristics are stochastically dependent and therefore during an univariate view problems can occur. For example the case can arise that all univariate process capabilities C p.j and C pk.j for all j 1,..., m quality characteristics are larger or equal 1 process seems to be capable), the products requirements however are not fulfilled. The reverse case arises, if at least one process capability C pk.j is less than 1, the product however still can be used. This misinterpretation can cause substantial costs. On the one hand products are used although they reach not their requirements, on the other hand products are scrapped or worked over again, although they meet their requirements. These statements lead back on Diplom-Betriebswirt FH) Lorenz Braun graduates at the Faculty of Economics and Social Sciences at the University of Hohenheim in Stuttgart Germany) 1

3 statistical regularities, which consider the structure of dependency of the quality characteristics. In order to avoid such misinterpretations, multivariate methods of the SPC were developed. To be called are the χ 2 - and T 2 -control chart, which is to be interpreted as multivariate extension of the average control chart. In this paper, as a generalization of the process capabilities C p and C pk, a new multivariate process capability and a new multivariate acceptance chart are introduced and examined regarding their properties. In addition some procedures are presented, which permit a sensitive analysis of the introduced multivariate methods of the SPC, e.g. if the multivariate process capabilities are too small, or the control limits of a multivariate control chart are exceeded. For the multivariate acceptance chart in addition an application in receiving inspection is presented. In the following firstly the fundamental problems of the multivariate statistical process control are discussed. Subsequently follows the introduction of the elliptical process capabilities EC p and EC pk 1. Before the multivariate acceptance chart MAC) is presented, the χ 2 - and the T 2 -chart are introduced briefly. Both charts are part of a global approach of the MSPC. In this global approach the quality characteristics are assumed to be multivariate normal distributed and therefore their probability region in the two-dimensional case corresponds to an ellipse and in the higher-dimension case to an hyper-)ellipsoid. From tolerance interval to tolerance ellipse In the univariate case each quality characteristic is regarded individually and independent of all different quality characteristics. The specifications for a quality characteristic are defined by a tolerance interval tolerance region), which is described by an upper T u ) and a lower specification limit T l ) and in its center the target value M is normally situated. Assuming normal distribution for the quality characteristics, the actual process spread process region) is defined according to the 3σ-rule with [µ ± 3σ] as an interval, which contains 99,73 per cent of all values of the quality characteristic. Thus in the univariate case the tolerance and the actual process spread are described by an interval. In the case of higher-dimension the shape of the tolerance region and the shape of the process region does not correspond. For the two-dimensional case this is to be clarified. The two tolerance intervals together yield a tolerance rectangle or in the standardized case a tolerance square. 2 Proceeded from a bivariate normal distribution of the two quality characteristics, in the independent case the shape of the process region is a circle, in the dependent case an ellipse. Within the process region the values of the two quality characteristics occur with a given probability of 1 α. Now however if the tolerance region is described by a square, it does not correspond to the shape of the elliptical process region. This leads to contradictory statements, which are to be clarified at a concrete case. If a pair of two quality characteristics is situated outside of the elliptical process region, but within the rectangular tolerance region see point P in Figure 1 - left side), then it cannot be decided whether the process is capable or not. The point P is situated within the tolerance region process is capable), but is situated at the same time outside of the 1-α) process region. This means, one has to assume another, shifted process region see Figure 1 - right side), which fits not completely into the rectangular tolerance region process is not capable). Therefore it will be assumed that with a given probability of 99,73 per cent one point P cannot occur if a process is capable. 1 Elliptical Capability of Process 2 Standardization in this context means that each quality characteristic has a expected value of zero, a variance of one and the same tolerance intervals 2

4 Figure 1: Tolerance and process region in the two-dimensional case For this reason the tolerance rectangle is to be replaced by a tolerance ellipse, considering the dependency of the two quality characteristics. The surface of the tolerance ellipse or the volume of a tolerance hyper-)ellipsoid is 3 V ol T ol Σ th yy 1/2 χ 2 m, π) m/2 [Γm/2 + 1)] 1, 1) whereby m is the number of quality characteristics, Γ corresponds to the gamma function and χ 2 m, is the 0,9973 percentile of the χ 2 distribution with m degrees of freedom. Σ th yy is the hypothetical variance matrix of the quality characteristics, which fits with a given probability of 99,73 per cent into the tolerance rectangle or in the tolerance hyper-)cube. The hypothetical variance matrix is calculated using the correlation matrix of the quality characteristics R yy by 4 Σ th yy diag T u T l 6 ) R yy diag T u T l 6 ). 2) diag... ) is a matrix, whose elements beside its main diagonal elements are equal to zero and whose main diagonal elements are, corresponding to the 3σ rule, the sixth part of the tolerance intervals of the quality characteristics. By formula 1) the volume of the tolerance ellipsoid can be calculated. A Comparison of this volume with the volume of the process region V ol P roz ) is the basic of the elliptical process capability. From uni- to multivariate process capability With the basic process capability C p the width of the distribution of a quality characteristic is set into relation to the tolerance interval. The width of the normal distribution is determined according to the 3σ rule in such a way that 99,73 per cent of all values of a quality characteristic are considered. The C p is 5 C p T u T l. 3) 6σ 3 see Kotz, S. and N. L. Johnson 1993), S. 184 and Taam, W., P. Subbaiah and J. W. Liddy 1993), S see Jahn, W. 1994), S. 440 ff. 5 see Montgomery, D. C. and G. C. Runger 1994), S. 851 ff. 3

5 The basic process capability does not consider a deviation between the target value M and the expected value µ. Therefore the basic process capability have to be corrected in the case of a deviation. The corrected process capability C pk is defined by a factor k with k M µ T u T l )/2, 4) C pk C p 1 k). 5) If C p < 1, then the process is not capable, since the spread of the quality characteristic does not fit into the tolerance interval. A process is capable, if C p > 1 and C pk > 1 applies. The more largely the difference between the expected value and the target value will become, the more the factor k will increase. If the expected value of a quality characteristic lies exactly on a specification limit, then applies k 1 and therefore C pk 0. In practical calculation the expected value and the standard deviation of the population are replaced by their estimated values. The basic elliptical process capability EC p compares the per cent concentration ellipsoid elliptical process region) of the vector Y T Y 1..., Y m ) of the quality characteristics with the tolerance ellipsoid elliptical tolerance region). That EC p is 6 7 EC p ) 1/m V ol T ol, V ol P roz m ) 1/m. Cp.j 6) j1 C p.j thereby corresponds to the univariate basic process capabilities of the m quality characteristics. The basic elliptical process capability is not influenced by the dependency structure of the quality characteristics. This is also meaningful, since otherwise EC p would depend on the degree of multicollinearity. On the other hand the deviation between the vector of expected values M T M 1,..., M m ) and the vector of the expected values µ T µ 1..., µ m ) must consider the dependency structure of the quality characteristics. The factor of correction of the elliptical process capability K E and the corrected elliptical process capability are ) T ) K E µ M Σ th 1 ) yy µ M, 7) χ 2 m, EC pk EC p 1 K E ). 8) The numerator in formula 7) corresponds to an elliptical equation. The denominator is the percentile of the χ 2 distribution with m degrees of freedom. The further the vector of expected values withdraw from the vector of target values, the larger the counter becomes in relation to the denominator. If a value K E of 1 arises, the vector of expected values lies exactly on the limit sphere) of the tolerance ellipsoid. If the vector of expected values is situated outside of the tolerance region, the EC pk becomes negative. Thus the EC p and the EC pk must be interpreted similar to the univariate process capabilities C p and C pk. 6 see Braun, L. 2002), S. 105 ff. 7 The derivation of this formula is in the appendix 4

6 Uni- and multivariate control charts By control charts it will be checked whether a quality characteristic of a capable process C pk > 1 or EC pk > 1) persists constantly over time Shewhart charts) or whether the quality characteristics stay within their specification limits over time acceptance charts). In the univariate case using Shewhart charts it is checked by a periodical draw of small samples usually sample size N 5) whether the expected value µ and the standard deviation σ of a quality characteristic persists constantly, respectively lies within a given 1 α) confidence interval. If this interval is exceeded, it has to be assumed that the quality characteristic and therefore also the process has changed. Ordinary a so-called double card is led, into which a value for the location average value or median) and a value for the dispersion range or standard deviation) are entered. The χ 2 - and the T 2 -charts can be regarded as the multivariate extension of the Shewhart mean chart. 8 In the χ 2 chart the values of the quadratic form N Y µ ) T ) Σ 1 yy Y µ 9) are plotted, whereby the control limit corresponds to the 1 α) percentile of the χ 2 distribution with m degrees of freedom. Y T Y 1..., Y m ) corresponds to the vector of means estimated from a sample of size N. If it is assumed that the matrix of variance Σ yy is unknown and must be estimated from the sample, a T 2 chart will be led. Into this chart the values N Y µ ) T ) S 1 yy Y µ, 10) are entered, whereby S yy corresponds to the variance matrix estimated from the sample. The control limit CL) depends in this case on sample size N and on the number of quality characteristics m and is calculated by the F distribution with CL N 1)m N m F m,n m),1 α. 11) From uni- to multivariate acceptance charts With a univariate acceptance chart it is to be checked whether a quality characteristic is situated within its specification limits or not. Assumed that the standard deviation of the quality characteristic is known and constant, the expected value however may vary. An upper UCL) and a lower control limit LCL) is calculated, which define an interval, within the sample means can move, without exceeding the specification limits by a given probability). The control limits are 9 UCL T u z 1 α1) σ z 1 α2) σ N, 12) LCL T l + z 1 α1) σ + z 1 α2) σ N. 13) 8 see Sparks, R. S. 1992), S. 375 ff. 9 see DGQ 1995), S. 69 ff. 5

7 z 1 α1) corresponds thereby to the 1 α) percentile of the standard normal distribution and represent the allowed dispersion of the individual values of a quality characteristic. Since the expected value also varies in the acceptance chart, the allowed dispersion of means is considered by z 1 α2). σ/ N corresponds thereby to the standard deviation of the mean, i.e. σ Y. If the standard deviation is unknown σ, it will be estimated from the sample by s. In this case the control limits are calculated by percentiles of the t distribution instead of the percentiles of the standard normal distribution, with t N 1,1 α1) respectively t n 1,1 α2) )/ N. For the multivariate acceptance chart MAC) a control limit must be found, which considers the dispersion of the individual values and of the vector of expected values of the quality characteristics in consideration of their dependency structure). In the case of the univariate acceptance chart this happens via a security interval add on the lower control limit respectively subtract from the upper control limit see formula 12) and 13). The control limit of the MAC is the squared generalization of the control limits of the univariate acceptance chart and depends on the simple elliptical process capability EC p. The control limit CL) for the MAC is 10 1/2 CL χm,0.9973) 2 EC p χ 2 m,1 α1)) 1/2 ) χ 2 1/2 2 m,1 α2). 14) N The term [ χ 2 m,0.9973) 1/2 EC p ] 2 corresponds to the theoretical χ 2 value of an elliptical equation in the case of a given variance matrix Σ yy, whose boundary is the same as the boundary of the tolerance ellipsoid. On the basis of this theoretical χ 2 value the allowed dispersion of the individual values and of the vector of expected values of the quality characteristics is subtracted. Into the MAR the result of the squared form ) T ) Y M Σ 1 yy Y M 15) is entered. If the result is situated outside the control limit, then similar to the univariate case it is assumed that values of the quality characteristics occur outside the tolerance ellipsoid with a given probability). Now it is assumed that the variance matrix Σ yy is unknown and will be replaced by its estimation S yy. Then the control limit is CL [ ) χ 1/2 2 1/2 N 1)m m,0.9973) an EC p N m F m,n m,1 α1) ) ] 1/2 2 N 1)m N m)n F m,n m,1 α2). 16) Thereby a N is a factor of correction which becomes necessary, because the standard deviations of the quality characteristics is systematically underestimated in particular using a small sample size and thus the elliptical process capability EC p is systematically overrated. The factor a N is 11 a N 10 The derivation of this formula is in the appendix 11 see Hartung, J. and B. Elpert 1984), S N 1 ΓN/2) ΓN 1)/2). 17) 6

8 Moreover F...) corresponds to the F distribution with the corresponding degrees of freedom. Apart from the classical application as acceptance chart the formula 16) can also be used for receiving inspection. It is not checked, whether a single quality characteristic lies into its specification limits, but whether the entire product meets its requirements. Analyze exceeding the control limit Starting from an elliptical tolerance region in MSPC, an error occurs whenever the tolerance ellipsoid is exceeded. Thus e.g. the corrected elliptical process capability EC pk becomes negative, if the vector of expected values is situated outside of the tolerance ellipsoid. In the case of the multivariate extensions of the mean chart T 2 - and χ 2 -chart) one intervenes, when the 1 α) probability ellipsoid of the vector of expected values is exceeded. In the case of the MAC one intervenes when the intervention ellipsoid control limit) is exceeded. The exceeding of the control limit in the multivariate case coincedes not necessarily with an exceeding of the control limits of an individual quality characteristic. A certain combination of the values of the quality characteristics can also be responsible for exceeding the multivariate control limit, although all univariate control limits are kept. Thus the univariate control limits can only give a first note in searching for the cause of exceeding the multivariate control limit. The so-called conditional control limits CL j of the individual quality characteristics offer a better interpretation. These indicate the allowed deviation between the target value and the expected value of a quality characteristic Y j, without exceeding the multivariate control limit, on condition that all other quality characteristics are constant and zero. The conditional control limits are 12 UCL j /LCL j M j ± CL σj/m j 2, 18) whereby σj/m j 2 is the variance of the jth quality characteristic, on condition that the remaining quality characteristics m-j) are constant. σj/m j 2 is the reciprocal value of the j th main diagonal element of the inverse variance matrix. The more narrow an interval [UCL j LCL j ] is in relation to the univariate interval [UCL j LCL j ], the more important is the quality characteristic for staying within the multivariate control limit. A further possibility to facilitate the interpretation of exceeding the multivariate control limit are the conditional elasticities ɛ j. They indicate a change of the result of the squared form Q in per cent, if the value of the j th quality characteristic increases by one per cent, on condition that all other quality characteristics are constant, but not necessarily zero. They are calculated by ɛ j Q Q Y j, 19) Y j whereby Y j is an increase of 1 per cent of the current deviation between Y j and the target value MAC) or the expected value χ 2 - or T 2 -chart). The larger ɛ j becomes, the more change of Q can be achieved by a change of the value of the j th quality characteristic. 12 see Jahn, W. 2001) and Wang, F. K. u.a. 2000), S

9 Examples of use Figure 2: Graphical representation of Example 1 In this section three possible application of this multivariate approach are presented. First the calculation of univariate and elliptical process capabilities is compared Example 1). In Example 2 the leading of a MAC is introduced an in Example 3 an application of the MAC in case of a receiving inspection is presented. Example l This example was taken from Taam, Subbaiah and Liddy 1993). 13 The structure of a module on an integrated circuit is described among other things by the quality characteristic width Y 1 ) and thickness Y 2 ). The target values of the two quality characteristics in µm are M T 4.5, 0.75) and the specification limits are T T l 4, 0.5) and T T u 5, 1). From a sample of N50 one obtains the vector of means Y T 4.3, 0.8) and the estimated variance matrix S yy ). 20) With the formulas 3), 4) and 5) follows the univariate process capabilities C p , C pk and C p , C pk The elliptical process capabilities are according to the formulas 6), 7) and 8) EC p and EC pk The relationship between the process and the tolerance region for this example is shown in Figure 2. Because the vector of means is situated only scarcely within the tolerance ellipse modified tolerance region), the corrected elliptical process capability EC pk becomes very small. Example 2 In the second example a drilling in a steel sheet is examined more exactly. On a workpiece the three quality characteristics distances X and Y axle are measured Y 1 and Y 2 ) as well as the 13 see Taam, W., P. Subbaiah and J. W. Liddy 1993), S. 343 ff. 8

10 Figure 3: Graphical representation of Example 2 diameter of the drilling Y 3 ). The specifications of the quality characteristics in millimeter are M T 25, 25, 20) and T T u 24.9, 24.9, 19.75) respectively T T o 25.1, 25.1, 20.25). From a large sample the variance matrix Σ ) was determined. From this using formula 6) EC p is calculated. According to formula 14) with α1) and α2) 0.05 a control limit of CL is calculated. Altogether 10 further samples were taken out of the process and the results squared statistics) were entered into the MAC in Figure 3. Figure 3 shows that between the sample number 5 and 6 theoretically a modification of the process took place. This modification could be lead back to a systematic deviation between the vector of target values and the vector of expected values. A more exact analysis was not made. Example 3 An enterprise manufacturing valves buys in addition valve covers. In the receiving inspection it is checked whether the valve cover fulfills its requirements and therefore will be assembled without problems. On the basis of the quality characteristics external dimensions 1 and 2 Y 1 and Y 2 ) and a diameter Y 3 the supply should be checked by a sample of the size N10. The valve cover with the quality characteristics is represented in figure 4. The specifications for the quality characteristics are M T 30, 27.5, 5.5) and T T l 29.5, 27, 5) respectively T T u 30.5, 28, 6). From a sample of N 10 follows the vector of means and the variance matrix Y T , , 5.607) 9

11 Figure 4: Graphical representation of Example 3 S yy ) According to formula 16) with α1) 0.05, α2) 0.05 and EC p one get a control limit of CL If during the receiving inspection a value of the squared form of more than 9,253 will be calculated, then the supply has to be rejected. In the example the result of the squared form is Q Therefore the supply has to be rejected because of the sample. A more exact analysis of the three quality characteristics follows on the basis of formulas 12) and 13) with the t distribution and 9 degrees of freedom k A ) for the upper and lower univariate control limits UCL T , , 5.781), LCL T , , 5.219). None of the individual quality characteristics is situated outside of their univariate control limits. Therefore a step forward the conditional control limit should be calculated according to formula 18). One get UCL T , , 5.713), LCL T , , 5.287). Moreover none of the quality characteristics is situated outside of its conditional control limit. Only the quality characteristic Y 1 is close to its conditional control limit. In order to meet still more exact statements the elasticities according to formula 19) with ɛ T 1.292, 0.376, 0.339) are calculated. It is shown that by an increase of the deviation between Y 1 and M 1 around 1 per cent the T 2 value will increase by per cent. An acceptance of future supplies requires therefore e.g. a reduction of the deviation between Y 1 and M 1. 10

12 Summary The presented methods are to be regarded as elements of a global approach of MSPC. In this approach both, the tolerance and the process region are defined as an ellipse or an hyper-)ellipsoid. Substantial elements of the MSPC are the elliptical process capabilities EC p and EC pk, which can be interpreted as the univariate capabilities C p and C pk. The multivariate extensions of the mean chart, the χ 2 and T 2 chart, follow also from an elliptical process region and are therefore elements of the approach. The MAC introduced in this work can be used classically as a control chart and also for receiving inspection. In addition some methods were shown for interpreting the exceeding of a multivariate control limit. Using an elliptical tolerance region in the multivariate case, a precise control of processes becomes possible, considering the dependency structure of the quality characteristics. This means that products can be judged as a whole regarding their requirements and not only as individual quality characteristics according to their specification, independently of all other quality characteristics. Appendix Elliptical Process Capability EC p ) 1/m V ol T ol V olp roc ) 1/2m Σ th yy Σ yy m ) 1/m C p.j j1 Σ th yy 1/2 χ 2 m, π) m/2 [Γm/2 + 1)] 1 Σ yy 1/2 χ 2 m, π)m/2 [Γm/2 + 1)] 1 m ) 1/m j1 σth j m j1 σ j m j1 T u.j T l.j ) m j1 6 σ j ) 1/m ) 1/m Principle of calculating the CL for MAC V ol P roz ) 1/m V olt ol ) 1/m Σ yy 1/2m χ 2 ) 1/2 m,1 α Σ th yy 1/2m χ 2 ) 1/2 m, χ 2 m,1 α χ2 m, ECp 2 References Braun, L. 2002): Statistisches Prozessmanagement - Modellierung betrieblicher Prozessnetzwerke mit multivariaten Methoden, Dissertation, to appear DGQ 1995): SPC II -Qualitaetsregelkartentechnik, Band 16-32, Frankfurt u.a. Hartung, J. and B. Elpert 1984): Multivariate Statistik, Muenchen, Wien, Oldenburg 11

13 Jahn, W. 1997): Prozesse sensibler steuern, in: Qualitaet und Zuverlaessigkeit, Jg. 42/4, S Jahn, W. 2001): Prozessverbesserung mit multivariaten Methoden, to appear Kotz, S. and N. L. Johnson 1993): Process capability indices, Great Britain Montgomery, D. C. and G. C. Runger 1994): Applied statistics and probability for engineers, New York u.a. Sparks, R. S. 1992): Quality control with multivariate Data, in: Austral. J. Statist., 343), S Taam, W., P. Subbaiah and J. W. Liddy 1993): A note on multivariate capability indices, in: Journal of Applied Statistics, Vol. 20, No. 3, S Wang, F. K. u. a. 2000): Comparison of three multivariate process capability indices, in: Journal of Quality Technology, Vol. 32, No. 3, July, S