A BAYESIAN APPROACH FOR PROCESS CAPABILITY ANALYSIS Steven J.J4ttig and C. Teresa Lam Department of Industrial and Operations Engineering The University of Michigan Ann Arbor, MI 48109-2117 Technical Report 93-6 January 1993

A BAYESIAN APPROACH FOR PROCESS CAPABILITY ANALYSIS C. Teresa Lam and Steven J. Littig Industrial and Operations Engineering Department University of Michigan Ann Arbor, MI 48109 Abstract A Bayesian approach is developed to perform capability analysis for a one-dimensional normal process with symmetrical bilateral design specifications. The performance of the process is evaluated based on three separate criteria: actual proportion nonconforming, potential proportion nonconforming and process centering. Capability indices based on these criteria are defined and discussed. The probability that the process is capable given an observed sample is derived. The minimum values required for capability indices estimated from a sample to ensure that the process is capable are also presented in both tabular and graphical formats. 1 Introduction Integrated manufacturing systems require that accurate information be exchanged and analyzed freely between many different departments. Specifically, a quantitative measure of production quality is needed that can be well understood by people throughout the organization. It is this need for improved communication that motivated the development of process capability indices. According to Kane (1986), process capability indices are intended to provide a common, easily understood language for quantifying the performance of a process. For this reason, these indices have become the predominate measure of process performance used in industry today. Intuitively, process performance can be defined by how well the process produces parts which conform to the engineering design specifications. Unfortunately, the capability indices cur1

rently used are frequently inconsistent and misleading measures of conformance even for simple one-dimensional process characteristics. When one considers the expanding use of multivariate tolerances to improve quality, the deficiencies with the current indices become even more alarming. Because of these limitations, we have introduced new capability indices which solve these problems and yet complement the current system in order to minimize disruption to industry standards and procedures. In this paper, we introduce a system of three indices for measuring process performance and provide a brief discussion and justification for their use. We then employ a Bayesian approach for the case of a one-dimensional normal process with symmetric bilateral design specifications. Using a noninformative prior, the probability that the process is capable given an observed sample is derived and can be computed numerically. We define a capable process as a process where the three indices simultaneously satisfy certain prespecified criteria. Numerical results summarized in tables and figures are given to demonstrate the method and its usefulness in providing sound statistically based inferences for process capability analysis. 2 Process Capability Indices Process capability indices have become widely used as an attempt to convey the process performance in a dimensionless and easily interpreted manner. The earliest form of this type of capability index is U-L Cp= 6 where U and L are the upper and lower specification limits and a is the standard deviation of the characteristic. In the standard case of a one-dimensional process with symmetric bilateral design specifications, the nominal target value is centered between the specifications. The process characteristic is generally assumed to be normally distributed and thus the process performance is a function of both the process mean and standard deviation. As explained in Kane (1986), it is apparent that Cp strictly measures the potential process performance since only the standard deviation 2

is related to the design specifications. Since the location of the process mean is not considered, it is possible to have any proportion of parts outside the specification limits for a given Cp by merely locating the process mean sufficiently close to, or outside, the specification limit. Thus Cp only quantifies thepotential performance of a process which is attained when the process mean is equal to the design target T = (U + L)/2. The Cp index has become a fixture in industry for measuring the potential capability of a process with perfect centering. A benchmark of 1.0 was chosen to relate Cp to the standard six sigma spread used on control charts. Therefore, a process with Cp = 1.0 with an underlying stable normal distribution will produce nonconforming parts at a rate of 0.27 percent (or 2700 parts per million) assuming the process mean is centered at the target value. Here, we use the ANSI (1982) definition of a nonconforming part defined as any part thatfalls outside the tolerance zones. Because of this one-to-one relationship between Cp and the proportion nonconforming, capability indices are commonly used as a minimum benchmarking criteria for manufacturing processes. In fact, both GM and Ford Motor Company specifically document the use of process capability indices for assessing process performance. For example, Ford's Q-101 manual states the requirement that all manufacturing processes must achieve minimum process capability indices Cp and Cpk of 1.33 or better (1.0 for previously tooled parts). This standardized interpretation has spread and most engineers and managers responsible for quality gauge their processes with process capability indices in comparison to this 1.33 (or 1.0) benchmark. One major use of process capability indices is enable decision makers with little statistical training to make informed statistical decisions regarding the purchase or removal of long term capital equipment An informed decision regarding the capability of a process must then consider both the potential and actual performance of the process. While process potential is well summarized by Cp, there exists some debate on how to effectively measure the actual process performance. The two most common measures that have been introduced are Cpk [Kane (1986)] and Cpm [Chan, Cheng and Spiring (1988)]. These indices simultaneously account for both the mean and standard deviation of the process in an attempt to reflect the actual process performance. Unfortunately, 3

neither of these two indices have direct physical meaning. It has been shown that both Cpk and Cpm are inconsistent measures of the actual proportion nonconforming thus making interpretation difficult [Boyles (1991), Lam and Littig (1992)]. Furthermore, these indices are unsuitable for extension to more complex multivariate processes. In order to solve this problem, Lam and Littig (1992) suggest reporting measures that have direct physical meaning and supply information relevant to process description and improvement. In particular, they recommend three separate criteria to judge or evaluate the performance of a manufacturing process. 1. p = Actual Proportion Nonconforming The actual proportion of nonconforming parts currently being produced by the process is often of primary interest. It is a function of both process mean and process variation and therefore reflects the current state of the process. This measure describes the expected performance assuming no actions are taken to shift the process mean or to reduce the process variation. 2. p* = Potential Proportion Nonconforming This measures the minimum possible proportion of nonconforming parts that can be achieved through simple location shifts of the process mean. In the case of a symmetric distribution such as normal considered here, this minimum value will be attained when the process mean is centered within the specification limits. 3. k = Process Centering This measures the deviation of the process mean from the design target with respect to the allowable tolerance. For symmetric bilateral design specifications, a natural quantitative measure of process centering is the k index defined in Kane (1986). Specifically, if g is the process mean, Kane (1986) defines k 2|T-pj U-L which is the ratio of the deviation of the process mean from the design target with respect to the allowable tolerance. 4

As discussed in Lam and Littig (1992), while it is informative to report actual and potential proportion nonconforming, it is not directly compatible with the current corporate requirements. It is therefore convenient to transform actual and potential proportion nonconforming into indices that retain their physical meaning and at the same time communicate these values in a numerical format that is familiar to engineers, managers and technicians. In particular, if p* and p are the potential and actual proportion nonconforming of a process, they define Cp* = - (1 - ) and Cpp = 1 ( - (-) (1) where ~(x) is the integral of a standardized normal density from minus infinity to any real number x. Note that Cp* in Equation (1) above is defined such that Cp* = Cp when the underlying process follows a one-dimensional normal distribution. Hence, the interpretation of Cp* is consistent with the traditional Cp. This is of practical importance since Cp has become a well known, understood, and deeply entrenched fixture in industry for measuring potential capability. While Cp is only applicable to one-dimensional processes and used in situations when normality or near-normality is assumed, Cp* and Cpp applies to all manufacturing processes provided that the actual and potential proportion nonconforming can be computed with sufficient accuracy. In particular, Littig and Lam (1993) compute Cp* and Cpp for nonsymmetric bilateral design specifications, and unilateral design specifications. Littig, Lam and Pollock (1992) apply Cp* and Cpp to multi-dimensional processes. In general, the authors recommend that Cp*, Cpp and the k indices should be reported for any process characteristic in order to consistently and effectively communicate the ability of a process to meet the design specifications. 3 Joint Posterior Density Function of Cp* and CPU Let X be the one-dimensional characteristic of interest and assume that it is normally distributed with mean gi and standard deviation a. Define CPU as given in Kane (1986), CPU =3a 5

The likelihood function of p and a based on a sample x1, x2,...., xn is given by L(pa) = (2c2)-n/2exp [ (xi - )2] i=l 22' Using a noninformative prior for p and a meaning that p. and loga are approximately locally uniform, i.e., p(g,) oa, the joint posterior density function is given in Box and Tiao (1973) fl(. (, ~,s)= -c-n+ [ (n - 1)s2 (n- 1)/2 e fl(ga Ix s) = c-n+1 ( "2 exp- 2- (n- )s2 + n(x-)2]} (2) where x and s are the sample mean and standard deviation respectively, (n 1)2 (n- 1-1 C 2d2n e it 2 se l and r(x) is the gamma function evaluated at x. Observe that Cp* and CPU are functions of p and a and we can uniquely solve for pg and a in terms of Cp* and CPU. In particular, < =U-L and.U U- L)CPU. (3) 6Cp* 2Cp* If Cp* and CPU are point estimates of Cp* and CPU when x and s are used to estimate g and a respectively, then s =C and =U- )CPU. (4) 6Cp* 2Cp* Using the change of variables given in Equation (3) above with Jacobian equal to (U-L)2/( 12 Cp*) and substituting Equation (4) into (2), it is readily verified that the joint posterior density function of Cp* and CPU is given by f2(Cp*, CPU I Cp*, CPU) n- 1 2i, 3c n-iCp*\ i [( J, *Cp*\ 2 CPU CPUi'1 = Cp* p*) exp 2 (n 1) cp*) +9nC Cp* - ) Cp 6

The marginal posterior density function of Cp* is therefore given by n- 1 2 l.Ji p-'lh(Cp* I Cp*) = C - ) exp -n-1 (n) )C 4 A Bayesian Approach to Analyze Cp*, Cpp and k Suppose that a process is considered capable whenever Cp* > cl, Cpp > c2 and k < ko where c1, c2 and ko are prespecified constants. Observe that k < ko is equivalent to Cp*(l- ko) < CPU < Cp*(1 + ko). Also using Equation (1), Cpp > c2 is equivalent to the actual proportion nonconforming p <2 [1 - (3c2)]. Since we assume that the underlying one-dimensional process is normal, it follows that 1 - p = 4D(3CPU) - ((3CPU - 6Cp*) and we require that 0(3CPU) - (D(3CPU - 6Cp*) > 2((3c2) - 1 for the process to be considered capable. Given any positive real number x, define a function g(x) to be the smallest real number such that 4(3g(x)) - 0(3g(x) - 6x) = 2D(3c2) - 1. Since the normal density is symmetric, it is clear that Cpp > c2 is now equivalent to g(Cp*) < CPU < 2Cp* - g(Cp*). We can now evaluate P(process is capable | an observed sample) 7

to be equal to q(cl,c2,ko) - P(Cp* > cl,Cpp > c2,k < ko I Cp*,CPU) 00 min{x(l+ko),2x-g(x)} IJmax{ x(-ko) g(x) f2(x,y I Cp*,CPU)dydx. In the special case when c2 = 0 and ko = oo, q(c1,0,oo) = P(Cp* > c Cp*) = h(x I Cp*)dx. l1 The Bayesian approach to analyze the capability of a normal process with symmetric bilateral design specifications has been studied before by Cheng and Spiring (1989) for Cp and Chan, Cheng and Spiring (1988) for Cpm with t = T. The minimum values of Cp* required to ensure that q(c,0, oo) = 0.9, 0.95 and 0.99 with cl = 1 are tabulated in Cheng and Spiring (1989). Hence, their results are a special case of the approach considered here. In Figure 1, the curves represent the minimum required values for Cp* and Cpp such that q(1.0, 1.0, 0.33) = 0.90, 0.95 and 0.99 for sample size n = 25, 50, 75,100, 150 and 200. Suppose the process is considered to be capable whenever Cp* > 1.0, Cpp > 1.0 and the process mean lies within the middle third of specification zone (i.e., k < ko = 0.33). For example if n = 50, Cp* = 1.5 and Cpp = 1.25, then the probability that the process is capable is greater than 0.95 but less than 0.99. This can be seen from Figure 1 since the point for the the sample values of Cp* and Cpp lies above the curve for n=50 at 0.95 but below the curve for n=50 at 0.99. Given an observed value of Cp*, Tables 1 and 2 provide the minimum required Cpp values such that q(c1, c2, ko) = 0.90, 0.95 and 0.99 for sample size n = 25, 50,75,100, 150 and 200, and various combinations of cl, c2 and ko. Suppose now that the process is considered to be capable whenever Cp > 1.33 and Cpp > 1.33. For example, if n = 100 and we observe that Cp* = 2.0, then the process is considered capable with probability 0.95 whenever Cpp > 1.508 from Table 2 with q(1.33, 1.33, oo) = 0.95. Note that in Figure 1, all the curves decrease and then increase sharply for large values of Cp*. This is because for large Cp* values, k < kg = 0.33 is the dominant constraint whereas for 8

smaller values of Cp*, the constraint Cpp > 1.0 dominates. In the case when k > 1.0, all the curves decrease to a constant value of Cpp as Cp* increases. This can be seen directly from Table 1 with q(1.0, 1.0, oo). For example when n = 50 at 0.99, as Cp* increases from 1.33 to 2.0, the minimum required Cpp decreases from 1.309 to the constant value of 1.292. Reference ANSI Standard Y14.5M. Dimensioning and Tolerancing, ASME (1982). Boyles, R. A. (1991). "The Taguchi Capability Index". Journal of Quality Technology 23, pp. 17-26. Box, G. E. P.; and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Addison-Wesley Publishing Company. Chan, L. K.; Cheng, S. W.; and Spiring, F A. (1988). "A New Measure of Process Capability: Cpm". Journal of Quality Technology 20, pp. 162-175. Cheng, S. W.; and Spiring, F. A. (1989). "Assessing Process Capability: A Bayesian Approach". lIE Transactions, 21, pp. 97-98. Kane, V. E. (1986). "Process Capability Indices". Journal of Quality Technology 18, pp. 41-52. Lam C. T.; and Littig, S. J. (1992). "A New Standard in Process Capability Analysis". Technical Report 92-23, University of Michigan. Littig, S. J.; Lam, C. T.; and Pollock, S. M. (1992). "Process Capability Measurements for a Bivariate Characteristics over an Elliptical Tolerance Zone". Technical Report 92-42, University of Michigan. Littig, S. J.; and Lam, C. T (1993). "Case Studies in Process Capability Measurement". To appear in The Proceeding and ASQC Quality Congress Transactions. 9

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q(1.0, 1.0, oo) q(1.0, 1.0, 0.33) q(1.0, 1.0, 0.2) Cp n 0.90 0.95 0.99 0.90 0.95 0.99 0.90 0.95 0.99 25 1.239 1.239 1.259 50 1.149 1.198 1.309 1.149 1.198 1.309 1.200 1.224 1.309 1.33 75 1.117 1.154 1.230 1.117 1.154 1.230 1.183 1.198 1.238 100 1.100 1.130 1.192 1.100 1.130 1.192 1.174 1.186 1.214 150 1.079 1.103 1.151 1.079 1.103 1.151 1.165 1.175 1.194 200 1.068 1.088 1.128 1.068 1.088 1.128 1.160 1.169 1.184 25 1.230 1.308 1.232 1.308 1.352 1.384 50 1.148 1.195 1.293 1.158 1.197 1.293 1.318 1.336 1.372 1.50 75 1.117 1.153 1.227 1.133 1.158 1.227 1.307 1.320 1.347 100 1.099 1.130 1.190 1.120 1.138 1.190 1.300 1.312 1.334 150 1.079 1.103 1.150 1.107 1.119 1.152 1.292 1.302 1.320 200 1.068 1.088 1.128 1.100 1.110 1.132 1.288 1.296 1.312 25 1.229 1.305 1.475 1.280 1.323 1.475 1.474 1.501 1.561 50 1.148 1.195 1.292 1.237 1.256 1.305 1.446 1.463 1.497 1.67 75 1.117 1.153 1.226 1.223 1.237 1.267 1.435 1.449 1.476 100 1.099 1.130 1.190 1.215 1.227 1.251 1.429 1.441 1.463 150 1.079 1.103 1.150 1.207 1.217 1.235 1.421 1.431 1.449 200 1.068 1.088 1.128 1.203 1.211 1.227 1.416 1.425 1.440 25 1.229 1.305 1.470 1.474 1.501 1.559 1.731 1.757 1.809 50 1.148 1.195 1.292 1.446 1.464 1.497 1.705 1.723 1.756 2.00 75 1.117 1.153 1.226 1.435 1.449 1.476 1.694 1.708 1.735 100 1.099 1.130 1.190 1.429 1.441 1.463 1.688 1.700 1.723 150 1.079 1.103 1.150 1.421 1.431 1.449 1.680 1.690 1.708 200 1.068 1.088 1.128 1.416 1.425 1.440 1.675 1.684 1.700 Table 1: Minimum Required Values for Cp* and Cpp such thatq(l.0, 1.0, k))= 0.90,0.95 and 0.99 for Various kc and n 11

q(1.33, 1.33, oo) q(1.33, 1.33, 0.2) q(1.33, 1.0, 0.25) Cp* n 0.90 0.95 0.99 0.90 0.95 0.99 0.90 0.95 0.99 25 1.458 50 1.535 1.604 1.536 1.604 1.372 1.396 1.67 75 1.491 1.541 1.651 1.492 1.541 1.651 1.356 1.371 1.410 100 1.467 1.508 1.593 1.469 1.508 1.593 1.349 1.361 1.385 150 1.440 1.472 1.536 1.445 1.472 1.536 1.341 1.350 1.369 _ 200 1.425 1.452 1.505 1.432 1.453 1.505 1.336 1.344 1.360 25 1.643 1.746 1.748 1.790 1.637 1.655 1.764 50 1.534 1.597 1.727 1.708 1.726 1.770 1.608 1.625 1.659 2.00 75 1.491 1.540 1.638 1.695 1.709 1.737 1.597 1.611 1.638 100 1.467 1.508 1.589 1.688 1.700 1.723 1.590 1.602 1.625 150 1.440 1.472 1.535 1.680 1.690 1.708 1.582 1.592 1.611 200 1.425 1.452 1.505 1.675 1.684 1.700 1.578 1.586 1.602 25 1.643 1.745 1.969 1.994 2.022 2.083 1.878 1.904 1.957 50 1.534 1.597 1.727 1.966 1.984 2.018 1.852 1.870 1.903 2.33 75 1.491 1.540 1.638 1.955 1.969 1.996 1.841 1.855 1.882 100 1.467 1.508 1.589 1.949 1.961 1.984 1.834 1.846 1.869 150 1.440 1.472 1.535 1.941 1.951 1.969 1.826 1.836 1.855 200 1.425 1.452 1.505 1.936 1.945 1.961 1.822 1.830 1.846 Table 2: Minimum Required Values for Cp* and Cpp such that q(1.33, c2, ko) = 0.90, 0.95 and 0.99 for Various c2, k and n 12