CHAPTER 5 ESTIMATION OF PROCESS CAPABILITY INDEX WITH HALF NORMAL DISTRIBUTION USING SAMPLE RANGE

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1 CHAPTER 5 ESTIMATION OF PROCESS CAPABILITY INDEX WITH HALF NORMAL DISTRIBUTION USING SAMPLE RANGE

2 In this chapter the use of half normal distribution in the context of SPC is studied and a new method of estimating σ for this distribution using sample range is discussed. The proportionality constant used to express σ as a function range is obtained by a simulation study with different samples of different sizes generated from half normal distribution. 5.1 INTRODUCTION Process capability studies have been gained increasing importance in process control. The process capability is expressed in terms of a Process Capability Index (PCI), a dimensionless measure for the capability of the process, also an important measure of performance of a manufacturing system, denoted by C p. We assume a continuous random variable X denoting the measurement on a continuous quality characteristic with the distribution function F(x) say. Given the upper and lower specification limits (USL and LSL) for X. We wish to estimate the process capability in relation to these limits by using C p = USLLSL, where σ denotes the process standard deviation. This formula is based on the principle of Shewhart s control chart in which the spread between control limits is 6σ. When the process characteristic X follows normal distribution σ is estimated by using sample standard deviation. Alternatively if there are m samples, each of size n = 4 or 5 say, then this σ can be estimated by using, σ R, where R denotes the average of ranges (maxmin) over m sample and d 2 can be found in the statistical table for the given sample n. Since a control chart is usually maintained for process control, it is always possible to get R. As done in the case of half-logistic distribution here we propose a new estimate σ R N N N where k, is the bias factor. The value of k, is estimated by simulation, with different sub group sizes and taking a large number of samples generated from a half-normal distribution with mean 0 and known σ. σ This chapter has been presented at International Conferences on Recent Advances in Statistics and Their Applications (ICRASTAT-2013), authored by Ramesh B. & Sarma K.V.S. 67

3 After estimating σ, the process capability is estimated for both centered and eccentric process and the results are compared with the true value by using the known σ. The effect of changing σ on C p and C pk is also demonstrated. 5.2 A MORE GENERAL FORM OF PROCESS CAPABILITY INDEX FOR NORMAL PROCESSES On the basis of Vannman(1995), Pearn and Chen(1997) have proposed a generalized index, which they called superstructure to include the four basic indices, C p, C pk, C pm and C pmk as special cases. The superstructure has been referred to as C p (u,v), which can be defined as, C p (u,v) = µ σ µt (5.1) where µ - is the process mean given by (USL+LSL)/2 σ - is the process S.D. d - is half of the length of the specification interval, given by d = USLLSL m - is the midpoint between the USL and LSL, given by (USL+LSL)/2 T is the target value given by (USL+LSL)/2 = µ (u,v) is the positive integers, takes values 0 and 1. By using different values for u and v, the superstructure C p (u,v) produces the following values. C p (u,v) = C p (0,0) = C p = USLLSL (5.2) σ C p (u,v) = C p (1,0) = C pk = min USLµ σ C p (u,v) = C p (0,1) = C pm = USLLSL σ µt, µlsl (5.3) σ (5.4) C p (u,v) = C p (1,1) = C pmk = min USLµ, µlsl σ µt σ µt The above four indices are also inter related as given below. (5.5) C pm = C p 1 µt / σ (5.6) 68

4 C pmk = C pk 1 µt / σ (5.7) These indices can be ranked in the sense of sensitivity to departure of process mean from the target value as C p < C pk < C pm < C pmk (5.8) For symmetric tolerances, we can show that, C pk = (1-k)C p (5.9) C pmk = (1-k)C pm (5.10) where k = µ T /d, is the departure ratio. If the process is on-target, which implies µ T then, we get k = 0. By substituting µ T in the equations (5.4) & (5.5) and k = 0 in equations (5.9) & (5.10), we get, C pm = C p C pmk = C pk C pk = C p C pmk = C pm C pmk = C pm = C pk = C p = USLLSL = (5.11) σ σ The estimators of the indices C p (u,v) may be obtained by replacing µ=x and σ 2 = S 2 where X X, and S 2 = X X in equation (5.1). For the normal distribution, these estimation based on X and S 2 are quite suitable, stable & reliable, whereas for non-normal distribution these are not suitable and unstable, since the distribution of the sample variance S 2, is sensitive to departure from normality. Since the basic indices C p (u,v) are in-appropriate for processes with non-normal distributions, we considered a two generalizations of C p (u,v) as C Np (u,v) and C N u, v, where N stands to indicate non-normality. 69

5 Following the logic used by Pearn and Kotz (1994) and Pearn and Chen (1995), new indicators of process capability were proposed by Pearn and Chen (1997) who used the subscript N to indicate non-normality of the underlying distribution. 5.3 COMPUTATION OF PROCESS CAPABILITY FOR NON-NORMAL PROCESSES USING PERCENTILES Two generalizations of the basic index C p (u,v) as, C Np (u,v) and C N u, v were proposed by Pearn and Chen (1997) in which σ is replaced by F.F. and the process mean μ is replaced by M (median) since the process median is a robust measure of the central tendency than the process mean, particularly for skew distributions with long tails. Then the resultant first generalization of C p (u,v) called as superstructure form of Chen method where F α is the α th percentile and is given by, C N u, v M F. F. where, M is the median of the distribution. m = [USL-LSL]/2. μ = process mean. σ = process standard deviation µt (5.12) u,v are the integers taking values 0 and 1. The generalizations of the basic indices proposed by Pearn and Chen(1997), are obtained by setting the respective parameter values in the above superstructure and the procedure is as follows, USLLSL C Np (u,v) = C Np (0,0) = C Np = (5.13) F. F. C Np (u,v) = C Np (0,1) = C Npk = min USLM, MLSL F. F. F. F. (5.14) C Np (u,v) = C Np (1,0) = C Npm = USLLSL F. F. MT (5.15) 70

6 C Np (u,v) = C Np (1,1) = C Npmk = min USLM F. F. MT, MLSL The above four indices can be related and established as, (5.16) F. F. MT C Npm = C Np 1 MT / σ C Npmk = C Npm 1 MT / σ (5.17) (5.18) These indices can be ranked in the sense of sensitivity to departure process mean from the target value, as C Np < C Npk < C Npm < C Npmk (5.19) For symmetric tolerances, we can show that, C Npk = (1-k)C Np (5.20) C Npmk = (1-k)C Npm (5.21) where k = M T /d is the departure ratio. If the process is on-target, which implies MT then, we get k = 0. By substituting MT in the equations (5.17) & (5.18) and k = 0 in equations (5.20) & (5.21), we get, C Npm = C Np C Npmk = C Np C Npk = C Np C Npmk = C Npm C Npmk =C Npm =C Npk = C Np = USLLSL σ = (5.22) σ The proposed estimators are essentially based on the estimates C p U and C p L for two percentiles F and F 0.135, utilizing estimates of the mean, standard deviation, skewness and kurtosis. Then the resultant second generalization of C p (u,v) called as 71

7 superstructure form of Clement s method where F α is the α th percentile and is given by the indices in which those estimators correspond to can be expressed as: (u,v) = 1 u C N USLLSL F. F. µt + (u) min USLM, F. M MT MLSL (5.23) MF. MT Again by setting the parameter values (u,v) = (0,0), (1,0), (0,1) and (1,1) and following similar procedure as above, we get: USLLSL C N (u,v) = C N (0,0) = C N = (5.24) F. F. C N (u,v) = C N (0,1) = C N = min USLM, F. M MLSL MF. (5.25) C N (u,v) = C N (1,0) = C N = USLLSL F. F. MT (5.26) C N (u,v)= C N (1,1)= C N = min USLM MLSL, F. M MT MF. MT (5.27) By comparing the equations (5.13) and (5.24) we get: C N = C Np also from equations (5.15) and (5.26) we get C N C Npm. If the underlying distribution is normal, then we get M=µ, F. F. 6, F. M3σ and [M F. 3σ. Clearly, the generalizations C N (u,v), again reduce to the basic indices C p (u,v) and the following results are obtained. C N = C p C N C pk C N = C pm C N = C pmk 72

8 These indices can be ranked in the sense of sensitivity to departure process mean from the target value, as C N < C N < C N <C N (5.28) When the process characteristic is not normally distributed Clement (1989) suggested the computation of PCR as, C = C = min{c p L, C P U}, where C p L = ξl and C P U = Uξ (5.29) ξξ L ξ U ξ When the underlying distribution is half normal with µ = 0 and σ = 1.5 the percentile values are as follows 1) ξ is the median of the distribution. i.e., ξ = M = at F. 2) ξ L is the lower capability limit. i.e. ξ L = at F. 3) ξ U is the upper capability limit. i.e. ξ U = 3.91at F. Table-5.1 gives the percentile values with respect to the different values of true σ, which are helpful to compute the C p and C pk values. Further we have brief computations with these values at different values of true σ. These values can be obtained from online calculator available in the website σ = 0.5 σ = 1.0 σ = 1.5 σ = 2.0 σ = 2.5 (M) at F L (F1) at F U (F2) at F Table-5.1: Percentile values of half normal distribution at different values of σ. Now, if we introduce a scaled parameter σ into the equation (5.29) then we get, C = C = min {C p L, C P U}, Where C p L = σξl and C P U = Uσξ (5.30) σξσξ L σξ U σξ By substituting σ in the place of σ, the new estimates of C p L and C p U are found by using the equation (5.30) and is given by, C = C = min {C p L, C P U}, Where C p L = σξl and C P U = Uσξ (5.31) σξσξ L σξ U σξ 73

9 5.4 HALF NORMAL DISTRIBUTION AND ITS PROPERTIES It is a special case of the folded normal distribution. Let X be a quality characteristic which possess standard normal distribution, no(0,σ 2 ) then Y= X is said to be half normal distribution. The probability density function of half normal distribution is given by; f y (y;θ) = e, y>0 where E(Y) = µ = Alternatively, using a scaled precision parameterization, obtained by setting θ = π, σ the probabilitiy density function is, f y (y;θ) = θ π exp π y θ, y > 0 ; where E(Y) = μ = θ (5.32) The Cumulative Distribution Function (CDF) is given by, F y (y;σ) = e σ σ π dx Suppose u is a random deviate from uniform distribution between 0 & 1. Then equation (5.32) will be applied to generate random deviates of Standard half normal distribution using the MS-Excel as, X ij = ABS(2*(NORMSINV(B2)) This is multiplied by σ and added with mean then the resultant formula will be, X ij = ABS(2*(NORMSINV(B2))*($AU$6))+($AU$5) Ramesh and Sarma (2013) have proposed a new estimator of σ for half normal N distribution by finding a constant k, such that, σ = R N, where R is the trimmed mean of sample range. They have also used a robust estimator using Tukey s M- N estimator in place of the trimmed mean of ranges. The value of k, for each n is N obtained by simulation such that, k, = R, where σ 0 is the target standard deviation σ for half normal distribution from which data is generated. Two different estimators, 74

10 are obtained for σ, half normal distribution one based on Trimmed mean estimator. The observations made from this experiment are reported in the subsequent sections. Using online calculator the percentiles of half normal distribution can be easily computed as shown in Figure-5.1. Since only a single value can be obtained each time from the online calculator we have used Excel functions to generate random samples and exploited the relationship between normal and half normal distributions to create samples of size n and thereby derived the value of k N,. The experiment is described in the following section. For each method the performance measures like Bias, MSE, MAD and Variance are found. We have conducted two experiments one with σ 0 = 1.5 and the other with σ 0 = 2.5 and estimated σ 0 by simulation and the results are presented in Table-5.2(a) and 5.2(b). distribution. N Figure-5.2 shows the Excel template to compute k, under half normal Figure-5.1: Online calculator for the percentiles of Half normal distribution. 75

11 Method n = 2,, = n = 3,, = Estimate: MSE Estimate: MSE N R/k, MOM BLUE Method n = 4,, = n = 5,, = Estimate: MSE Estimate: MSE N R/k, MOM BLUE Table-5.2(a): Comparison of, method with the other methods of estimation for σ 0 = 1.5. Method n = 2,, = n = 3,, = Estimate: MSE Estimate: MSE N R/k, MOM BLUE Method n = 4,, = n = 5,, = Estimate: MSE Estimate: MSE N R/k, MOM BLUE Table-5.2(b): Comparison of, method with the other methods of estimation for σ 0 =

12 Figure-5.2: Computation of, under half normal distribution using MS-Excel for n = 4. 77

13 N The estimate of k, from 10,000 samples each of size n is obtained by two methods; a) By using Trimmed mean of R N., b) By using the Tukey s M-estimator. The Tukey s M- estimator is a robust estimator and is found very helpful when the data has contamination. The definition of M-estimators was motivated by robust statistics, which contributed new types of M-estimators. The statistical procedure of evaluating an M-estimator on a data set is called M-estimation. It is also defined as zero of an estimating function. There are four methods of M-estimators as shown in the Table-5.3 (Coakes (2007)). The SPSS syntax for obtaining M estimators is shown below. N EXAMINE VARIABLES=HND_k, /PLOT BOXPLOT STEMLEAF /COMPARE GROUPS /MESTIMATORS HUBER(1.339) ANDREW(1.34) HAMPEL(1.7,3.4,8.5) TUKEY(4.685) /STATISTICS NONE /MISSING LISTWISE /NOTOTAL. N A sample output of M-Estimator of k, with different methods is shown in Table-5.3. M-Estimators Huber's M- Estimator a Tukey's Hampel's M- Biweight b Estimator c Andrews' Wave d N HND_k, a. The weighting constant is b. The weighting constant is c. The weighting constants are 1.700, 3.400, and d. The weighting constant is 1.340*pi. Table-5.3: Output of Tukey s M-Estimator using SPSS for n = 2. N The computation of the value of bias factor k, for σ 0 = 1.5 and σ 0 = 2.5 using M-Estimator with one of the four (Huber, Andrew, Hampel and Tukey) methods is shown in Table

14 Sample size (n) Tukey s method, Trimmed Mean method Table-5.4: Values of, using Tukey s M-Estimator and Trimmed mean. We have used the Trimmed mean method in further calculations. Computation of k2,n with σ = 1.5 k2,n Tukey s method Trimmed Mean method sample size Figure-5.3: Comparison between the Tukey s and Trimmed mean of,. From Figure-5.3, it is seen that when the sub group size is large, we need a higher value of k N,. 79

15 5.5 ESTIMATION OF PROCESS CAPABILITY BY RANGE METHOD USING CHEN AND CLEMENT S GENERALIZATIONS Using the method of ranges to estimate σ 0 in case of half normal distribution, we have calculated the process capability index using different indices discussed in section 5.2. For convenience we reproduce (5.12) below known as generalization of the Chen method for evaluation with different parameters. du M m C N u, v 3 F. F. 6 vµ T The computation and comparison of two methods viz., Chen method-c N u, v and Clement s method-c N u, v with the superstructure C u, v under the condition of normal and half-normal distributions are as follows; For Normal case A separate Excel template has been developed to evaluate the process C p, C pk etc., values for which the inputs are shown in Table-5.5. When σ 0 is known we get the true value of C p. This is compared with the C p obtained by using d 2,n method. The computation and comparison of all these generalizations are shown in the excel template as Figure-5.4 and the results are shown in the Table-5.6 for n=3. We have conducted simulated trails with 30 samples of different sizes from Normal mean = 15 and σ 0 = 0.5 (Gupta (2006)). We have estimated σ 0 by the method of Ranges using the general d 2,n. Assuming that the process specification limits are LSL = 14 and USL = 16 we have obtained C p as (<1) which means the process is not capable of meeting the specifications. However for smaller values of σ 0, C p values are found to be >1. (Czarski (2008)). We have worked out the expressions for C p and C pk using the superstructure formula given in (5.13) to (5.16) and (5.23) to (5.27). It should be noted that when the distribution is taken as normal the following reduction takes place in the expressions F. F. 6σ F. M3σ 80

16 [MF. 3σ The Table-5.5 gives the values obtained under the assumption of normal distribution. Input values USL = 16, LSL = 14 µ=15, σ = 0.50 k N, = 3.86 for n = 3 Derived values T =15, m = 15, µ = , M = σ = 0.50, F. F. = F. M = 1.501, [MF. = Table-5.5: Values obtained under the condition of normal distribution for n=3. We have compared the results by using the superstructure method and the results obtained are shown in Table-5.6 to Table 5.7 under the condition of normal distribution and Table-5.8 to Table-5.9 under the condition of half normal distribution. The Excel template used to perform the calculations for normal case is shown in Figure-5.4. It is observed that the values of all the generalizations of C p (u,v), C Np (u,v), C' Np (u,v) are equal and are presented in the Table-5.6. It is also observed that by changing the values of u and v the capability indices obtained are similar when we compare all the generalizations of C p (u,v), C Np (u,v), C' Np (u,v) which can verify from the Table-5.7 Thus we observe from the Table-5.6 and 5.7 that all the generalizations C p (u,v), C Np (u,v), C' Np (u,v) are equal, which was verified under the condition that the distribution is normal. For Non-Normal case The case of half normal distribution is carried out for n=3 with σ = 0.5, 1.0, 1.5, 2.0 and 2.5 and compared with the other generalizations viz., Chen method-c Np (u,v) and Clement s method-c' Np (u,v). The results can be observed from the Table-5.8 to 5.9. The computation of these generalizations by using MS-Excel template is shown in Figure-5.5 at different values of σ for n=3. 81

17 We have considered the computation of Chen method with the target σ and estimated σ and the results can be found in the Table-5.10, which shows the minimum bias value as the value of σ increases. The Table-5.11 gives the comparison of Chen method with the target σ and estimated σ with increase in the sample size n and thus the computation shows that the bias values of C Npk and C Npmk at n=3 are equal. The Table-5.12 and 5-13 gives the comparison of Chen and Clement s method at different sample sizes and thus we found there exists a bias between these two methods of computation except at n=3 and also we found that values of C Np = C Np and C Npm = C Npm. The Table-5.14 gives the formulae to compute the superstructure using R-code and the Figure-5.6 shows the R-code template. 82

18 Figure-5.4: Computation of C p under the assumption that the distribution is normal. 83

19 σ C p (u,v) C Np (u,v) C' Np (u,v) C p C pk C pm C pmk C Np C Npk C Npm C Npmk C' Np C' Npk C' Npm C' Npmk Table-5.6: Comparison of superstructure under the assumption that the distribution is normal for n = 3. (u,v) C p (u,v) C Np (u,v) C' Np (u,v) (0,0) = C p = C p = C p (1,0) = C pk = C pk = C pk (0,1) = C pm = C pm = C pm (1,1) = C pmk = C pmk = C pmk Table-5.7: Verification of superstructure with different values of u and v with σ = 0.5 under the assumption that the distribution is normal. 84

20 C p (u,v) Chen method C Np (u,v) Clements method C' Np (u,v) σ C p C pk C pm C pmk C Np C Npk C Npm C Npmk C' Np C' Npk C' Npm C' Npmk Table-5.8: Computation of C p with half normal distribution for n=3. (u,v) C p (u,v) C Np (u,v) C' Np (u,v) (0,0) = C p = C Np = C' Np (1,0) = C pk = C Npk = C' Npk (0,1) = C pm = C Npm = C' Npm (1,1) = C pmk = C Npmk = C' Npmk Table-5.9: Verification of superstructure with different values of u and v under half normal for n=3. 85

21 Figure-5.5: Excel template used to perform the calculations for half normal distribution. 86

22 Computation with target σ Computation with estimated σ ( ) Chen method C Np (u,v) Chen method C Np (u,v) σ C Np C Npk (C p L, C p U) C Npm C Npmk (C pm U, C pm L) C Np C Npk (C p L, C p U) C Npm C Npmk (C pm U, C pm L) Bias 0f C Npk Bias 0f C Npmk (3.0693,3.0704) (0.5095,1.0260) ( ,0.6863) ( ,0.5802) ( ,0.5369) (3.0693,3.0704) (0.4025,08111) ( ,0.4768) ( ,0.3779) ( ,0.3367) (3.0657,3.0679) (0.5085,1.0255) ( ,0.6861) ( ,0.5801) ( ,0.5369) (3.0657,3.0679) (0.4018,0.8104) ( ,0.4765) ( ,0.3778) ( ,0.3366) Table-5.10 : Comparison of C Np (u,v) with target σ and for n = 3 at different standard deviations

23 Computation with target σ Computation with estimated σ ( ) σ = 0.5 Chen method C Np (u,v) Chen method C Np (u,v) n C Np C Npk (C p U, C p L) C Npm C Npmk (C pm U, C pm L) C Np C Npk (C p U, C p L) C Npm C Npmk (C pm U, C pm L) Bias 0f C Npk Bias 0f C Npmk (2.8978,3.2419) (2.5859,2.8809) (3.2601,2.9207) (2.6028,2.9053) (3.0704,3.0693) , (3.0679,3.0657) (3.0657,3.0679) (4.1042,2.0355) (0.6845,1.2589) (4.9691,2.6517) (0.7331,1.3738) (3.3351,2.8045) (2.4347,2.6095) (3.8090,3.3637) (2.7971,3.1674) Table-5.11 : Comparison of C Np (u,v) with target σ and with σ = 0.5 at different sizes of n

24 n σ = 1.00 Chen method with estimated σ (σ ) C Np (u,v) Clements method with estimated σ (σ ) C Np (u,v) Bias of C Npk (C Npk - C Npk ) Bias of C Np (C Np - C Np ) σ C Np L C Np U C Npk C Np C Np L C Np U C Npk C Np Table-5.12: Computation of C pk and C p for k 2,n with σ =

25 σ = 2.50 Chen method with estimated σ (σ ) Clements method with estimated σ (σ ) C Np (u,v) C Np (u,v) n C Np C Npk (C p U, C p L) C Npm C Npmk (C pm U, C pm L) C Np C Npk (C p U, C p L) C Npm C Npmk (C pm U, C pm L) Bias 0f C Npk Bias 0f C Npmk ( ,0.5715) ( ,0.3412) ( ,1.1059) ( ,0.3970) ( ,0.5369) ( ,0.3366) ( ,1.0388) ( ,0.3990) ( ,0.7982) ( ,0.3661) ( ,1.5445) ( ,0.3981) ( ,0.6020) ( ,0.3540) ( ,0.1648) ( ,0.4097) Table-5.13: Comparison of C Np (u,v) with C Np (u,v) using σ = 2.50 at different sizes of n. 90

26 R-code for the computation of superstructure. >USL<-16.0;LSL<-14.0; SIG<-0.50; Mu<-15.0 >Mu_C< ;SIG_K<-0.50 >T<-(USL+LSL)/2 >d=(usl-lsl)/2;m=(usl-lsl)/2 >F1<-0.001;F2<-1.304;M< >u<-0;v<-0 >Cp_K=(USL-LSL)/(6*SIG_K) >CpU=(USL-Mu_C)/(3*SIG_K) >CpL=(Mu_C-LSL)/(3*SIG_K) >Cpk=min(CpU,CpL) >Cpm=(USL-LSL)/(6*sqrt(((SIG_K)^2)+((Mu_C-T)^2))) >CpmU=(USL-Mu_C)/(3*sqrt(((SIG_K)^2)+((Mu_C-T)^2))) >CpmL=(Mu_C-LSL)/(3*sqrt(((SIG_K)^2)+((Mu_C-T)^2))) >Cpmk=min(CpmU,CpmL) >CNp=(USL-LSL)/(F2-F1) >CNpU=(USL-M)/((F2-F1)/2) >CNpL=(M-LSL)/((F2-F1)/2) >CNpk=min(CNpU,CNpL) >CNpm=(USL-LSL)/(6*sqrt(((F2-F1)/6)^2+(M-T)^2)) 91

27 >CNpmU=(USL-M)/(3*sqrt(((F2-F1)/6)^2+(M-T)^2)) >CNpmL=(M-LSL)/(3*sqrt(((F2-F1)/6)^2+(M-T)^2)) >CNpmk=min(CNpmU,CNpmL) >C'Np=(USL-LSL)/(F2-F1) >C'NpU=(USL-M)/((F2-M) >C'NpL=(M-LSL)/((M-F1) >C'Npk=min(C'NpU,C'NpL) >C'Npm=(USL-LSL)/(6*sqrt(((F2-F1)/6)^2+(M-T)^2)) >C'NpmU=(USL-M)/(3*sqrt(((F2-M)/3)^2+(M-T)^2)) >C'NpmL=(M-LSL)/(3*sqrt(((M-F1)/3)^2+(M-T)^2)) >C'Npmk=min(C'NpmU,C'NpmL) >Cp(u,v)=(d-(u*abs(Mu_C-m)))/(3*sqrt(((SIG_K)^2)+(v*(Mu_C-T)^2))) >CNp(u,v)=(d-(u*abs(Mu_C-m)))/(3*sqrt(((F2-F1)/6)^2+(v*(M-T)^2))) >C'Np(u,v)=(1-u)*((USL-LSL)/(6*sqrt(((F2-F1)/6)^2+(v*(M-T)^2)))) +(u*(min((usl-m)/(3*sqrt(((f2-m)/3)^2+(v*(m-t)^2))),(m- LSL)/(3*sqrt(((M-F1)/3)^2+(v*(M-T)^2)))) >C_Np_uvU<-(USL-M)/(3*sqrt(((F2-M)/3)^2+(v*(M-T)^2))) >C_Np_uvL<-(M-LSL)/(3*sqrt(((M-F1)/3)^2+(v*(M-T)^2))) >C_Np_uv<-(1-u)*((USL-LSL)/(6*sqrt(((F2-F1)/6)^2+(v*(M- T)^2))))+(u*(min(C_Np_uvU,C_Np_uvL))) Table-5.14: R-code for the computation of superstructure. 92

28 Figure-5.6: R-Coding used to perform the superstructure calculations. 93

29 Thus it is shown that in the case of half normal distribution the process standard deviation can be estimated by the method of ranges with suitable bias factor. It is also seen that the derived C p values are close to the target value obtained when the process is centered at the middle of the specification limits. In the next chapter an experiment on the estimation of variance components in Gage R&R studies using Ranges and ANOVA is reported. 94