IJQRM 18,1. The current issue and full text archive of this journal is available at erald-library.com /ft

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1 The research register for this journal is available at /research_registers The current issue and full text archive of this journal is available at erald-library.com /ft IJQRM 18,1 50 A computerized approach to calculate the nonconforming rate for product interference problems J.N. Pan Department of Statistics, National Chen-Kung University, Tainan, Taiwan, People s Republic of China International Journal of Quality & Reliability Management, Vol. 18 No. 1, 2001, pp # MCB University Press, X Keywords Product quality, Reliability, Computing, Manufacturing Abstract It is very common that most product assemblies in industry require two mating parts where one must fit into the other. However, we may observe interference if the shaft does not fit within or over the shaft, or we may experience excessive clearance. In calculating the percentage of interference or excessive clearance, most industries usually assume normal distribution for quality characteristics of mating parts and use a distribution of their difference to analyze the interference or clearance. However, it has not been proved that the collected data matches the assumption of normality. Therefore, it is doubtful that the calculated nonconforming rate truly reflects the performance of the assembly process. This research focuses on the clearance and interference analysis for the non-normal data of product assembly. Comparisons have also been made between actual nonconforming rate and the assumed normal percentage of nonconformance by overlooking their theoretical distributions of two mating parts. Finally, a computer program has been proposed to simplify tedious calculation process for estimating the nonconforming rate under non-normal distribution. Introduction In the process of manufacturing and assembly of industrial products, it is very common to have problems of interference and excessive clearance. In the past, authors (Devor et al., 1992; Kolarik, 1995; Mitra, 1992; Phadke, 1999; Taylor, 1992) assumed that the quality characteristics of assembled parts follow normal distributions. Based on this assumption, the percentage of interference and excessive clearance on product assembly are calculated. For example, Kolarik (1995) used the example of a lid fitting into a cup to calculate the rate of interference/excessive clearance assuming normal distribution. However, in conducting process capability studies, we often find that the collected data does not follow the normal assumption (Pan and Wu, 1997). The following example was brought up by Taguchi (Devor et al., 1992; Taylor, 1992): The color-intensity quality characteristic of TV sets produced in the USA followed a uniform distribution. However, it was noted that the quality characteristic of the TV sets produced by the Japanese company followed a normal distribution, which nearly stayed within the specification. One can show that the loss value for the process approximated by uniform distribution is three times higher than that approximated by normal distribution. Apparently, when the quality characteristics of assembled parts are not under normal distributions, possible over or under estimation of the manufacturing

2 costs will occur. The purpose of this study is to propose a general interferenceclearance analysis of product assembly by relaxing the normal assumption, in order to estimate the nonconforming rate of product assembly under nonnormal distributions. The research objectives of this study are briefly listed as below:. To solve the interference problems of product assembly under nonnormal conditions.. To establish a method for calculating the accurate nonconforming rate of mating parts under non-normal distributions.. To compare the difference between actual and the assumed normal nonconforming rate by overlooking the theoretical distribution of mating parts.. To develop a computer program using Fortran programming language to simplify the tedious calculation process of nonconforming rate. A computerized approach 51 Calculation of the nonconforming rate for product assembly under non-normal distribution In this paper, we consider the quality characteristics of two mating parts follow but not limited to the following five common probability distributions: (1) Exponential (for modeling the case of truncated parts). (2) Uniform (for modeling the case when machine is probably out of calibration). (3) Normal (for modeling the symmetric shape of parts). (4) Lognormal (for modeling the right skewed process). (5) Weibull distribution (for its flexibility in fitting various shapes of parts). The 15 combinations of two mating parts under these five probability distributions are listed in Table I, where two mating parts X and Y are random variables. The interference-clearance distribution (Y-X) of eight combinations (see Table II for details) can be derived analytically. A decision flow chart for calculating the nonconforming rate of product interference is illustrated in Figure 1. Notice that if the data of mating parts does not follow one of the five common distributions, one might consider to use the Johnson s system of Part Y Part X Exponential Uniform Normal Lognormal Weibull Exponential Uniform Normal Lognormal 3 3 Weibull 3 Table I. Various combinations of two mating parts under five common probability distributions

3 IJQRM 18,1 52 Table II. Eight interferenceclearance distributions which can be derived analytically distributions (Johnson, 1949) to transform the non-normal data to a normal one. For the rest of the seven combinations, we have developed a Fortran computer program based on this algorithm to estimate the nonconforming rate of these product assemblies due to the intractability of their mathematical integration involved. Part Y Part X Exponential Uniform Normal Lognormal Weibull Exponential Uniform Normal 3 Lognormal Weibull Figure 1. A decison flow chart for calculating the nonconforming rate of product interference

4 In order to verify the accuracy of the developed computer program (written in Fortran), we use exponential distribution as a baseline for comparing the theoretical (exponential) versus empirical (Weibull) nonconforming rate since exponential is a special case of Weibull distribution. After performing the computer simulation for 100,000 times, the results are shown in Table III, where P E denotes the nonconforming rate under exponential distribution, and P W denotes the nonconforming rate under Weibull distribution. Thus, the relative error of the empirical (Weibull) and theoretical (exponential) values can be defined as: A computerized approach 53 P W P E P E 100%: Given 1 = 1 for part X and 2 = 1 for part Y, the simulation results indicate that in all cases the relative errors are less than 2 per cent. Thus the accuracy of our Fortran computer program has been further verified. Comparison of the assumed-normal and actual nonconforming rate for mating parts If the collected data follow specific distributions, we have compared the actual (non-normal) nonconforming rate with the assumed normal nonconforming rate by overlooking their theoretical distributions of two mating parts. Given the parameters of two specific distributions and the upper specification limit (USL), lower specification limit (LSL), as well as target value of the interferenceclearance distribution, if we treat the expected mean and variance of a specific distribution as the parameters of normal distribution, then the assumed normal nonconforming rate can be calculated. On the other hand, its actual (nonnormal) nonconforming rate can be determined by either using the developed computer program or mathematical computation. The exponential and Weibull distribution shown in Tables IV and V have further demonstrated the difference between actual and the assumed normal nonconforming rate. Table IV and Figure 2 show that if the quality characteristic of mating parts follow exponential distributions, but we view them as being the results of normal USL = 6, LSL = ± 6, target = 0 Exp vs Exp Weibull vs Weibull Relative error 1 vs 2 P E ( 1 ; 1 ut vs ( 2 ; 2 ut P W (%) (2,1) (1.5,1) ± (2,1) (2,1) ± (2,1) (2.5,1) ± (2,1) (3,1) ± (2,1) (3.5,1) (2,1) (4,1) (2,1) (4.5,1) (2,1) (5,1) Table III. The simulation results of two mating parts follow Weibull distribution

5 IJQRM 18,1 USL = 6, LSL = ± 6, target = 0 Exponential vs exponential Normal vs normal Proportion of Proportion of 1 vs 2 nonconformance nonconformance Difference 54 Table IV. Comparison of exponential vs the assumed normal nonconforming rates ± ± ± ± USL = 3, LSL = ± 3, target = 0 Weibull vs Weibull Normal vs Normal Proportion of Proportion of ( 1 ; 1 ut vs ( 2 ; 2 ut nonconformance nonconformance Difference Table V. Comparison of Weibull vs the assumed normal nonconforming rate (3,2) (4,1.5) ± (3,2) (4,2) ± (3,2) (4,2.5) ± (3,2) (4,3) ± (3,2) (4,3.5) ± Figure 2. Comparison of exponential vs the assumed normal nonconforming rate

6 distributions, the nonconforming rate may be under or overestimated. Table V and Figure 3 show that if the quality characteristic of mating parts follow Weibull distribution, but we view them as being the results of normal distributions, the nonconforming rate may be under-estimated. However, if we fix the scale parameter ( ), and increase the value of shape parameter ( ) from 1.5 to 3.5, the Weibull nonconforming rate will be close to normal nonconforming rate since the Weibull distribution approximates the normal distribution when is between 3.5 and 4.0. A computerized approach 55 A numerical example and its application An explaination of the over and under estimation of the nonconforming rate assuming normal distribution was made as a baseline for the calculation of interference or excessive clearance. We used a set of realistic data from the De- Fung manufacturing plant in Taiwan, to discuss the interference problem which occurred in the proper fitting of bearing and shaft. A total of 42 sets of data were analyzed. The engineering specifications of bearing and shaft are listed as follows:. the interior diameter of the bearing is mm;. the exterior diameter of the shaft is mm. The data was input into the computer and calculations were performed using the attached Fortran program (see Appendix). The results show uniform distribution for the bearing with an upper limit of 25.98mm and a lower limit of 25.85mm; normal distribution for the shaft with a sample mean of mm and a sample variance of mm. The process of statistical data analysis is illustrated as follows: (1) Input the relevant data. A total of 42 sets of data were input into the computer. Figure 3. Comparison of Weibull vs the assumed normal nonconforming rate

7 IJQRM 18,1 56 (2) Perform statistical calculations The basic statistics of 42 bearings are printed as follows: Mean Variance Std dev. Skewness Kurtosis E E ± Minimum Maximum Range Coef. var. Count E The basic statistics of 42 shafts are printed as follows: Mean Variance Std dev. Skewness Kurtosis E E ± Minimum Maximun Range Coef. Var. Count E (3) Perform K-S goodness of fit test. After inputting the significance level ( ) and the sample size (n), the output of K-S statistics and its critical values for the five common probability distributions can be obtained. One can determine the proper distributions for the quality characteristics of mating parts. In our example, if we let = 0.05 and n = 42, then the critical value of K-S test, a ˆ D 0:05 ˆ 0: and the smallest K-S statistics (uniform) for the 42 bearings, D c = Since D c < D a, one can conclude that the collected data follow uniform distribution. Moreover, the smallest K-S statistics normal for the 42 shafts, D c = Since D c < D a, one can conclude that the collected data follow normal distribution. (4) Input the upper and lower specification limits as well as the target value of the part s fitness. The specification limits for the bearings are mm, the specification limits for the shafts are mm, the USL for the part s fitness = , the LSL for parts fitness = ± , and the target value = 0. (5) Calculate the nonconforming rate for product assembly. Using the Fortran computer program, the actual nonconforming rate between proper fitting of bearings (uniform) and shafts (normal) distribution shall be equal to per cent. If we overlook the uniform distribution and assume normal distribution, the nonconforming rate becomes per cent. In this example, we have shown a case of over-estimation of per cent. Suppose 1,000,000 units of the product assembly are manufactured each year and the manufacturing cost of each assembly = $100, $30,000 over estimation can be expected approximately. From the management point of view, if the nonconforming rate cannot be accurately estimated, we will have a higher inventory cost due to over estimation of nonconforming rate. If it is under-estimated, then we will face the problem of shortage cost. Therefore, it becomes very important to estimate accurately the nonconforming rate of product interference, since controlling manufacturing cost is crucial to the improvement of product quality and productivity.

8 Conclusions Mating parts assembly of industrial products play an important role in product quality. The continuing efforts on improving quality of the assembled products present a daily challenge. If one cannot accurately estimate the nonconforming product assembly, it will result in inaccurate estimation of the manufacturing costs. When estimating the nonconforming rate of assembled products, most companies make the assumption that the key quality characteristics of mating parts follow normal distribution; whether the rate of interference and/or excessive clearance can truly reflect the actual nonconforming rate of product assembly is very questionable. In this paper, a computerized approach has been proposed to estimate accurately the nonconforming rate of product assembly in which mating parts do not follow normal distributions. The research contributions of this paper can be summarized as follows:. For the five most common distributions used in modeling the quality characteristics of mating parts: exponential, uniform, normal, lognormal, Weibull, a general method has been established to accurately estimate the nonconforming rate of product interference when mating parts do not follow normal distributions.. This paper makes a comparison between actual (non-normal) and the assumed normal nonconforming rate for mating parts. When the measurements of quality characteristics show non-normal distributions, and we overlook these facts, the nonconforming rate will often be either over or underestimated. Therefore, we suggest that specific distributions of the collected data should be determined in advance in order to accurately estimate the actual nonconforming rate for mating parts.. This paper performs a statistical analysis for various interference problems of product assembly under non-normal conditions, and provides a method for calculating actual nonconforming rate of product interference. A computer program was developed using Fortran language to simplify the tedious calculation process, which can be a useful reference for engineers and management.. A realistic example has been given to demonstrate the usefulness of the proposed computer program. The accuracy and validity of this computer program have also been verified. A computerized approach 57 References Devor, R.E., Chang, T.H. and Suthewand, J.W. (1992), Statistical Quality Design and Control, Macmillan, New York, NY. Johnson, N.L. (1949), ``Systems of frequency curves generated by methods of translation, Biometrika, Vol. 36, pp Kolarik, W.J. (1995), Creating Quality, McGraw-Hill, New York, NY. Mitra, A. (1992), Fundamentals of QC Improvement, Macmillan, New York, NY.

9 IJQRM 18,1 58 Pan, J.N. and Wu, S.L. (1997), ``Process capability analysis for non-normal relay test data, International Journal of Microelectronics and Reliability, Vol. 37 No. 3, pp Phadke, M.S. (1989), Quality Engineering Using Robust Design, Prentice-Hall, Englewood Cliffs, NJ. Taylor, W. (1992), Optimization and Variation Reduction in Quality, McGraw-Hill, New York, NY. Further reading Daniel, W.W. (1990), Applied Nonparametric Statistics, 2nd ed., PWS-KENT Publishing Company, Boston, MA. D Agostino, R.B. and Stephens, M.A. (1986), Goodness-of-Fit Techniques, Marcel Dekker, New York, NY. Engelhardt, B. (1992), Introduction to Probability and Mathematical Statistics, 7th ed., Duxbury Press, CA. Grant, E.L. and Leavenworth, R.S. (1996), Statistical Quality Control, 7th ed., McGraw-Hill, New York, NY. Kapur, K.C. and Lamberson, L.R. (1977), Reliability in Engineering Design, John Wiley & Sons, New York, NY. Appendix Program listing-two different combinations of pdf shown in the numerical example are included. The Fortran program for calculating the nonconforming rate of product interference I NT E GE R S E L E CT 1, S E L E CT 2, NOUT, I RUL E, I S E E D, NR, I NDE X, I RE AL PR, ANORDF, C, D, E, G, VAL UE, VAL UE 1, Z, Z 1, A, B, E RRABS, E RRE S T, E RROR RE AL E RRRE L, E RR, R( ), R1 ( ), P( ), XM, XS T D, T HE T A, E 1, E 2 RE AL F, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11, F 12 RE AL ANS WE R, RE S UL T, RE S UL T 1, XM1, XS T D1 E XT E RNAL UMACH, ANORDF, QDAG, RNS E T, RNL NL, RNE XP, RNWI B, RNNOA, & F, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 10, F 11, F 12 wr i t e ( *, *) * ******************************************* wr i t e ( *, *) * wr i t e ( *, *) *!! We l c ome t o us e t hi s pr ogr a m!! wr i t e ( *, *) * wr i t e ( *, *) * T hi s pr ogr a mc omput e t h e pe r c e nt a g e of wr i t e ( *, *) * i nt e r f e r e nc e a n d/ or e x c e s s i v e c l e a r a nc e wr i t e ( *, *) * ( i. e. t he pe r c e nt a g e of non - c onf or mi ng) wr i t e ( *, *) * f or 15 di f f e r e nt c ombi n a t i ons of t he wr i t e ( *, *) * f ol l owi ng f i v e c ommon di s t r i b ut i ons : wr i t e ( *, *) * wr i t e ( *, *) * ( 1) E x p one nt i a l

10 wr i t e ( *, *) * ( 2) Un i f or m wr i t e ( *, *) * ( 3) No r ma l wr i t e ( *, *) * ( 4) L o gnor ma l wr i t e ( *, *) * ( 5) We i bul l wr i t e ( *, *) * wr i t e ( *, *) * ******************************************* wr i t e ( *, *) wr i t e ( *, *)!! E nt e r t he p df o f t wo ma t i ng pa r t s ( 1, 2, 3, 4, 5) : RE AD( *, *) S E L E CT 1, S E L E CT 2 I F ( ( S E L E CT 1. E Q. 1). AND. ( S E L E CT 2. E Q. 1) ) T HE N CAL L E XPONE NT I AL ( PR) E L S E I F ( ( ( S E L E CT 1. E Q. 1 ). AND. ( S E L E CT 2. E Q. 2) ). OR. & ( ( S E L E CT 1. E Q. 2). AND. ( S E L E CT 2. E Q. 1) ) ) T HE N CAL L E XPUNI ( PR) E L S E I F ( ( ( S E L E CT 1. E Q. 1 ). AND. ( S E L E CT 2. E Q. 3) ). OR. & ( ( S E L E CT 1. E Q. 3). AND. ( S E L E CT 2. E Q. 1) ) ) T HE N CAL L E XPNOR( PR) E L S E I F ( ( ( S E L E CT 1. E Q. 1). AND. ( S E L E CT 2. E Q. 4) ). OR. & ( ( S E L E CT 1. E Q. 4). AND. ( S E L E CT 2. E Q. 1) ) ) T HE N CAL L E XPL OG( PR) E L S E I F ( ( ( S E L E CT 1. E Q. 1). AND. ( S E L E CT 2. E Q. 5) ). OR. & ( ( S E L E CT 1. E Q. 5). AND. ( S E L E CT 2. E Q. 1) ) ) T HE N CAL L E XPWE I ( P R) E L S E I F ( ( S E L E CT 1. E Q. 2). AND. ( S E L E CT 2. E Q. 2) ) T HE N CAL L UNI F ORM( PR) E L S E I F ( ( ( S E L E CT 1. E Q. 2). AND. ( S E L E CT 2. E Q. 3) ). OR. & ( ( S E L E CT 1. E Q. 3 ). AND. ( S E L E CT 2. E Q. 2) ) ) T HE N CAL L UNI NOR( P R) E L S E I F ( ( ( S E L E CT 1. E Q. 2). AND. ( S E L E CT 2. E Q. 4) ). OR. & ( ( S E L E CT 1. E Q. 4 ). AND. ( S E L E CT 2. E Q. 2) ) ) T HE N CAL L UNI L OG( P R) E L S E I F ( ( ( S E L E CT 1. E Q. 2). AND. ( S E L E CT 2. E Q. 5) ). OR. & ( ( S E L E CT 1. E Q. 5 ). AND. ( S E L E CT 2. E Q. 2) ) ) T HE N CAL L UNI WE I ( P R) E L S E I F ( ( S E L E CT 1. E Q. 3 ). AND. ( S E L E CT 2. E Q. 3) ) T HE N CAL L NORMAL ( PR) E L S E I F ( ( ( S E L E CT 1. E Q. 3). AND. ( S E L E CT 2. E Q. 4) ). OR. & ( ( S E L E CT 1. E Q. 4). AND. ( S E L E CT 2. E Q. 3 ) ) ) T HE N CAL L NORL OG( PR) E L S E I F ( ( ( S E L E CT 1. E Q. 3). AND. ( S E L E CT 2. E Q. 5) ). OR. & ( ( S E L E CT 1. E Q. 5). AND. ( S E L E CT 2. E Q. 3 ) ) ) T HE N CAL L NORWE I ( PR) E L S E I F ( ( S E L E CT 1. E Q. 4). AND. ( S E L E CT 2. E Q. 4) ) T HE N CAL L L OGNORMAL ( PR) E L S E I F ( ( ( S E L E CT 1. E Q. 4 ). AND. ( S E L E CT 2. E Q. 5) ). OR. & ( ( S E L E CT 1. E Q. 5). AND. ( S E L E CT 2. E Q. 4 ) ) ) T HE N CAL L L OGWE I ( PR) A computerized approach 59

11 IJQRM 18,1 60 E L S E I F ( ( S E L E CT 1. E Q. 5). AND. ( S E L E CT 2. E Q. 5) ) T HE N CAL L WE I BUL L ( PR) E NDI F E ND C ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐC C UNI F ORM& NORMAL DI S T RI BUT I ON C C ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐC S UBROUT I NE UNI NOR( P R) I NT E GE R I RUL E, NOUT RE AL A, B, E RRABS, E RRE S T, E RROR, E RRRE L, F 8, PR, C, D, E, G, VAL UE, VAL UE 1 E XT E RNAL F 8, QDAG, UMACH COMMON C, D, E, G WRI T E ( *, *) e nt e r t he l o we r l i mi t s of uni f or mdi s t r i but i on X: RE AD ( *, *) C WRI T E ( *, *) e nt e r t he up pe r l i mi t s of uni f or mdi s t r i but i on X: RE AD ( *, *) D WRI T E ( *, *) e nt e r t he me a n v a l ue of nor ma l di s t r i but i on Y: RE AD ( *, *) E WRI T E ( *, *) e nt e r t he v a r i a nc e of nor ma l di s t r i b ut i on Y: RE AD ( *, *) G CAL L UMACH( 2, NOUT ) WRI T E ( *, *) e nt e r t h e l owe r l i mi t of nor ma l i nt e gr a l : RE AD ( *, *) VAL UE WRI T E ( *, *) e nt e r t h e upp e r l i mi t of nor ma l i nt e gr a l : RE AD ( *, *) VAL UE 1 A=VAL UE B=VAL UE 1 E RRABS =0. 0 E RRRE L = I RUL E =1 CAL L QDAG ( F 8, A, B, E RRABS, E RRRE L, I RUL E, PR, E RRE S T ) WRI T E ( NOUT, ) PR, E RRE S T F ORMAT ( Pr o ba bi l i t y =, F 9. 6, 13X, e r r or e s t i ma t e d=, F 10. 7) E ND RE AL F UNCT I ON F 8( X) RE AL X, C, D, E, G COMMON C, D, E, G Z =ANORDF ( ( D+X- E ) / S QRT ( G) ) - ANORDF ( ( C+X- E ) / S QRT ( G) ) F 8=( 1/ ( D- C) ) *Z RE T URN E ND C ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐC C NORMAL & NORMAL DI S T RI BUT I ON C C ÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐÐC S UBROUT I NE NORMAL ( PR) I NT E GE R NOUT RE AL ANORDF, C, D, E, G, VAL UE, VAL UE 1, Z, Z 1, PR E XT E RNAL UMACH, ANORDF WRI T E ( *, *) e nt e r t he me a n v a l ue of n or ma l di s t r i but i on X: RE AD ( *, *) C WRI T E ( *, *) e nt e r t he v a r i a nc e of nor ma l d i s t r i b ut i on X: RE AD ( *, *) D WRI T E ( *, *) e nt e r t he me a n v a l ue of n or ma l di s t r i but i on Y:

12 RE AD ( *, *) E WRI T E ( *, *) e nt e r t he v a r i a nc e of nor ma l d i s t r i b ut i on Y: RE AD ( *, *) G CAL L UMACH( 2, NOUT ) WRI T E ( *, *) e nt e r t he l o we r l i mi t of nor ma l i n t e gr a l : RE AD ( *, *) VAL UE WRI T E ( *, *) e nt e r t he up pe r l i mi t of nor ma l i n t e gr a l : RE AD ( *, *) VAL UE 1 Z =( VAL UE - ( C- E ) ) / S QRT ( D+G) Z 1=( VAL UE 1- ( C- E ) ) / S QRT ( D+G) PR=ANORDF ( Z 1) - ANORDF ( Z ) WRI T E ( *, *) pr oba bi l i t y =, PR E ND A computerized approach 61