Evaluating Process Capability Indices for some Quality Characteristics of a Manufacturing Process

Transcription

1 Journal of Statistical and Econometric Methods, vol., no.3, 013, ISSN: (rint version), (online) Scienress Ltd, 013 Evaluating Process aability Indices for some Quality haracteristics of a Manufacturing Process Kayode S. Adekeye 1 and Joshua O. Ogundele Abstract The effectiveness of a manufacturing roduct roduced deends on whether the secified standards by the management / roduction engineers are met or not. It is ossible that a rocess will be stable but will not meet the secification set by the roduction engineer or management. In order to determine whether a rocess meets the secification of the rocess engineer or management, rocess caability indices (PIs) are often used. This study is aimed at evaluating the caability and the ercentage of nonconforming unit of a manufacturing rocess. Data on two quality characteristics were extracted from the records of Evans Medical PL, Agbara, Nigeria. The rocess caability indices were used to determine the caability of the rocess and the ercent non-conforming was used to check the amount of the rocess that does not meet the required standard. The obtained results reflect the amount of the roducts roduced during the eriod of study that do not conform to secification and the gain of testing for the normality assumtion as reflected in the arts er million defects for the rocess data. The confidence interval derived for the rocess indices can be used to imrove the caability of the rocess. Keywords: Process, Stability, aability Index, Defect Parts Per Million (PPM), Percent Non-onforming 1 Introduction The challenge in today s cometitive markets is to be on the leading edge of roducing high quality roducts at minimum costs. A rocess is a unique combination of machines, materials, methods, and eole engaged in roducing a measurable outut (Oakland, 00). The Process aability is a measurable roerty of a rocess to the secification. The outut of this measurement can be used to redict how many arts will be roduced out of 1 Deartment of Mathematical Sciences, Redeemer s University, Redemtion am, Nigeria. Deartment of Mathematics/Statistics, Rufus Giwa Polytechnic, Owo, Nigeria. Article Info: Received : May 31, 013. Revised : July 4, 013. Published online : Setember 9, 013

2 106 Kayode S. Adekeye and Joshua O. Ogundele secification. Process caability analysis together with statistical rocess control and design of exeriments are statistical methods that have been used for decade with urose to reduce the variability in industrial rocesses and roducts (Albing, 006). Process caability analysis deals with how to assess the caability of a manufacturing rocess, where information about the rocess is used to imrove the caability. One can determine how well the rocess will erform relative to roduct requirement or secifications. Before assessing the caability of a rocess, it is imortant that the rocess is stable and reeatable. To check if a rocess is stable, statistical rocess control is usually alied in order to be able to detect and eliminate assignable cause of variation and control charts are usually used in order to determine if the rocess is in statistical control and reveal systematic atterns in rocess outut (Montgomery, 008). The most frequent tools used when erforming a caability analysis is some kind of rocess caability index. A rocess caability index is a numerical summary that comares the behavior of a roduct or rocess characteristic to engineering secification (Kotz and Johnson, 1993). A rocess caability index is a unit-less measure that quantifies the relationshi between the actual erformance of the rocess and its secified requirements. (Mcoy, 1991) described rocess caability indices as the common way to summarize the rocess erformance. The most widely used rocess caability indices in the industry today analyze the caability of a rocess under the assumtion that the rocess is stable and that the studied characteristics is normally distributed (Siring, 1991). Process caability studies are conducted on rocesses which have all assignable causes eliminated, and insofar as ossible, have all common causes reduced to a minimum. By definition, these conditions exist when a rocess is in statistical control. A large value of the index indicates the current rocess is caable of roducing arts that, in all likelihood, will meet or exceed the customer s requirements. A caability index is convenient because it reduces comlex information about the rocess to a single number. The indices are used to communicate how well the rocess has erformed. For stable or redictable rocess, it is assumed that these indices also indicate exected future erformance. Suliers may also use caability indices for different characteristics to establish riorities for imrovement activities. Similarly, the effect of a rocess change can be assessed by comaring caability indices calculated before and after the change. In the literature, there are many rocess caability indices available. These are, k, m and mk. However, these indices oerate under the assumtion of normality (Lee, 001). lements (1989) used a Pearson distribution curve to estimate the non-normal rocess caability index, Vännman and Albing (007) roosed family of quantile based rocess caability indices for situation when the rocess data is non-normal, zarski (008) used what he called the exact method to estimate the quantiles roosed by Vännman and Albing (007). Thus, rocess caability indices should be determined based on the distribution of the rocess data. In this aer, the caability of the manufacturing rocesses of ofta tablet roduction is considered using the most common rocess caability indices. The normality assumtion was first tested for the quality characteristics under study, thereafter the test for the stability of the rocess was carried out. If the outut of the characteristic is stable, then the caability was then evaluated using the indices of rocess caability to ascertain the amount of nonconforming roduct that may reach the factory floor. Furthermore, the confidence intervals for the caability indices were constructed.

3 Evaluating Process aability Indices of a Manufacturing Process 107 Process aability Indices Process caability indices (PIs) can be categorized under two cases. These are for situations when the distribution of the rocess data is normal and non-normal. In this aer, the indices considered are the rocess caability ratio ( ) and k..1 Process aability Indices For Normal Process The Index is a rocess caability index that indicates the rocess otential erformance by relating the natural rocess sread to the secification (tolerance) sread. It is often used during the roduct design hase and ilot roduction hase. The index simly makes a direct comarison of the rocess natural tolerance to the engineering requirements. Assuming the rocess distribution is normal and the rocess average is exactly centered between the engineering requirements, a P index of 1 would give a "caable rocess." However, to allow a bit of room for rocess drift, the generally acceted minimum value for P is In general, the larger the P is the better (Merton, 003). The index is ( USL LSL) (1) 6 where, USL is the Uer Secification Limit, LSL is the Lower Secification Limit, and σ is the standard deviation of the characteristic under study. In this aer, the concet of confidence interval roosed by Kotz and Johnson (1993) is incororated. The 100(1- α) % confidence interval for the index is ˆ /, n1 ˆ 1 /, n1 () n1 n1 where ˆ. It should be noted that if the samle size is large enough (e.g. n > 100), n 1 n 1 then E ˆ. The k index is the measurement of erformance of a rocess considering both the location and the disersion information about it. It is known as the shorter standardized distance from the center of the rocess to either USL or LSL (Alsu and Watson, 1993). The k index indicates the rocess actual erformance by accounting for a shift in the mean of the rocess toward either the uer or lower secification limit. It is often used during the ilot roduction hase and during routine roduction hase (Kane, 1986). A k of at least + 1 is required, and is referred. Note that k is closely related to P, and that the difference between k and reresents the otential gain to be had from centering the rocess. The comutation formula is USL X X LSL k min, 3 3 (3) Lee (001) resented an alternative formula for comuting the k index. The formula is

4 108 Kayode S. Adekeye and Joshua O. Ogundele T k (1 k) (4) 6 * X where k, X and are the mean and standard deviation resectively, T USL LSL calculated from a random samle of size n, and T USL LSL In this aer, the Lee (001) aroach was adoted for the calculation of k. It should be noted that k can be calculated even if only one secification limit exists or if a minimum or maximum is secified. The k(max), that is k for Uer Secification Limit is determined using: k (max) USL X (5) 3 Similarly, the k for Lower Secification Limit is determined by: k (min) X LSL (6) 3 Bissel (1990) constructed a confidence interval for using Taylor s series under the k assumtion of normal distribution of ˆ k. The Bissel confidence interval is defined as ˆ ˆ ˆ 1 k ˆ 1 k k Z( / ), k Z( / ) 9n ( n 1) 9n ( n 1) (7) where Z / is the ( / ) uer quantile of standard normal distribution.. Process aability Indices For Non-Normal Process If the rocess data is non-normal, calculating and k directly based on the raw data may lead to over estimation of the PIs. One ossible way to deal with such roblem is to transform the data to normal distribution using data transformation such as Bob-ox transformation. In some cases, the rocess data distribution is so far from normal distribution and can not be transformed to normal. lements(1989) roosed a method based on quantile of the data. Actually these methods are suitable for any rocess distributions. So it is called the robust method for PA. Pan and Wu (1997) used the S-lus comuter language to simlify the calculation of non-normal rocess caabilities indices. The Pan and Wu (1997) formula for the common PIs are resented below:

5 Evaluating Process aability Indices of a Manufacturing Process 109 USL- LSL = q q and USL q q LSL k min, q q0.50 q0.50 q (8) (9) Peng (010) used Vännman (1995) unified four basic PIs to estimate PIs. The non-normal rocess caability indices can be obtained using: ( u, v) d u M m q q v M 3 T 6 th where q is the quantile of the rocess. i.e., P( X q ), M is the median of the data, T is the target value, d is the half length of the secification interval, and m is the midoint of the secification interval. That is d = (USL LSL)/ and m = (USL + LSL)/. From Equation (10), the two caability indices used in this aer can be deduced as follows:, if u v 0 ( u, v) (11) k, if u 1 and v 0 It should be noted that the Pan and Wu (1997) Equations and Equation (10) are equivalent and will therefore, yield the same indices values..3 Evaluating Parameters.3.1 Percentage Non-conforming (PN) The rocess ercent nonconforming (PN) is the long term rocess yield that can be exected from the rocess if it is allowed to oerate at the caability. For rocess with two sided secification limits, the ercentage of non-conforming items (PN) can be calculated as PN ( X USL) ( X LSL) On the assumtion of normality of the rocess, the ercentage of non-conforming items can be exressed as USL LSL PN 1 (1) Where (. ) is the cumulative distribution function of the standard normal distribution and µ and σ are the rocess mean and standard deviation, resectively. Pearn and Kotz (006) noted that if the rocess is erfectly centered at the secification range(µ = m), then the PN can be exressed as ( 3 ) and when µ m, it only rovides a lower bound on (10)

6 110 Kayode S. Adekeye and Joshua O. Ogundele PN. Thus, The rocess ercent nonconforming and rocess yield for the index is defined as: PPN = [(1- (3 ))]100 (13) The rocess ercent nonconforming and rocess yield for the k index is defined as: PPN = [ (-3 ) + (-3( - )]100 (14) k k k It should be noted that the ercent nonconforming for is twice the ercent nonconforming for k (Albing, 006).3. Defects arts er million (m) The defect arts er million (m) is obtained directly from the ercentage non-conforming defined by: m = 10000* PN For examle, if = 1.00, the corresonding PN = 0.7 which gives 700 arts er million. 3 Main Results Data from a manufacturing comany that roduces ofta tablet is used in this aer. ofta tablet is roduced with many quality characteristics among them are weight, thickness, Ammonium chloride content and the hardness of the tablet. Two out of these imortant characteristics of ofta tablet were considered in this aer. The two characteristics are the weight and thickness of the tablet. The secification limits for the weight is (mg) while the secification limits for the thickness is (mm). The Data were collected from the record book of the comany for the eriod of 30 months with the samle size of 10 er month. Thus, there are 300 data oints for each of the characteristic under study. 3.1 Weight of Tablet The secification limits for this variable are USL = 1050 and LSL = 950. The data was first tested for normality. The histogram of the data is well sread and aroximately normal. However, the cdf comare with the normal distribution shows a wide variation with the distribution of the data. This is an indication that the data is non-normal. Further analysis using the Kolmogorov-Smirnov Goodness-of-Fit Test and the Shairo-Wilk test was conducted. The Kolmogorov-Smirnov Goodness-of-Fit Test gives the following results: KS = and = The Shairo-Wilk test gives W = 0.93 with = Therefore, the weight rocess data can be adjudged to be non- normal. To test for stability of the rocess data, the X and S control charts was used. From the data X= 995.9, S= and the samle standard deviation (S) = For n =10, A 3 = 0.975, B 4 = 1.716, and B 3 = 0.84 are read off from SQ table (see e.g. Montgomery, 010). The lot of the control charts for X and S charts shows that no oints fall outside the control limits. Therefore, the rocess of the weight of ofta tablet is stable.

7 Evaluating Process aability Indices of a Manufacturing Process 111 Since the oulation standard deviation is unknown, then the samles estimate of the oulation standard deviation (S) equals was used. Thus using Equation (1), the index is obtained to be: = = = ( ) To determine the k index, we know that USL = 1050 and LSL = 950, then , T and k Substituting these values into Equation (4), we have: 100 k = ( ) = ( ) Using Equations (13) and (14) resectively, the ercent nonconforming units are 0.011and 0.04 for and k. To determine the confidence intervals for the comuted indices, Equations () and (7) were used resectively for and k for a situation when the data is assumed normal. Using the obtained values of = 1.9 and k = 1.17, the confidence intervals are for the index and for the k index However, the data was confirmed to be non-normal, therefore, the above indices values can be misleading. Thus, using Equations (10) and (11), the non-normal based PIs for the data were comuted. From the data q = 105, q = 953, and q 0.50 = 998. Substituting these values into Equations (10) and (11) we have = = k and k min, min1.93, Using Equations (13) and (14), the ercent non-conforming unit are and resectively, for and k. with confidence intervals for the index and 1.10 k 1.30 for the k index. 3. Thickness of Tablet The secification limits of the thickness quality characteristic are USL = 3.97 and LSL = Therefore,

8 11 Kayode S. Adekeye and Joshua O. Ogundele , T and k = = The data was first tested for normality using the Kolmogorov-Smirnov Goodness-of-Fit Test and the Shairo-Wilk test. The results of the Kolmogorov-Smirnov Goodness-of-Fit Test is KS = with = Also the Shairo Wilk test gives W = 0.90 with = Thus, the thickness characteristic data is adjudged to be non-normal. From the data, X 3.799, S , and S = The lot of X and S control charts indicate that no oints fall outside the control limits, and no evidence of systematic change. This is an indication that the rocess is stable. Hence we roceed to determine the caability of the characteristic rocess The index using Equation (1) is = = » 0.51, and the k index using Equation (4) is ( ) 0.4 k = ( ) = 0.499» ( ) The ercent nonconforming roducts using the above comuted indices are 1.60 and for and k, resectively. Also the confidence intervals for the comuted indices are for the index and for the k index Since the data is tested to be non-normal then the above comutation can lead to over estimation of the index. Using Equations (10) and (11), the non-normal based PIs for the data were determined to be and k min, min0.93, The ercent nonconforming roducts using the above comuted indices are 0.33 and 0.36 for and k, resectively. Also the confidence intervals for the comuted indices are for the index and for the k index k k 4 Discussion of Result The summary of the caability indices, the ercent non-conforming roducts, the art er million that are non-conforming and the length of the intervals for the two characteristics under study are resented in Table 1.

9 Evaluating Process aability Indices of a Manufacturing Process 113 Table 1: Summary of aability Indices for Weight and Thickness Normal based PIs Non-normal based PIs haracteristic Index PNm Limit Interval Weight = 1.9 k = Index PNm Limit Interval = 1.39 k = Thickness = 0.51 k = = 0.98 k = The rocess caability indices for the weight of ofta tablet are very close under the two conditions that were used in this study as reflected in Table 1. This may be due to the fact that the distribution of the data is not far from normal distribution. However, the arts er million defects for the non-normal based indices show an areciable gain over the normal based indices. The results for the thickness shows that the rocess for the thickness is not caable and not well centered since the and k are not equal. Also the non-normal based indices show areciable gain in the arts er million defects. The confidence interval for the and k were also determined for the urose of future management of the weight and thickness of the tablet rocess. However, the obtained and k for the thickness are not caable of roducing according to the secifications. Hence, the confidence intervals obtained based on the comuted values can not be used for future management. 5 onclusion The two quality characteristics of ofta tablet considered in this study reflect that the rocess were not caable and not well centered. The areciable gain obtained by using the non-normal based caability indices is a reflection of the imortance of testing the data first for the normality assumtion. If the normal based caability indices had been used to study the caability of the rocess, wrong and invalid inference would have been made and therefore a wrong conclusion would have been drawn and thus faulty signal of higher arts er million defects will be reorted. Based on the obtained results, it will be recommended that the comany should adjust the secification limits for the weight and more imortantly for the thickness of the tablets in order to increase the tolerance band which will lead to the imrovement of the recision of the rocess. References [1] M. Albing. Process aability Analysis with Focus on Indices for One-sided Secification Limits. Licentiate Thesis, Lulea University of Technology. (006), [] F. Alsu, and R.M. Watson. Practical Statistical Process ontrol, Van Nostrand Reinhold, NY. (1993) [3] A.F. Bissel. How Reliable is Your aability Index? Journal of the Royal Statistical Society,. 39(3), (1990), [4] J. A. lements. Process aability alculations For Non-normal Distributions. Quality Progress, (1989),

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