A MODIFIED APPROACH FOR ESTIMATING PROCESS CAPABILITY INDICES USING IMPROVED ESTIMATORS

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1 Pak. J. Statist. 017 Vol. 33(), A MODIFIED APPROACH FOR ESTIMATING PROCESS CAPABILITY INDICES USING IMPROVED ESTIMATORS Seem Şaha Vahaplar 1 ad Özlem Ege Oruç Departmet of Statistics, Dokuz Eylül Uiversity, Türkiye 1 seem.saha@deu.edu.tr; ozlem.ege@deu.edu.tr Correspodig author ABSTRACT I this paper, ew estimators for process capability idices have bee proposed usig improved estimators of populatio mea ad variace. The proposed ad classical idices are compared with respect to mea square error. A umerical example is cosidered to illustrate the proposed idices ad to judge the merits of the proposed estimators. KEYWORDS Statistical process cotrol, Process capability idices, Improved estimatio, Mea square error. 1. INTRODUCTION Process capability aalysis has bee studied by may researchers i both theoretical ad applied fields. Process capability idices quatify how much the process is able to produce required items. Kotz ad Lovelace (1998) ad Wu et al. (009) defie the geeral idea of process capability idices as comparig what the process should do with what the process is actually doig. Kae (198) states that these idices relate to the atural tolerace ad egieerig specificatios. Details of these idices are give by Kotz ad Johso (1993) ad Kotz ad Lovelace (1998). I statistical process cotrol, two assumptios have to be satisfied for coductig a process capability aalysis; (i) the process should be i a state of statistical cotrol, (ii) the quality characteristic should be ormally distributed (Motgomery, 009; Stoumbos, 00; Ali et al., 015; Väma ad Kotz, 1995; Kae, 198; Nagata, 1995; Maiti et al., 010). Process capability idices give reliable results whe these assumptios are satisfied. Improved estimatio plays a importat role i statistical iferece. The mai purpose of this estimatio techique is to examie the coditios uder which biased estimators ca lead to a improvemet over the covetioal ubiased procedures. Give additioal iformatio such as coefficiet of variatio, kurtosis or skewess, the problem has bee studied extesively by Searls (194), Sigh et al. (1973), Bibby ad Touteburg (1974) ad Arholt ad Hebert (1995). Compared to the usual procedures, improved estimatio provides more efficiet estimators with respect to mea square error. 017 Pakista Joural of Statistics 411

2 41 A Modified Approach for Estimatig Process Capability Idices. The aim of this paper is to itroduce ew estimators of process capability idices by usig improved estimators of populatio mea ad variace. It should be oted that improved estimators of populatio parameters are geerally biased estimators but they have lower mea square error (MSE).So, process variability is cosidered to be reduced sice proposed estimators are biased with lower mea square error with regard to the classical estimators of process capability idices. The paper is orgaized as follows. First, we provide a overview of process capability idices ad improved estimators of populatio mea ad variace. The we propose improved estimators of process capability idices. Fially, we compare them with the classical estimators of process capability idices ad evaluate the results.. PROCESS CAPABILITY INDICES C p seems to be the first process capability idex proposed i the literature (Jura, 1974; Kae, 198). It assesses whether the atural tolerace of a process is withi the specificatio limits. This meas comparig allowable process spread with the actual process spread as give i equatio (1), C p allowable process spread USL LSL actual process spread where USL ad LSL are upper ad lower specificatio limits, respectively (Jura, 1974; Wu et al., 009; Kae, 198). C p is stated to be iadequate for off-ceter processes. For a off-ceter process; the process mea is ot equal to the target value (T). For such processes, C pk idex provides a good alterative. It takes both the process spread ad the departure of the process mea from the target value ito cosideratio. C pk USL LSL d m mi, d is the half specificatio width defied i (3) ad m is the midpoit betwee upper ad lower specificatio limits as i (4). USL LSL d (3) USL LSL m (4) Despite beig more explaatory tha C p idex, (1) () C pk still has a disadvatage. It caot provide iformatio about the locatio of process mea i the tolerace iterval LSL, USL. Also, C ad C are ot related to the cost of failig to meet the target p pk

3 Şaha Vahaplar ad Ege Oruç 413 requiremet of the customers. For this reaso, Hsiag ad Taguchi (1985) ad Cha et al. (1988) proposed C pm idex, C pm 3 USL LSL d T (5) where is a measure of the average product deviatio from the target value. E X T T Pear et al. (199) itroduced C pmk idex havig all the properties of previous idices. Two importat compoets of this idex ca be summarized as (i) variatio relative to the process mea (σ ) ad (ii) deviatio of the process mea from the target T. Therefore, C pmk is more sesitive tha previous idices (Vama, 1995). () C pmk USL LSL d m mi, T T T (7) 3. PROPOSED ESTIMATORS OF PROCESS CAPABILITY INDICES I this sectio, alterative estimators of four process capability idices are proposed. It is assumed that the process is uder statistical cotrol ad approximately ormally distributed. Let x1, x,, x be a radom sample from a ormally distributed process with ukow mea µ ad variace σ. It is well kow that the ubiased estimators of µ ad σ are x ad s, respectively. 1 x i 1 i x (8) 1 s x i x 1 (9) i1 Searls (194) defied a improved estimator of the populatio mea as give, x 1 x i i 1 (10) v where v is the kow coefficiet of variatio of the distributio. It ca be see that MSE x is always less tha MSE x. Sigh et al. (1973) defied a improved estimator of the populatio variace as,

4 414 A Modified Approach for Estimatig Process Capability Idices. s x i x (11) 3 1 i1 where, the coefficiet of kurtosis, is kow as a priori iformatio. It is well kow that the coefficiet of kurtosis ca be calculated through equatio (1) by usig the fourth momet aroud the origi. It ca be see that MSE s is always less tha MSE s. 4 4 (1) Usig the improved estimators of populatio mea ad variace, the estimators of process capability idices are obtaied as follows. p USL LSL USL LSL ˆ x i x 3 1 i1 (13) Here, it was formerly stated that the process is ormally distributed. Besides, is assumed to be kow as a prior iformatio about the process. As the value of coefficiet of kurtosis is 3 for ormal distributio, it is substituted i equatio (13) ad (14) is obtaied. p USL LSL i1 x i x 1 (14) The estimator of C pk is obtaied as pk xi d m d ˆ m v 3ˆ xi x 3 1 where v is the coefficiet of variatio for ormal distributio which is v. (15) The estimator of C pm idex is give below. pm USL LSL USL LSL ˆ T x x x ˆ i i T 1 v (1)

5 Şaha Vahaplar ad Ege Oruç 415 Fially, the estimator of C pmk is give i equatio (17). pmk xi d m d ˆ m v ˆ T x x x ˆ i i T 1 v (17) 4. COMPARISON OF CLASSICAL AND PROPOSED INDICES I order to research the behavior of classical ad proposed idices, a applicatio is realized usig Miitab. For this applicatio, a sample data set from Motgomery (009) is used. Data set cosists of flow width measuremet data of the resistace i semicoductor maufacturig. The target value of flow width is 1,5 micros. Also, lower ad upper specificatio limits are 1 ad micros, respectively. Histogram of data with USL, LSL ad target value is give i Figure 1. Figure 1: Histogram of Flow Width Data with LSL, USL ad Target Value

6 41 A Modified Approach for Estimatig Process Capability Idices. As metioed before, there are two assumptios of process capability aalysis. So we eed to check whether the assumptios are satisfied or ot. Aderso-Darlig ormality test shows that the data are distributed ormally. The output of the test is give i Figure. Figure : Output of the Normality Test for Flow Width Data x & s charts i Figure 3 show that oe of the observatios fall beyod the cotrol limits ad the process is i statistical cotrol.

7 Sample StDev Sample Mea Şaha Vahaplar ad Ege Oruç 417 Xbar-S Chart of x1;...; x5 1,7 UCL=1,7091 1, 1,5 _ X=1,54 1,4 1, Sample LCL=1,3438 0,3 UCL=0,73 0, 0,1 _ S=0,180 0,0 LCL= Sample Figure 3: x & s Cotrol Charts for Flow Width Data Both assumptios are satisfied so the estimates of idices are calculated as i Table 1. It ca be see that proposed estimates for all four idices are greater tha their classical estimates. MSE of proposed estimators are lower tha classical estimators. This meas that the variability of proposed estimators are lower. So the estimatios obtaied by proposed estimators ca be more reliable tha the classical estimators because of lower variability. Table 1 Classical ad Proposed Estimates of Process Capability Idices for Flow Width Data Process Capability Idex Classical Estimate Proposed Estimate C p C pk C pm Cpmk CONCLUDING REMARKS I this paper, our focus has bee itroducig ew estimators of process capability idices by usig improved estimatio of populatio mea ad variace. The proposed estimators are biased but they have lower mea squared error.

8 418 A Modified Approach for Estimatig Process Capability Idices. To compare the estimators, a sample data set is used. First, ormality test is realized to see if the process is ormally distributed or ot. The, quality cotrol charts are used to see whether the process is i a state of statistical cotrol. Fially, classical ad proposed estimates of C p, C pk, C pm ad C pmk idices are obtaied. Accordig to the results, proposed estimates for all four idices are greater tha their classical estimates. REFERENCES 1. Ali S., Aslam, M., Abbas, N., Kazmi, S.M.A. ad Hasa, T. (015). O process capability ad system availability aalysis of the iverse Rayleigh distributio. Pakista Joural of Statistics ad Operatio Research, XI(1), Cha, L.K., Cheg, S.W. ad Spirig, F.A. (1988). A ew measure of process capability C pm. Joural of Quality Techology, 0(3), Hsiag, T.C. ad Taguchi, G. (1985). A tutorial o quality cotrol ad assurace - The Taguchi methods. I: ASA Aual Meetig, Las Vegas. 4. Jura, J.M. (1974). Quality Cotrol Hadbook, 3 rd editio. McGraw-Hill, New York. 5. Kae, V.E. (198). Process capability idices. Joural of Quality Techology, 18(1), Kotz, S. ad Johso, N.L. (1993). Process Capability Idices. Chapma & Hall, Lodo. 7. Kotz, S. ad Lovelace, C. (1998). Process Capability Idices i Theory ad Practice. Arold, Lodo. 8. Maiti, S.S., Saha, M. ad Nada, A.K. (010). O geeralizig process capability idices. Quality Techology & Quatitative Maagemet, 7(3), Motgomery, D.C. (009). Itroductio to Statistical Quality Cotrol, th Editio. Wiley, Uited States of America. 10. Nagata, Y. (1995). 永田靖 - 岡山大学経済学会雑誌, (3-4), Pear, W.L., Kotz, S. ad Johso, N.L. (199). Distributioal ad iferetial properties of process capability idices. Joural of Quality Techology, 4(4), Searls, D.T. (194). The utilizig of a kow coefficiet of variatio i estimatio procedure. Joural of America Statistical Associatio, 59, Sigh, J., Padey, B.N. ad Hirao, K. (1973). O the utilizig of a kow coefficiet of variatio of kurtosis i estimatio procedure of variace. Aals of the Istitute of Statistical Mathematics, 5, Stoumbos, Z.G. (00). Process capability idices: overview ad extesios. Noliear Aalysis: Real World Applicatios, 3, Väma, K. ad Kotz, S. (1995). A superstructure of capability idices - Distributioal properties ad implicatios. Scadiavia Joural of Statistics, (4), Väma, K. (1995). A uified approach to capability idices. Statistica Siica, 5, Wu, A.W., Pear, W.L. ad Kotz, S. (009). A overview of theory ad practice o process capability idices for quality assurace. Iteratioal Joural of Productio Ecoomics, 117,