Economic Specification Limits Setting for Rectifying Inspection Plan with Inspection Error

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1 Eonomi Spifiation imits Stting for Rtifying nsption Plan with nsption Error Chung-Ho Chn Dpartmnt of Managmnt and nformation Thnology Southrn Taiwan nivrsity 1 Nan-Tai Strt, Yung-Kang City Tainan 71, Taiwan hnh@mail.stut.du.tw Abstrat n this papr, w prsnt a modifid Pulak and Al-Sultan s modl for dtrmining th onomi spifiation limits undr th singl sampling rtifying insption plan with insption rror. Th 1% insption and prft rplamnt of produts ar onsidrd in th rjtd lot and th non-onforming itms in th sampl of aptd lot ar not rplad but liminatd from th lot. Taguhi s symmtri quadrati quality loss funtion is applid for valuating th produt quality. A numrial xampl and th snsitivity analyss of paramtrs ar providd for illustration. Kywords: Rtifying nsption Plan, nsption Error, Spifiation imits, Taguhi s Quality oss Funtion. 1. ntrodution Taguhi [18] proposd th quadrati quality loss funtion rdfind by th produt quality is th total loss to th soity. Th quadrati quality loss funtion an b applid in statistial pross ontrol, th onomi dsign of sampling plan, th onomi sltion of spifiation limits, and othr work of on-lin and off-lin quality ontrol. Kapur and Wang [9], Kapur [1], Kapur and Cho [11, 1], and Fng and Kapur [5, 6] hav addrssd th problm of stting th onomi spifiation limits for th quality haratristi whn pross is not apabl of spifiations in th short trm. Kapur and Wang [9, p. 8] pointd out that suppos w an t improv th prsnt pross, thn a short trm approah to dras varian of th itms shippd to th ustomr is to put spifiation limits on th pross and trunat th distribution by insption. Th 1% insption is adoptd and th quadrati quality loss funtion is applid to dsign th onomi spifiation limits of th quality haratristi. Th quality loss is valuatd from th viwpoint of both th produrs and th ustomrs. Th loss to th ustomrs is du to variability from th targt valu and to th produrs ar inurrd du to insption and srap. Th onomi - 1 -

2 spifiation limits must b dtrmind on th basis of minimizing total loss, whih inluds insption ost, srap/rwork ost and loss to th ustomrs as wll as to th produrs. nsption rror usually ours whn th produt is masurd. Prvious rsarhrs,.g., Tang [16], Chn and Chung [, 3], Hong and Elsayd [8], Frrll and Chhokr [7], Markowski and Markowski [13], Duffuaa and Siddiqui [4], Fng and Kapur [6], and Tasli and Koksal [15] hav proposd th fft of insption rror on th problm of statistial quality ontrol. Tang and Tang [17] disussd th gnral srning produr for produt quality. Srning is boming an attrativ prati for rmoving non-onforming itms. n rtifying insption plan, on usually nds to do 1% insption for th produt of th rjtd lot and rpla th non-onforming itms by onforming ons. Pulak and Al-Sultan [14] prsntd a rtifying insption plan for dtrmining th optimum pross man. Howvr, thy did not onsidr th quality ost for th produt within th spifiation limits and did not point out whthr th non-onforming itms in th sampl of aptd lot is rplad or liminatd from th lot. Chn [1] propos a modifid Pulak and Al-Sultan s modl for dtrmining th optimum pross man and standard dviation undr th rtifying insption plan with avrag outgoing quality limit prottion. Th quality ost within spifiation limits has bn addrssd in Chn s modl. n this papr, w prsnt a modifid Pulak and Al-Sultan s [14] modl for dtrmining th optimum onomi spifiation limits undr th rtifying insption plan with insption rror. Th singl attribut sampling plan is usd in srning th lot of produt. Th 1% insption and prft rplamnt of produts ar onsidrd in th rjtd lot and th non-onforming itms in th sampl of aptd lot ar not rplad but liminatd from th lot. Th quadrati quality loss funtion is applid for valuating th produt quality. Th numrial xampl and th snsitivity analyss of paramtrs ar providd for illustration.. Modifid Pulak and Al-Sultan s Modl with nsption Error Considr a produt with both-sidd spifiation limits. Th insption rror ours baus of imprft masurmnt systms. Th tru valu of quality haratristi, Y, is normally distributd with man µ and standard dviationσ. Dnot th obsrvd valu of Y as X. Assum that th onditional distribution of X givn Y is normally distributd with man y and standard dviationσ. Hn, th obsrvd valu X is normally distributd with man µ and standard dviation 1/ α + 1/ β, whr α = 1/ σ and β = 1/ σ. Th xptd ost pr itm for modifid Pulak and Al-Sultan s modl nds to inlud th quadrati quality loss of produt. Th onditional xptd ost funtion is - -

3 n + ( N D µ + ( N n oss( y f( y dy+ oss( y h( x, y dydx ( n D, if D d hx (, ydydx F1 = (1 R[ E( D D > d + p( N n] + N + Nµ + oss( y h( x, y dydx N, if D > d hx (, ydydx whr n is th sampl siz; N is th lot siz; D is th numbr of non-onforming itms found in th sampl siz n; d is th allowan numbr of non-onforming itms found in a sampl siz n; Y is th tru valu of quality haratristi; f(y is th probability dnsity funtion of Y; oss( y = k( y y is th quality loss funtion of Y; k is th quality loss offiint; y is th targt valu of produt; X is th obsrvd valu of quality haratristi; h( x, y is th joint distribution of X and Y; = y σ is th lowr spifiation limit of produt; = y + σ is th uppr spifiation limit of produt; is th insption ost pr itm; R is th xptd ost of rplaing all rjtd itms found in a rjtd lot ( = R drl ; R is th ost of rplaing a dftiv itm by an aptabl itm; d rl is th xptd numbr of dftiv itms in a rjtd lot (= th xptd numbr of dftivs found in th sampl, givn that th lot was rjtd + th xptd numbr of dftivs in th non-sampl portion of th lot, drl = ED ( D> d + p ( N n ; is th ost of prossing pr itm; p is th apparnt probability of a dftiv itm µ µ ( = 1 Φ ( +Φ( ; P( D d is th probability of aptan for 1/ α + 1/ β 1/ α + 1/ β lot (= 1 P( D > d. Hn, th xptd valu of unonditional marginal ost for a lot siz N givn th stpoints ar nd,, µ, σ,and σ is givn by - 3 -

4 1 { ( µ ( ( ( ETC = n + N D + N n k y y f y dy + ( (, ( n D } P( D d + { R [ E( D D > d + p ( N n] + hx (, ydydx ( (, N + Nµ + N } P( D > d k y y h x y dydx k y y h x y dydx hx (, ydydx ( t = ( ( A k y y f y dy and A 1 = k y y h x y dydx ( (,. hx (, ydydx Hn, th xptd valu of unonditional marginal ost for a lot siz N an b rwrittn as ETC = [ n + ( N D µ + ( N n A + ( n D A ] P( D d ( N + Nµ + R + NA [1 P( D d ] 1 = R + N + Nµ + NA ( A + µ npp( D d [ R + ( N n + ( N n( A1 A ] P( D d (3 Thus, th xptd ost pr itm is R ( A + µ npp( D d 1 ETC µ A N N R n n [ + (1 + (1 ( A1 A ] P( D d N N N 1 = (4 whr ED ( D> d = d 1 np d ( np np[1 ] d = d! d np d ( np 1 d! d = (5-4 -

5 d np d ( np PD ( d = (6 d! d = k( y y f( y dy = k[ σ + ( µ y ] (7 From Fng and Kapur [6], w hav following Eqs. (8-(13: p µ µ = 1 Φ ( +Φ( 1/ α + 1/ β 1/ α + 1/ β (8 h( x, y dydx = Φ( µ Φ( 1/ α + 1/ β µ 1/ α + 1/ β (9 1 k y y h x y dydx ( (, α α βµ + β + + = k xmxdx ( + k ( y xmxdx ( + ( α α β α β βµ 1 [( ] ( α + β α + β + k y + m x dx (1 µ µ µ xmxdx ( = µ [ Φ( Φ( ] µ 1/ α + 1/ β [ ϕ( 1/ α + 1/ β 1/ α + 1/ β 1/ α + 1/ β µ µ µ ϕ( ] + (1 / α + 1/ β {[ ϕ( (11 1/ α + 1/ β 1/ α + 1/ β 1/ α + 1/ β µ µ µ µ ϕ( ] +Φ( Φ( } 1/ α + 1/ β 1/ α + 1/ β 1/ α + 1/ β 1/ α + 1/ β µ µ xm( x dx = µ [ Φ( Φ( ] 1/ α + 1/ β 1/ α + 1/ β µ µ 1 / α + 1 / β [ ϕ( ϕ( ] 1/ α + 1/ β 1/ α + 1/ β (1 µ µ m( x dx = Φ( Φ( 1/ α + 1/ β 1/ α + 1/ β (13-5 -

6 whr ϕ( is th probability dnsity funtion of standard normal variabl and Φ ( is th umulativ distribution funtion of standard normal variabl. t is diffiult to show that th sond ordr drivativ of in Eq. (4 is positiv and it has a losd-form solution. For th givn paramtrs, w may apply th dirt sarh mthod for obtaining th optimum * (th optimum spifiation limits * = y * σ and = y + σ with th minimum xptd ost pr itm. On an st = 4(.14 for * * sarhing th minimum ETC. 3. Numrial Exampl and Snsitivity Analyss of Paramtrs Considr that th quality haratristi is normally distributd with known man µ = 1 and standard dviation σ =. 5. Th pross is undr ontrol but not apabl of spifiations. W adopt th rtifying insption plan for valuating th quality of lot. Th 1% insption and prft rplamnt of produts ar onsidrd in th rjtd lot and th non-onforming itms in th sampl of aptd lot ar not rplad but liminatd from th lot. Assum that th standard dviation of insption rror σ =. 5 du to imprft masurmnt mahin. t th lot siz N = 1, th insption ost pr itm = 1, th ost of prossing pr itm = 5, and th ost of rplaing a dftiv itm by an aptabl itm R = 3. Assum that th quality loss offiint k = 5, th targt valu of th produt y = 11, th sampl siz n = 36, and th aptan numbr d =. By solving th modifid modl (4, w hav th optimum * =. 43with ETC = Hn, w hav th onomi lowr and uppr spifiation limits S = and S = 11.15, rsptivly. Tabls 1-9 list th fft of th paramtrs on th optimum solution. From Tabls 1-9, w hav th following onlusions: 1. Th lot siz N, th insption ost, and th prossing ost pr itm hav no fft on th optimum. Th standard dviation of insption rror σ has a littl fft on th optimum. Th pross standard dviation σ, th pross man µ, th rplamnt R, th quality loss offiint k, and th targt valu t hav a major fft on th optimum.. Th lot siz N has no fft on th optimum ETC. Th pross standard dviation σ, th standard dviation of insption rror σ, th insption ost, th rplamnt R, th quality loss offiint k, and th targt valu t hav a littl fft on th optimum ETC. Th prossing ost pr itm and th pross man hav a modrat fft on th optimum ETC

7 4. Conlusions n this papr, w hav prsntd th onomi spifiation limits stting basd on singl sampling rtifying insption plan with insption rror. This work is an xtnsion of Pulak and Al-Sultan s [14] on. Furthr dirtion of study will xtnd this mthod to th skwd quality haratristi or othr attribut rtifying insption plans. Rfrns 1. Chn, C.H., Th modifid Pulak and Al-Sultan s modl for dtrmining th optimum pross paramtrs, Communiations in Statistis Thory and Mthods, 35, (6.. Chn, S.. and Chung, K. J., nsption rror ffts on onomi sltion of targt valu for a prodution pross, Europan Journal of Oprational Rsarh, 79, ( Chn, S.. and Chung, K. J., Sltion of th optimal prision lvl and targt valu for a prodution pross: th lowr-spifiation-limit as, E Transations, 8, ( Duffuaa, S. O. and Siddiqui, A.W., Pross targting with multi-lass srning and masurmnt rror, ntrnational Journal of Prodution Rsarh, 41, (3. 5. Fng, Q. and Kapur, K. C., Eonomi dvlopmnt of spifiations for 1% insption basd on asymmtri quality loss funtions, E Transations, 38, (6a. 6. Fng, Q. and Kapur, K. C., Eonomi dsign of spifiations for 1% insption with imprft masurmnt systms, Quality Thnology & Quantitativ Managmnt, 3, (6b. 7. Frrll, W. G. and Chhokr, A., Dsign of onomially optimal aptan sampling plans with insption rror, Computrs and Oprations Rsarh, 9, (. 8. Hong, S. H. and Elsayd, E. A., Th optimum man for prosss with normally distributd masurmnt rror, Journal of Quality Thnology, 31, ( Kapur, K. C. and Wang, C. J., Eonomi dsign of spifiations basd on Taguhi's onpt of quality loss funtion, in Quality: Dsign, Planning, and Control, DVor, R. E. and Kapoor, S. G., Ed., Boston, Th Wintr Annual Mting of th Amrian Soity of Mhanial Enginrs, 3-36 ( Kapur, K. C., An approah for dvlopmnt of spifiations for quality improvmnt, Quality Enginring, 1, ( Kapur, K. C. and Cho, B.-R., 1994, Eonomi dsign and dvlopmnt of spifiations, Quality Enginring, 6, ( Kapur, K. C. and Cho, B.-R., Eonomi dsign of th spifiation rgion for multipl quality haratristis, E Transations, 8, (

8 13. Markowski, E. P. and Markowski, C. A., mprovd attribut aptan sampling plans in th prsn of mislassifiation rror, Europan Journal of Oprational Rsarh, 139, (. 14. Pulak, M. F. S., Al-Sultan, K. S., Th optimum targting for a singl filling opration with rtifying insption, Omga, 4, ( Tasli, A. and Kohsal, G., Th fft of insption rror and rwork on quality loss, Th 35th ntrnational Confrn on Computrs and ndustrial Enginring, ( Tang, K., Th ffts of insption rror on a omplt insption plan, E Transations, 19, ( Tang, K. and Tang, J., Dsign of srning produr: a rviw, Journal of Quality Thnology, 6, 9-6 ( Taguhi, G., ntrodution to Quality Enginring, Asian Produtivity Organization, Tokyo, Japan (1986. Tabl 1 Th fft of lot siz N N ETC Tabl Th fft of pross standard dviation σ σ ETC Tabl 3 Th fft of standard dviation of insption rror σ ETC Tabl 4 Th fft of insption ost ETC

9 Tabl 5 Th fft of pross man µ µ ETC Tabl 6 Th fft of prossing ost ETC Tabl 7 Th fft of rplaing ost R ETC R Tabl 8 Th fft of quality loss offiint k k ETC Tabl 9 Th fft of targt valu y y ETC

10 具檢驗誤差之選剔檢驗計劃的經濟規格界限設定陳忠和南台科技大學管理與資訊系摘要本研究提出一具檢驗誤差的修正 Pulak 與 Al-Sultan 模式以決定經濟規格界限, 在選剔檢驗計劃中拒收批將採全數檢驗並考慮以完全置換方式來更換不良品, 至於接受批中樣本所發現不良品將加以廢棄田口對稱二次品質損失函數將用來衡量產品的品質, 文中將舉數值例子並進行參數敏感度分析加以說明關鍵字 : 選剔檢驗計劃檢驗誤差規格界限田口品質損失函數 - 1 -

生物統計教育訓練 - 課程. Introduction to equivalence, superior, inferior studies in RCT 謝宗成副教授慈濟大學醫學科學研究所. TEL: ext 2015

生物統計教育訓練 - 課程. Introduction to equivalence, superior, inferior studies in RCT 謝宗成副教授慈濟大學醫學科學研究所. TEL: ext 2015 生物統計教育訓練 - 課程 Introduction to equivalence, superior, inferior studies in RCT 謝宗成副教授慈濟大學醫學科學研究所 tchsieh@mail.tcu.edu.tw TEL: 03-8565301 ext 2015 1 Randomized controlled trial Two arms trial Test treatment