Bayesian Economic Cost Plans II. The Average Outgoing Quality
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1 Eltro. J. Math. Phs. Si. 22 Sm Eltroi Joural of Mathmatial ad Phsial Sis EJMAPS ISS: Basia Eoomi Cost Plas II. Th Avra Outoi Qualit Abraham F. Jalbout 1*$ Hadi Y. Alkahb 2 Fouad. Jalbout 3 Abdulla Darwish 3 1 Dartmt of Chmistr Uivrsit of w Orlas w Orlas LA USA Ajalbout@jmas.or 2 Dartmt of Mathmatis Dillard Uivrsit w Orlas LA 7112 USA 3 Dartmt of Phsis ad Eiri Dillard Uivrsit w Orlas LA 7112 USA *Author to whom orrsod should b addrssd. $ Sakr at th 16 th Aual Cofr o Alid Mathmatis (CAM Uivrsit of Ctral Oklahoma Fbruar Rivd: 14 Dmbr 21 / Atd: 5 Jauar 22/Pubishd: 15 Jauar 22 Abstrat: I rt ars rsarhrs i various qualit otrol rodurs osidr th ossibilit of istio rrors as a imortat issu. Th rs of ths rrors lads to has i th so-alld oratioal haratristi (O.C. otrol urv ad as a rsult th avra outoi qualit of a idustrial ross. W rst a w mathmatial modl that a b alid to alulat suh quatitis as th td umbr of dftiv itms rlad i a atd lot ad othr futios of this ross. Kwords: Istio Errors idustrial ross Basia mthods statistis AMS Mathmatial Subjt Classifiatio: P3 22 b EJMAPS (htt:// Rrodutio for oommrial uross rmittd
2 Eltro. J. Math. Phs. Si. 22 Sm Itrodutio I attribut samli las th rrors ar rall of two kids: T I rror: a ood itm is lassifid as bad with a robabilit 1 T II rror: a bad itm is lassifid as ood with a robabilit 2 Collis ad Cas [1] drivd a rssio for th robabilit of ata udr istio rrors. A rssio was latr drivd for th marial distributio of th obsrvd dftivs. (Hald [2-5] Cas Btt ad Shmidt [6-7] dvlod formulas for alulati th avra outoi qualit (AOQ wh attribut istio is subjt to T I ad T II istio rrors. i diffrt rtifiatio istio oliis ar osidrd. Ths oliis wr first itrodud b Wortham ad Mo [8]. Bai ad Cas [9] latr ralizd ths modls ad th dvlod i diffrt saml/rst-oflot disositio oliis for sil ad doubl samli alo liitl dvlod AOQ modls. I this work both th attribut variabl las ar osidrd [1] ad a Basia thiqu is dvlod to stimat diffrt aramtrs. Adi A lists all of th otatios usd i this work. Althouh mor th mor rt work of Johso Kotz ad Wu [11] dsribs som mor modr idustrial aroahs to istio rrors with attributs i qualit otrol th still fort to ilud th Basia mthods a ovl aroah whih is rstd i this work. 2. Mathmatial Dvlomt Hald [2-5] has drivd th followi form of th marial distributio of : ( (1 1 L (2.1 udr th assumtio that th umbr of dftivs i a lot siz is biomiall
3 Eltro. J. Math. Phs. Si. 22 Sm distributd with a.d.f: (2.2 f 1 (1 ( L whr is th ross fratio dftiv. Th sod assumtio of quatio (2.1 is that th umbr of dftivs i a saml siz iv is hromtri: f ( (2.3 thus this rovs that th Hald s [2-5] drivatios of th biomial distributio is rrodud b hromtri samli. Thus for th Basia orati haratristi (BOC urv th robabilit of lot ata is drivd from th abov quatios as: ( (1 ( a L whr is th ata umbr. For th istio rror aalsis th obsrvd dftivs from a saml is rlad b obsrvd umbr of dftivs. Th robabilit of lot ata iv i (2.4 will b rdud: a ( (2.5 whr: 1 (1 ( L (2.6
4 Eltro. J. Math. Phs. Si. 22 Sm Equatio (2.4 ivs th robabilit of lot ata for rft istio. Th robabilit of lot ata wh istio rrors ar rstd as a i quatio (2.5 ad usi quatio (2.6 w a driv: a (1. (2.7 W a ow ddu a rssio for th avra outoi qualit (AOQ. Th AOQ a b dfid as: td umbr of dftiv itms rmaii aftr istio AOQ total umbr of itms i th lot ( Pa. (2.8 A rssio fro AOQ a b drivd b itrodui th followi trms: ( M th umbr of dftivs i th uistd ortio of a atd ( M 2 th umbr of dftiv of dftiv itms lassifid as bi ood i th srd ortio of th rjtd lot. 2 is th umbr of dftiv itms lassifid as ood i th saml DITR is th umbr of dftiv itms itrodud throuh rlamt ito th lot. For a atd lot th td umbr of dftiv itms rlad i th lot is: Th robabilit that a itm is lassifid as bi ood is th:. (2.9 P ( 1 ( (2.1 A st of 1 itms ar sltd at radom tstd ad lassifid as ood or bad. A total of itms wr dd to rla th dftiv itms i th atd lot. This rodur
5 Eltro. J. Math. Phs. Si. 22 Sm of samli dfis a ativ biomial ross. Th td umbr of itms tstd to obtai itms whih ar ood is th:. (2.11 P Th td umbr of dftiv itms rlad i a atd lot is th: 2 DITR a P (2.12 P. Th td umbr of dftiv itms rlad i a rjtd lot whih is srd is: DITR s ( 2. (2.13 Th td umbr of itms to b rlad is: ( DITR 2 + P2 (1 P P P 2 [ + ( (1 Pa ]. P a (2.14 Th rssio AOQ is th: ( Pa + ( (1 Pa DITR AOQ. (2.15 Erssio (14 a b rdud to th form: 2 + ( (1 Pa + ( (1 Pa 2 AOQ. (2.16 (1 Similarl th AOQ rssio for samli with o rlamt a b drivd as: ( Pa + ( (1 Pa AOQ. (2.17 (1 P ( This is tru so o dftivs ar itrodud throuh th rlamt ross. a
6 Eltro. J. Math. Phs. Si. 22 Sm Colusios I this work rssios for th avra outoi qualit wr drivd for both th modl ivolvi rlamt of dftiv itms i th lot ad wh th itms ar ot rlad. Th ottial aliatio of this work lis i th abilit of idustrial rsarhrs to alulat both of ths quatitis ad did th loss of suh vts whih ar so vr ommo i ral lif istios. This siml modl is ovl ad will b fruitful for a idustrial ross ivolvi a ostat istio ross. (s Jalbout Alkahb [12] REFERECES 1. R.D. Collis K.E. Cas AIIE Trasatios ( A. Hald Statistial Thor of Samli Istio b Attributs Part Uivrsit of Coha Istitut of Mathmatial Statistis Dmark 3. A. Hald Statistial Tabls for Samli Istio b Attributs 1976 Uivrsit of Coha Istitut of Mathmatial Statistis Dmark 4. A. Hald Statistial Thor of Samli Istio b Attributs Part Uivrsit of Coha Istitut of Mathmatial Statistis Dmark 5. J.L. Hrbrt Commuiatios i Statistis. Thor ad Mthods K.E. Cas J.W. Shmidt G.K. Btt AIIE Trasatios ( R.D. Collis K.E. Cas G.K. Btt Itratioal Joural of Produtio Rsarh ( A.W. Wortham J. Mo Joural of Qualit Tholo 197 2( I. Bai K.E. Cas Joural of Qualit Tholo ( J.W. Shmidt K.E. Btt Joural of Qualit Tholo ( L. Johso S. Kotz. Wi Istio Errors for attributs i qualit otrol 1991 Chama ad Hall Ltd (Lodo; w York 12. F.. Jalbout H.Y. Alkahb Elt. J. Diff. Eqs Cof. 2 15
7 Eltro. J. Math. Phs. Si. 22 Sm Adi A: otatios Usd i tt otatio Dfiitio Fratio of itms dftiv Aart fratio dftiv 1 T I istio rror 2 T II istio rror Lot siz Valu for masurabl qualit haratristi i variabl samli las for fratio dftiv Saml Saml siz P a Probabilit of ata for a sil variabl samli la for fratio dftiv. Ata umbr i Distributio of atual dftivs a Probabilit of ata wh rrors ar rst Obsrvd umbr of dftivs AOQ Avra outoi qualit AOQ Avra outoi qualit rror
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