FUZZY ACCEPTANCE SAMPLING AND CHARACTERISTIC CURVES

Itratioal Joural of Computatioal Itlligc Systms, Vol. 5, No. 1 (Fbruary, 2012), 13-29 FUZZY ACCEPTANCE SAMPLING AND CHARACTERISTIC CURVES Ebru Turaoğlu Slçu Uivrsity, Dpartmt of Idustrial Egirig, 42075, Koya, Tury İhsa Kaya* Yıldız Tchical Uivrsity, Dpartmt of Idustrial Egirig, 34349, Yıldız, Istabul, Tury *Corrspodig Author s E-mail: ihaya@yildiz.du.tr; iaya@yahoo.com Cgiz Kahrama Istabul Tchical Uivrsity, Dpartmt of Idustrial Egirig, 34367, Maca, Istabul, Tury Abstract Accptac samplig is primarily usd for th ispctio of icomig or outgoig lots. Accptac samplig rfrs to th applicatio of spcific samplig plas to a dsigatd lot or squc of lots. Th paramtrs of accptac samplig plas ar sampl sizs ad accptac umbrs. I som cass, it may ot b possibl to dfi accptac samplig paramtrs as crisp valus. Ths paramtrs ca b xprssd by liguistic variabls. Th fuzzy st thory ca b succssfully usd to cop with th vaguss i ths liguistic xprssios for accptac samplig. I this papr, th mai distributios of accptac samplig plas ar hadld with fuzzy paramtrs ad thir accptac probability fuctios ar drivd. Th th charactristic curvs of accptac samplig ar xamid udr fuzziss. Illustrativ xampls ar giv. Kywords: Accptac Samplig, Fuzzy Sts, Charactristic Curvs, Samplig Pla 1. Itroductio Wh ispctio is for th purpos of accptac or rjctio of a product, basd o adhrc to a stadard, th typ of ispctio procdur mployd is usually calld accptac samplig. It is widly usd i idustry for cotrollig th quality of shipmts of compots, supplis, raw matrials, ad fial products. Accptac samplig plas ar practical tools for quality assurac applicatios ivolvig quality cotract o product ordrs ad it is a importat aspct of statistical quality cotrol. Accptac samplig ca b prformd durig ispctio of icomig raw matrials, compots, ad assmblis, i various phass of i-procss opratios, or durig fial product ispctio. Accptac sampls of icomig matrials may b rquird to vrify coformity to thir rquird spcificatios. I a wll-dvlopd quality systm, supplirs masurmts ca b rlid upo, which miimizs th amout of accptac samplig rquird, thus rducig rdudat costs i th valu-addig chai from supplir to producr. Th samplig plas provid th vdor ad buyr with dcisio ruls for product accptac to mt th prst product quality rquirmt. Accptac samplig prtais to icomig batchs of raw matrials (or purchasd Publishd by Atlatis Prss 13

E. Turaoğlu t al. parts) ad to outgoig batchs of fiishd goods. It is most usful wh o or mor of th followig coditios is prst: a larg umbr of itms must b procssd i a short tim; th costs of passig dfctiv itms is low; dstructiv tstig is rquird; or th ispctors may xpric bordom or fatigu i ispctig larg umbrs of itms. Th schm by which rprstativ sampls will b slctd from a populatio ad tstd to dtrmi whthr th lot is accptabl or ot is ow as a accptac pla or samplig pla. Thr ar two major classificatios of accptac plas: basd o attributs ad basd o variabls. Samplig plas ca b sigl, doubl, multipl, ad squtial (Kahrama ad Kaya, 2010). I rct yars, thr ar som studis coctratd o accptac samplig i th litratur. Kuo (2006) dvlopd a optimal adaptiv cotrol policy for joit machi maitac ad product quality cotrol. H icludd th itractios btw th machi maitac ad th product samplig i th sarch for th bst machi maitac ad quality cotrol stratgy for a Marovia dtrioratig, stat uobsrvabl batch productio systm. H drivd svral proprtis of th optimal valu fuctio, which hlpd to fid th optimal valu fuctio ad idtify th optimal policy mor fficitly i th valu itratio algorithm of th dyamic programmig. Par ad Wub (2007) itroducd a ffctiv samplig pla basd o procss capability idx, C, to dal with product accptac dtrmiatio for low fractio o-coformig products. Th proposd w samplig pla was dvlopd basd o th xact samplig distributio rathr tha approximatio. Practitiors could us this proposd mthod to dtrmi th umbr of rquird ispctio uits ad th critical accptac valu, ad ma rliabl dcisios i product accptac. Tsai t al. (2009) dvlopd ordiary ad approximat accptac samplig procdurs udr progrssiv csorig with itrmittt ispctios for xpotial liftims. Th proposd approach allowd rmovig survivig itms durig th lif tst such that som xtrm liftims could b sought, or th tst facilitis could b frd up for othr tsts. Jozai ad Miramali (2010) dmostratd th us of maxima omiatio samplig (MNS) tchiqu i dsig ad valuatio of sigl AQL, LTPD, ad EQL accptac samplig plas for attributs. Thy xploitd th ffct of sampl siz ad accptac umbr o th prformac of thir proposd MNS plas usig opratig charactristic (OC) curv. Aslam t al. (2010) dvlopd th doubl samplig pla ad dtrmid th dsig paramtrs satisfyig both th producr s ad cosumr s riss simultaously for th spcifid rliability lvls i trms of th ma ratio to th spcifid lif. Thy proposd th doubl samplig ad group samplig plas dsigd usig th two-poit approach udr p th assumptio that th liftim of a product follows th Birbaum Saudrs (BS) distributio with ow shap paramtrs. Aslam ad Ju (2010) dvlopd a doubl accptac samplig pla for th trucatd lif tst assumig that th liftim of a product follows a gralizd loglogistic distributio with ow shap paramtrs. Mrg ad Dligöül (2010) proposd a w idicator, ma squard ocoformac (MSNC), to gaug th prformac of sigl accptac samplig plas for attributs by usig th distributio of fractio ocoformac. This proposd idicator improvd th masur by icorporatig mor iformatio by th us of a custom-tailord prior distributio which i tur improvs prcisio. Tsai ad Li (2010) ivstigatd th dsig of lif tst plas udr progrssivly itrval csord tst. Basd o th lilihood ratio, th proposd lif tst plas ar stablishd so that th rquird producr ad cosumr riss ca b satisfid simultaously. Th fuzzy st thory which was itroducd by Zadh (1965) provids a strict mathmatical framwor i which vagu cocptual phoma ca b prcisly ad rigorously studid. It is a importat mthod to provid masurig th ambiguity of cocpts that ar associatd with huma bigs subjctiv judgmts icludig liguistic trms, satisfactio dgr ad importac dgr that ar oft vagu. A liguistic variabl is a variabl whos valus ar ot umbrs but phrass i a atural laguag. Th cocpt of a liguistic variabl is vry usful i dalig with situatios, which ar too complx or ot wll dfid to b rasoably dscribd i covtioal quatitativ xprssios (Zimmrma, 1991). I rct yars, som of th accptacs samplig studis hav coctratd o fuzzy paramtrs. Sadghpour-Gildh t al. (2008) aalyzd th accptac doubl samplig pla wh th fractio of dfctiv itms is a Fuzzy umbr. Jamhah t al. (2009) itroducd avrag outgoig quality (AOQ) ad avrag total ispctio (ATI) for sigl samplig ad doubl samplig plas wh proportio ocoformig was a triagular fuzzy umbr (TFN). Thy showd that AOQ ad ATI curvs of th pla wr li a bad havig a high ad low boud. Ajorlou ad Ajorlou (2009) proposd a mthod for costructig th mmbrship fuctio of th grad of satisfactio for th sampl siz basd o th shap of th samplig cost. Th proposd mthod fids a rasoabl solutio to th trad-off btw rlaxig th coditios o th actual riss ad th sampl siz. I this study accptac samplig plas ar aalyzd wh thir mai paramtrs ar fuzzy ad thir mai curvs ar obtaid udr fuzzy viromt. Jamhah t al. (2010) prstd th accptac sigl samplig pla wh th fractio of ocoformig itms is a fuzzy umbr ad big modld basd Publishd by Atlatis Prss 14

o th fuzzy Poisso distributio. Jamhah t al. (2011a; 2011b) dsigd a accptac sigl samplig pla with ispctio rrors wh th fractio of dfctiv itms ad th proportio of ocoformig products ar a fuzzy umbr. Thy show that th opratig charactristics curv of this pla was li a bad havig high ad low bouds, its width dpds o th ambiguity of proportio paramtr i th lot wh th sampls siz ad accptac umbrs wr fixd. Th rst of this study is orgaizd as follows: Th crtai importat trms rlvat to accptac samplig plas ar discussd i Sctio 2. Som dfiitios about charactristic curvs ar providd i Sctio 3. Discrt fuzzy probability distributios ad accptac probability fuctios of Biomial ad Poisso distributios with fuzzy paramtrs ar drivd i Sctio 4. Opratig charactristic curv (OC), avrag outgoig quality (AOQ), avrag sampl umbr (ASN), ad avrag total ispctio (ATI) ar drivd for sigl ad doubl samplig plas udr fuzzy viromt i Sctio 5. Sctio 6 icluds coclusios ad futur rsarch dirctios. 2. Accptac Samplig Plas A accptac samplig pla tlls you how may uits to sampl from a lot or shipmt ad how may dfcts you ca allow i that sampl. If you discovr mor tha th allowd umbr of dfcts i th sampl, you simply rjct th tir lot. Th pricipl of accptac samplig to cotrol quality is th fact that w do ot chc all uits (N), but oly slctd part (). Accptac samplig pla is a spcific pla that clarly stats th ruls for samplig ad th associatd critria for accptac or othrwis. Accptac samplig plas ca b applid for ispctio of (i) d itms, (ii) compots, (iii) raw matrials, (iv) opratios, (v) matrials i procss, (v) supplis i storag, (vi) maitac opratios, (vii) data or rcords ad (viii) admiistrativ procdurs. Thr ar a umbr of diffrt ways to classify accptac-samplig plas. O major classificatio is by attributs ad variabls. Accptac-samplig plas by attributs: (i) Sigl samplig pla, (ii) doubl samplig pla, (iii) multipl-samplig pla, ad (iv) squtial samplig pla. Th sigl-samplig pla is a basic to all accptac samplig. Th simpl accptac samplig procds as follows: From th whol lot cosistd from N uits w choos a slctio of uits. I th scod stp w must chc ths uits, if thy satisfy quality rquirmts. As a rsult, w gt a umbr of spoild uits d. If this d is gratr tha th accptac umbr c, th th lot will b rjctd, othrwis th lot will b accptd. Oft a lot of itms is so good or so bad that w ca rach a coclusio about its quality by taig a smallr sampl tha would hav b usd i a sigl samplig pla. If th umbr of dfcts i this first Fuzzy Accptac Samplig ad Charactristic Curvs sampl (d 1 ) is lss tha or qual to som lowr limit (c 1 ), th lot ca b accptd. If th umbr of dfcts first ad scod sampl (d 2 ) xcds a uppr limit (c 2 ), th whol lot ca b rjctd. But if th umbr of dfcts i th 1 sampl is btw c 1 ad c 2, a scod sampl is draw. Th cumulativ rsults dtrmi whthr to accpt or rjct th lot. Th cocpt is calld doubl samplig. Multipl samplig is a xtsio of doubl samplig, with smallr sampls usd squtially util a clar dcisio ca b mad. I multipl samplig by attributs, mor tha two sampls ca b ta i ordr to rach a dcisio to accpt or rjct th lot. Th chif advatag of multipl samplig plas is a rductio i sampl siz for th sam protctio. Sigl, doubl, ad multipl plas assss o or mor succssiv sampls to dtrmi lot accptability. Th most discrimiatig accptac samplig procdur ivolvs maig a dcisio as to dispositio of th lot or rsampl succssivly as ach itm of th sampl is ta. Calld squtial samplig, ths mthods may b rgardd as multipl-samplig plas with sampl siz o ad o uppr limit o th umbr of sampls to b ta. Wh uits ar radomly slctd from a lot ad tstd o by o, with th cumulativ umbr of ispctd pics ad dfcts rcordd, th procss is calld squtial samplig. Udr squtial samplig, sampls ar ta, o at a tim, util a dcisio is mad o th lot or procss sampld. Aftr ach itm is ta a dcisio is mad to (1) accpt, (2) rjct, or (3) cotiu samplig. Sampls ar ta util a accptac or rjctio dcisio is mad. Thus, th procdur is op dd, th sampl siz ot big dtrmid util th lot is accptd or rjctd (Kahrama ad Kaya, 2010). 3. Charactristic Curvs A importat masur of th prformac of a accptac-samplig pla is th opratigcharactristic (OC) curv. Th opratig charactristic (OC) curv dscribs how wll a accptac pla discrimiats btw good ad bad lots. A curv prtais to a spcific pla, that is, a combiatio of (sampl siz) ad c (accptac umbr). Th curv shows th ability of a samplig pla to discrimiat btw high quality ad low quality lots. With accptac samplig, two partis ar usually ivolvd: th producr of th product ad th cosumr of th product. Wh spcifyig a samplig pla, ach party wats to avoid costly mistas i accptig or rjctig a lot. Th producr wats to avoid th mista of havig a good lot rjctd (producr s ris) bcaus h or sh usually must rplac th rjctd lot. Covrsly, th customr or cosumr wats to avoid th mista of accptig a bad lot bcaus dfcts foud i a lot that has alrady b accptd ar usually th rsposibility of th customr Publishd by Atlatis Prss 15

E. Turaoğlu t al. (cosumr s ris). Th producr's ris is th probability of ot accptig a lot of accptabl quality lvl (AQL) quality ad th cosumr's ris is th probability of accptig a lot of limitig quality lvl (LQL) quality. Aothr accptac samplig curv is Avrag Outgoig Quality (AOQ) curv. Th avrag outgoig quality (AOQ) ca b dfid as th xpctd quality of outgoig product followig th us of a accptac samplig pla for a giv valu of th icomig quality. Th avrag sampl umbr (ASN) curv is dfid as th curv of th avrag umbr of sampl uits pr lot usd for dcidig accptac or oaccptac. For a sigl samplig pla, o tas oly a sigl sampl of siz ad hc th ASN is simply th sampl siz. I sigl samplig, th siz of th sampl ispctd from th lot is always costat, whras i doubl-samplig, th siz of th sampl slctd dpds o whthr or ot th scod sampl is cssary. Aothr importat masur rlativ to rctifyig ispctio is th total amout of ispctio rquird by th samplig program. Th avrag total ispctio (ATI) curv ca b dfid as th curv of th avrag umbr of uits ispctd pr lot basd o th sampl for accptd lots ad all ispctd uits i lots ot accptd. If th lots P p q p p, q q, 0 1 cotai o dfctiv itms, o lots will b rjctd, ad th amout of ispctio pr lot will b th sampl siz. 4. Discrt Fuzzy Distributios Th two importat distributios usd i samplig plas to calculat th accptac probability ar Biomial ad Poisso distributios. I this sctio th paramtrs of ths two distributios ar aalyzd udr fuzzy viromt. Thir procdur for calculatig th accptac probability is drivd wh th mai paramtrs of thm ar fuzzy. 4.1. Fuzzy Biomial Distributio A major assumptio of samplig pla is that fractio of dfctiv itms p is crisp. Howvr, somtims w ar ot abl to obtai xact umrical valu for p. May tims this valu is stimatd or it is providd by xprimt. Assum that i ths trials P S is ot ow prcisly ad ds to b stimatd or is obtaid from xprt opiios. This p valu is ucrtai ad is dotd as p. Thrfor P rprsts th fuzzy probability of succsss i idpdt trials ad ca b calculatd as follows (Kahrama ad Kaya, 2010): (1) P Pl, P r Pl mi p q p p, q q, Pr max p q p p, q q (2) If p valu is dfid as triagular fuzzy umbrs (TFNs) li p p1, p2, p 3, its cuts ca b drivd as follows: p p1 p2 p1, p3 p2 p 3 (3) If p valu is dfid as trapzoidal fuzzy umbrs (TrFNs) li p p1, p2, p3, p 4, its cuts ca b drivd as follows: p p1 p2 p1, p4 p3 p 4 (4) Fuzzy Numbr of Trials Th umbr of trials ca b dfid by liguistic variabls. TFNs or TrFNs ca b usd to dfi ths liguistic variabls. Assum that, umbr of trials is dfid by TFN 1, 2, 3 or TrFN 1, 2, 3, 4. Thir alpha cuts ca b drivd from th followigs quatios, rspctivly: Publishd by Atlatis Prss 16

1 2 1, 3 2 3 (5) 1 2 1, 4 3 4 (6) Fuzzy Accptac Samplig ad Charactristic Curvs Th th fuzzy probability of succsss P ca b calculatd as follows (Kahrama ad Kaya, 2010): P p q (7) P p q p p, q q,, 0 1 (8) or P Pl, P r (9) Pl mi p q p p, q q,, Pr max p q p p, q q, (10) Fuzzy Numbr of Succss Aothr situatio which should b ta ito accout is to dfi th umbr of succss by liguistic variabls. Fuzzy umbrs ca b usd to rprst this dfiitio succssfully. Assum that th umbr of succss is dfid as TFN 1, 2, 3 or TrFN 1, 2, 3, 4, th th P ca b calculatd as follows (Kahrama ad Kaya, 2010): P p q (11) P p q p p, q q,,, 0 1 (12) or P Pl, P r (13) Pl mi p q p p, q q,,, Pr max p q p p, q q,, (14) Th ad valus should b itgr umbrs i th classical biomial distributio. Thrfor th umbrs with dcimal poits should b limiatd from cuts. 4.2. Fuzzy Poisso Distributio Assum that th p valu is ucrtai ad is dotd as p. I this cas, λ is also dotd as.thrfor P rprsts th fuzzy probability of Publishd by Atlatis Prss 17

E. Turaoğlu t al. vts i vts ad ca b calculatd as follows (Kahrama ad Kaya, 2010): f, 0,1,2,..., ad 0,! (15) f l,, f r,, mi max!! (18) f, whr p! (16) P fl ;, fr ; (17) Fuzzy umbr of vts Th othr two paramtrs of Poisso distributio ad ca b also valuatd as fuzzy umbrs. Thir -cuts ca b asily valuatd basd o ithr TFNs or TrFNs. Th th fuzzy probability of vts ca b drivd as follows (Kahrama ad Kaya, 2010): f,!, 0,1,2,..., ad 0 f,!,, whr p (19) (20) P fl ;, fr ; (21) f mi l,,!,, f max r,,!,, (22) 5. Fuzzy Accptac Samplig Plas Somtims th paramtrs of samplig plas caot b xprssd as crisp valus. Thy ca b statd as approximatly, aroud, or btw. Fuzzy st thory is a vry usabl tool to covrt ths xprssios i to mathmatical fuctios. I this cas, accptac probability of samplig plas should b calculatd with rspct to fuzzy ruls. I th prvious sctio, biomial ad Poisso distributio hav b aalyzd wh thir paramtrs ar fuzzy. I this sctio sigl ad doubl samplig plas ar aalyzd by taig ito accout ths two fuzzy discrt distributios. 5.1. Fuzzy Sigl Samplig Assum that a sampl whos siz is a fuzzy umbr is ta ad 100% ispctd. Th fractio ocoformig of th sampl is also a fuzzy umbr p. Th accptac umbr is dtrmid as a fuzzy umbr c. Th accptac probability for this sigl samplig pla ca b calculatd as follows (Kahrama ad Kaya, 2010): P P d c, c, p a whr p. d c 0 d d! (23) P P, P (24) a al, d, ar, d, Publishd by Atlatis Prss 18

c d P mi,, c c al, d, d! d 0 c d P max,, c c ar, d, d! d 0 Fuzzy Accptac Samplig ad Charactristic Curvs (25) If th biomial distributio is usd, accptac probability ca b calculatd as follows (Kahrama ad Kaya, 2010): P a d c 0 d d d p q (26) c c d d d d Pa p q p q p p, q q,, c c d d d 0 d 0 (27) Pa Pal, P ar (28) c d d Pal mi p q p p, q q,, c c, d d 0 c d d Par max p q p p, q q,, c c d d 0 (29) AOQ valus for fuzzy sigl samplig ca b calculatd as follows (Kahrama ad Kaya, 2010): AOQ Pa p (30) AOQ AOQ, AOQ (31) l r AOQ mi P p p p, P P, l a a a AOQ max P p p p, P P r a a a (32) ATI curv ca also b calculatd as follows (Kahrama ad Kaya, 2010): ATI 1 Pa N (33) ATI ATI, ATI (34) l r ATI mi 1 P N p p, P P, p N, N N, l a a a ATI max 1 P N p p, P P, p N, N N r a a a (35) I Figur 1, vry poit o th fuzzy OC curv is rprstd by triagular fuzzy umbrs. Ths poits o th curv ar calculatd usig th fuzzy paramtrs,,, ad. Publishd by Atlatis Prss 19

E. Turaoğlu t al. Fig. 1 Fuzzy OC curv with th paramtrs,,, ad I Figur 2, vry poit o th fuzzy AOQ curv is rprstd by triagular fuzzy umbrs. Ths poits o th curv ar calculatd usig th fuzzy paramtrs,,, ad. Fig. 2 Fuzzy AOQ curv with th paramtrs,,, ad I Figur 3, vry poit of th fuzzy ATI curv is rprstd by triagular fuzzy umbrs. Ths poits o th curv ar calculatd usig th fuzzy paramtrs,,, ad. Publishd by Atlatis Prss 20

Fuzzy Accptac Samplig ad Charactristic Curvs Fig. 3 Fuzzy ATI curv with th paramtrs,,, ad I Figur 4, vry poit of th fuzzy ASN curv is rprstd by triagular fuzzy umbrs. Ths poits o th curv ar calculatd usig th fuzzy paramtrs,,, ad. Fig. 4 Fuzzy ASN curv with th paramtrs,,, ad A Illustrativ Exampl 1: Suppos that a product is shippd i lots of siz Approximatly 500. Sic th viromt is fuzzy, th four xprts of th firm hav th diffrt suggstios as i Tabl 1. Th avrag of ths suggstios is a sampl siz of Approximatly 48 ad a accptac umbr of approximatly 1. Lt us assum that th fractio of ocoformig for th icomig lots is approximatly 0.046. Basd o Eq. (25), th accptac probability of th samplig pla is calculatd as a = 0.0952, 0.3526, 0.6583) ad its mmbrship fuctio is show i Figur 5. Tabl 1. Th suggstios of four diffrt xprts of th firm about th paramtrs of sigl samplig plas Th paramtrs of samplig plas Exprts p c N E-1 Approximatly 5% Approximatly 50 Approximatly 1 Approximatly 500 E-2 Approximatly 4.5% Approximatly 45 Approximatly 2 Approximatly 500 E-3 Approximatly 4% Approximatly 50 Approximatly 1 Approximatly 500 E-4 Approximatly 4.8% Approximatly 45 Approximatly 1 Approximatly 500 Avrag Approximatly 4.6% Approximatly 48 Approximatly 1 Approximatly 500 Publishd by Atlatis Prss 21

E. Turaoğlu t al. Fig. 5 Mmbrship fuctio of accptac probability for sigl samplig AOQ is calculatd as by usig Eq. (32). ATI is also calculatd as by usig Eq. (35) ad its mmbrship fuctio is illustratd i Figur 6. Fig. 6 Mmbrship fuctio of ATI for sigl samplig To obtai th largst possibl accptac probability, th followig combiatio of th paramtrs,,, ad giv i Tabl 1 is usd: =TFN( 0.039, 0.04, 0.051), =TFN( 44, 45, 46), =TFN( 1, 2, 3) ad =TFN(490, 500, 510). Th obtaid rsult is =TFN (0.4377, 0.7306, 0.9044). To obtai th last possibl accptac probability, th followig combiatio of th paramtrs,,, ad giv i Tabl 1 is usd: =TFN(0.048, 0.05, 0.052), =TFN( 49, 50, 51), =TFN( 0, 1, 2) ad =TFN(490, 500, 510). Th obtaid rsult is =TFN(0.0705, 0.2873, 0.5823). Th othr possibl curvs li btw th bold OC curvs i Figur 7. Publishd by Atlatis Prss 22

Fig. 7 OC Curvs wh th paramtrs N, p,, ad c ar fuzzy for sigl samplig pla Th AOQ at th curv s maximum is th avrag outgoig quality limit (AOQL). I Figur 8 AOQL max idicats th largst possibl worst quality lvl ad AOQL mi idicats th last possibl worst quality lvl. As it ca b s from Figur 7, AOQL max is about 0.042 ad AOQL mi is about 0.003. Th othr possibl curvs li btw th bold AOQ curvs i Figur 8. Th largst possibl worst outgoig quality is obtaid by th followig combiatio of th paramtrs,,, ad giv i Tabl 1: =TFN( 0.043, 0.045, 0.047), =TFN( 44, 45, 46), =TFN( 1, 2, 3) ad =TFN(490, 500, 510). Th obtaid rsult is =TFN (0.016, 0.03, 0.041). Th last possibl worst outgoig quality is obtaid by th followig combiatio of th paramtrs,,, ad giv i Tabl 1 is usd: =TFN(0.048, 0.05, 0.052), =TFN( 49, 50, 51), =TFN( 0, 1, 2) ad =TFN(490, 500, 510). Th obtaid rsult is =TFN (0.003, 0.014, 0.03). Fig. 8 AOQ max. ad mi. poits wh th paramtrs N, p,, ad c ar fuzzy for sigl samplig pla To obtai th largst possibl avrag total ispctio th followig combiatio of th paramtrs,,, ad giv i Tabl 1is usd: =TFN( 0.049, 0.05, 0.051), =TFN( 49, 50, 51), =TFN( 0, 1, 2) ad =TFN(490, 500, 510). Th obtaid rsult is =TFN (232.371, 344.141, 479.495). To obtai th last possibl avrag total ispctio th followig combiatio of th paramtrs,,, ad giv i Tabl 1 is usd: =TFN(0.038, 0.04, 0.042), =TFN( 44, 45, 46), =TFN( 1, 2, 3) ad =TFN(490, 500, 510). Th obtaid rsult is =TFN (83.536, 167.576, 314.079). As it is s from Figur 9, th diffrc btw ad givs us th largst possibl rag of avrag total ispctio umbrs for sigl samplig. I our cas this rag is from 83.536 to 479.495. Th othr possibl curvs li btw th bold ATI curvs i Figur 9. Publishd by Atlatis Prss 23

E. Turaoğlu t al. Fig. 9 ATI Curvs wh th paramtrs N, p,, ad c ar fuzzy for sigl samplig pla 5.2. Fuzzy Doubl Samplig Assum that w will us a doubl samplig pla with fuzzy paramtrs 1, c1, 2, c 2. N ad p ar also fuzzy. If th Poisso distributio is usd, th accptac probability of doubl samplig ca b calculatd as follows (Kahrama ad Kaya, 2010): P P d c P c d c P d d c (36) a 1 1 1 1 2 1 2 2 P a d! d! d! c1 d1 1 p c2 d1 1 p c2 d1 d2 2 p d1 0 1 d1 c1 1 d2 0 2 (37) P P, P (38) a al, d; ar, d; P al, d ; P ar, d ; mi max d! d! d! c1 d1 1 p c2 d1 1 p c2 d1 d2 2 p d1 0 1 d1 c1 1 d2 0 2 d! d! d! c1 d1 1 p c2 d1 1 p c2 d1 d2 2 p d1 0 1 d1 c1 1 d2 0 2 (39) whr p p,, ad c c. If th biomial distributio is usd, accptac probability ca b calculatd as follows (Kahrama ad Kaya, 2010): Pa p 1 p p 1 p p 1 p d d d c1 c2 c2 d1 1 d 1 1 d1 1 d 1 1 d1 2 d 2 2 d2 d1 0 1 d1 c1 1 d2 0 2 (40) c1 c2 c2 d 1 1 1 1 d 1 1 1 1 d d d 1 2 d 2 2 d2 Pal mi p 1 p p 1 p p 1 p, d d d d1 0 1 d1 c1 1 d2 0 2 Par max p 1 p p 1 p p 1 p d d d c1 c2 c2 d1 1 d 1 1 d1 1 d 1 1 d1 2 d 2 2 d2 d1 0 1 d1 c1 1 d2 0 2 (41) whr p p, q q, 1 1, c1 c1, 2 2, ad c2 c 2. Publishd by Atlatis Prss 24

AOQ valus for fuzzy doubl samplig ca b calculatd as i Sctio 5.1. ASN curv for doubl samplig ca b calculatd as follows (Kahrama ad Kaya, 2010): ASN P 1 P 1 I 1 2 1 P 1 2 I I (42) ASN ASN, ASN (43) l r ASN mi 1 P p p,,, P P, l 1 2 I 1 1 2 2 I I ASN max 1 P p p,,, P P r 1 2 I 1 1 2 2 I I (44) ATI curv for fuzzy doubl samplig ca also b calculatd as follows (Kahrama ad Kaya, 2010): ATI ASN N 1 P d1 c2 N 1 2 P d1 d2 c 2 (45) ATI ATI, ATI (46) l r ATI mi ASN N P d c N P d d c, l 1 1 2 1 2 1 2 2 ATI max ASN N P d c N P d d c r 1 1 2 1 2 1 2 2 (47) Whr p p, ASN ASN, 1 1, N N, 2 2, ad c2 c 2. I Figur 10, vry poit of th fuzzy ASN curv is rprstd by triagular fuzzy umbrs. Ths poits o th curv ar calculatd usig th fuzzy paramtrs,,, ad. Fig. 10 Fuzzy ASN for doubl samplig A Illustrativ Exampl 2: Suppos that a product is shippd i lots of siz Approximatly 500. Sic th viromt is fuzzy, th four xprts of th firm hav th diffrt suggstios as i Tabl 2. Th avrag of ths suggstios is with sampl sizs dtrmid as Approximatly 50 for th first ad scod sampls. Also th avrags of accptac umbrs ar dtrmid as Approximatly 1 ad Approximatly 3 for th first ad scod sampls, rspctivly. Publishd by Atlatis Prss 25

E. Turaoğlu t al. Tabl 2. Th suggstios of four diffrt xprts of th firm about th paramtrs of doubl samplig plas Exprts Th paramtrs of samplig plas p 1-2 c 1 -c 2 N E-1 Approximatly 5% Approximatly 50 Approximatly 1-3 Approximatly 500 E-2 Approximatly 4.5% Approximatly 45 Approximatly 2-3 Approximatly 500 E-3 Approximatly 4% Approximatly 50 Approximatly 1-2 Approximatly 500 E-4 Approximatly 4.8% Approximatly 45 Approximatly 1-3 Approximatly 500 Avrag Approximatly 4.6% Approximatly 48 Approximatly1-3 Approximatly 500 Basd o Eq. (39), accptac probability of th doubl samplig pla is calculatd as follows: = Its mmbrship fuctio is show i Figur 11. [( Fig. 11 Mmbrship fuctio of accptac probability for doubl samplig ASN is calculatd as by usig Eqs. (42-44). Also AOQ is calculatd as 0.0054, 0.0212, 0.0449) ad ATI is calculatd as 87.2622, 322.4731, 685.7813) by usig Eqs. (45-47). To obtai th largst possibl accptac probability th followig combiatio of th paramtrs,,, ad giv i Tabl 2 is usd: =TFN( 0.038, 0.04, 0.042), =TFN( 49, 50, 51), =TFN( 49, 50, 51), =TFN( 0, 1, 2), =TFN( 2, 3, 4) ad =TFN(490, 500, 510). To obtai th last possibl accptac probability th followig combiatio of th,,, ad giv i Tabl 2 is usd: =TFN( 0.048, 0.05, 0.052), =TFN( 49, 50, 51), =TFN( 49, 50, 51), =TFN( 0, 1, 2), =TFN( 2, 3, 4) ad =TFN(490, 500, 510). As it is s from Figur 12, th diffrc btw ad givs us th largst possibl rag of accptac probability for doubl samplig. I our cas, this rag is from 0.0922 to 0.999. Th othr possibl curvs will li btw th bold OC curvs i Figur 12. Publishd by Atlatis Prss 26

Fig. 12 OC Curvs with th paramtrs,,, ad To obtai th largst possibl worst outgoig quality th followig combiatio of th paramtrs,,, ad giv i Tabl 2 is usd: =TFN( 0.046, 0.048, 0.05), =TFN( 44, 45, 46), =TFN( 44, 45, 46), =TFN( 0, 1, 2), =TFN( 2, 3, 4) ad =TFN(490, 500, 510). To obtai th last possibl worst outgoig quality th followig combiatio of,,, ad giv i Tabl 2 is usd: =TFN( 0.048, 0.05, 0.052), =TFN( 49, 50, 51), =TFN( 49, 50, 51), =TFN( 0, 1, 2), =TFN( 2, 3, 4) ad =TFN(490, 500, 510). As it ca b s from Figur 13, AOQL max is about 0.0479 ad AOQL mi is about 0.0042. Th othr possibl curvs li btw th bold AOQ curvs i Figur 13. Fig. 13 AOQ max. ad mi. poits wh th paramtrs N, p,, ad c ar fuzzy for doubl samplig To obtai th largst possibl avrag total ispctio umbr, th followig combiatio of th paramtrs,,, ad i Tabl 2 is usd: =TFN( 0.048, 0.05, 0.052), =TFN( 49, 50, 51), =TFN( 49, 50, 51), =TFN( 0, 1, 2), =TFN( 2, 3, 4) ad =TFN(490, 500, 510). To obtai th last possibl avrag total ispctio umbr, th followig combiatio of th paramtrs,,, ad i Tabl 2 is usd: =TFN( 0.038, 0.04, 0.042), =TFN( 49, 50, 51), =TFN( 49, 50, 51), =TFN( 0, 1, 2), =TFN( 2, 3, 4) ad =TFN(490, 500, 510). As it ca b s from Figur 14, th diffrc btw ad givs us th largst possibl rag of avrag total ispctio umbrs for doubl samplig. I our cas this rag is from 80.508 to 716.788. Th othr possibl curvs li btw th bold ATI curvs i Figur 14. Publishd by Atlatis Prss 27

Fig. 14 ATI Curv with th paramtrs,,, ad To obtai th largst possibl avrag sampl umbr th followig combiatio of th paramtrs,,, ad giv i Tabl 2 is usd: =TFN( 0.038, 0.04, 0.042), =TFN( 49, 50, 51), =TFN( 49, 50, 51), =TFN( 0, 1, 2), =TFN( 2, 3, 4) ad =TFN(490, 500, 510). To obtai th last possibl avrag sampl umbr th followig combiatio of th paramtrs,,, ad giv i Tabl 2 is usd: =TFN( 0.038, 0.04, 0.042), =TFN( 49, 50, 51), =TFN( 49, 50, 51), =TFN( 1, 2, 3), =TFN( 2, 3, 4) ad =TFN(490, 500, 510). As it s from Figur 15, th diffrc btw ASN max ad ASN mi givs us th largst possibl rag of th avrag umbr of sampl uits pr lot usd for maig dcisios (accptac or o accptac) for doubl samplig. I our cas, this rag is from 46.693 to 79.545. Th othr possibl curvs li btw th bold ASN curvs i Figur 15. Fig. 15 ASN max. ad mi. poits for doubl samplig pla 6. Coclusios Accptac samplig is a practical, affordabl altrativ to costly 100 % ispctio. It offrs a fficit way to assss th quality of a tir lot of product ad to dcid whthr to accpt or rjct it. Th applicatio of accptac samplig allows idustris to miimiz product dstructio durig ispctio ad tstig, ad to icras th ispctio quatity ad ffctivss. Dspit of th usfulss of accptac samplig, it has a mai difficulty i dfiig its paramtrs as crisp valus. Somtims it is asir to dfi ths paramtrs by usig liguistic variabls. For ths cass, th fuzzy st thory is th most suitabl tool to aalyz accptac samplig plas. Th fuzzy st thory givs a flxibl dfiitio to sampl siz, accptac umbr, ad fractio of ocoformig. I this papr, w aalyzd th accptac sigl ad doubl samplig plas wh th paramtrs N, p,, ad c ar fuzzy ad accptac probability, Publishd by Atlatis Prss 28

opratig charactristic (OC) curv, avrag sampl umbr (ASN), avrag outgoig quality limit (AOQL), ad avrag total ispctio umbr (ATI) wr also aalyzd with fuzzy paramtrs. Th obtaid fuzzy rsults show th whol possibilitis of ATI, ASN, AOQ, ad OC. For futur rsarch, th ffcts of fuzzy paramtrs ca b aalyzd for multipl samplig plas. Rfrcs 1. Ajorlou, S., Ajorlou, A. (2009). A fuzzy basd dsig procdur for a sigl-stag samplig pla. FUZZ- IEEE, Kora, August 20-24. 2. Aslam, M., Ju, C.H. (2010). A doubl accptac samplig pla for gralizd log-logistic distributios with ow shap paramtrs. Joural of Applid Statistics, 37(3),405 414. 3. Aslam, M., Ju, C.H., Ahmad, M. (2010). Nw accptac samplig plas basd o lif tsts for Birbaum Saudrs distributios. Joural of Statistical Computatio ad Simulatio, DOI:10.1080/00949650903418883 4. British Stadard (2006). Accptac samplig procdurs by attributs BS 6001. 5. Burr, J.T. (2004). Elmtary statistical quality cotrol, CRC Prss. 6. Duca, A.J. (1986). Quality cotrol ad idustrial statistics, Irwi Boo Compay. 7. ISO 2859-1 (1999). Samplig procdurs for ispctio by attributs. 8. Jamhah, E.B., Sadghpour-Gildh, B., Yari, G. (2009). Prparatio importat critria of rctifyig ispctio for sigl samplig pla with fuzzy paramtr. Procdigs of World Acadmy of Scic, Egirig ad Tchology, 38, 956-960. 9. Jamhah, E.B., Sadghpour-Gildh, B., Yari, G. (2010). Accptac sigl samplig pla by usig of poisso distributio. Joural of Mathmatics ad Computr Scic, 1(1), 6-13. 10. Jamhah, E.B., Sadghpour-Gildh, B., Yari, G. (2011a). Ispctio rror ad its ffcts o sigl samplig plas with fuzzy paramtrs. Struct Multidisc Optim, 43, 555 560. 11. Jamhah, E.B., Sadghpour-Gildh, B., Yari, G. (2011b). Accptac sigl samplig pla with fuzzy paramtr. Iraia Joural of Fuzzy Systms, 8(2), 47-55. 12. Joh, P.W.M. (1990). Statistical mthods i girig ad quality assurac, Joh Wily & Sos. 13. Jozai, M.J., Miramali, S.J. (2010). Improvd attribut accptac samplig plas basd o maxima omiatio samplig. Joural of Statistical Plaig ad Ifrc, 140, 2448 2460. 14. Jura, J.M., Godfry, A.B. (1998). Jura s quality hadboo. McGraw-Hill. 15. Kahrama, C., Kaya, İ. (2010). Fuzzy accptac samplig plas. I C. Kahrama & M. Yavuz, (Eds.), Productio girig ad maagmt udr fuzziss (pp. 457-481). Sprigr. 16. Kuo, Y. (2006). Optimal adaptiv cotrol policy for joit machi maitac ad product quality cotrol. Europa Joural of Opratioal Rsarch, 171, 586 597. Fuzzy Accptac Samplig ad Charactristic Curvs 17. Mrg, A.E., Dligöül, Z.S. (2010). Assssmt of accptac samplig plas usig postrior distributio for a dpdt procss. Joural of Applid Statistics, 37( 2), 299 307. 18. MIL STD 105E. (1989). Military Stadard-Samplig Procdurs ad Tabls for Ispctio by Attributs. Dpartmt of Dfs, Washigto, DC 20301. 19. Mitra, A. (1998). Fudamtals of quality cotrol ad improvmt. Prtic Hall. 20. Motgomry, D.C. (2005). Itroductio to statistical quality cotrol. Wily. 21. Par, W.L., Chi-Wi, W. (2007). A ffctiv dcisio maig mthod for product accptac. Omga, 35, 12 21. 22. Sadghpour-Gildh, B., Yari, G., Jamhah, E.B., (2008). Accptac doubl samplig pla with fuzzy paramtr, Procdigs of th 11th Joit Cofrc o Iformatio Scics, 1-9. 23. Schillig, E.G. (1982). Accptac samplig i quality cotrol. CRC Prss. 24. Schillig, E.G., Nubau, D.V. (2008). Accptac samplig quality i cotrol. CRC Prss. 25. Tsai, T.R., Chiag, J.Y. (2009). Accptac samplig procdurs with itrmittt ispctios udr progrssiv csorig. ICIC Exprss Lttrs, 3(2), 189 194. 26. Tsai, T.R., Li, C.W. (2010). Accptac samplig plas udr progrssiv itrval csorig with lilihood ratio. Statistical Paprs, 51( 2), 259-271. 27. Zadh, L.A. (1965). Fuzzy sts. Iformatio ad Cotrol, 8,338 353. 28. Zimmrma, H.J. (1991). Fuzzy st thory ad its applicatios. Kluwr Acadmic Publishrs. Publishd by Atlatis Prss 29