Acceptance Double Sampling Plan with Fuzzy Parameter

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1 ccetnce Double Smlng Pln wth Fuzzy Prmeter Bhrm Seghour-Gleh, Gholmhossen Yr, Ezztllh Blou Jmhneh,* Dertment of Sttstcs, Fculty of Bsc Scence, Unversty of Mznrn Bblosr, Irn Irn Unversty of Scence n Technology, Tehrn, Irn, c. r Dertment of Sttstcs, Scence n Reserch Brnch, Islmc z Unversty, Tehrn, Irn. e_ yhoo.com. bstrct In ths resent er we hve rgue the ccetnce ouble smlng ln when the frcton of efectve tems s fuzzy number. we hve shown tht the oertng chrcterstc (oc curves of the ln s le bn hvng hgh n low bouns whose wth eens on the mbguty roorton rmeter n the lot when tht smle sze n ccetnce numbers s fxe. Fnlly we comlete scuss onon by numercl exmle. Keywors: Sttstcl qulty control, ccetnce ouble smlng, fuzzy number.. Introucton Smlng for ccetnce or rejectng lot s n mortnt fel n sttstcl qulty control. ccetnce ouble smlng ln s one of the smlng methos for ccetnce or rejecton whch s long wth clsscl ttrbute qulty chrcterstc. In fferent ccetnce smlng lns the frcton of efectve tems, s consere s recse vlue, but sometmes we re not ble to obtn exct numercl vlue, n there lso * Corresonng uthor exst some uncertnty n the vlue of obtne from exerment, ersonl jugment or estmton. However the qulty chrcterstc n lot s not often exct n certn. The theory of fuzzy sets n wely use n solvng roblems n whch rmeter or qunttes cn t be exresse recsely. The theory s owerful n well-nown tool to formulte n nlyze the uncertnty resultng from mbguty n ersonl jugment. In elng wth the bove roblem we tre to restore the uncertnty exstng n the roblem by efnng the mrecse rmeter s fuzzy number, n cheve result wth hgher certnty. Wth ths efnton, the number of efectve tems n the smle hs fuzzy bnoml robblty strbuton. Clsscl ccetnce smlng lns hve been stue by mny reserchers. They re thoroughly elborte by Schllng (8. Sngle smlng by ttrbutes wth relxe requrements were scusse by Oht n Ichhsh (88 ngw n Oht (0, Tm Kngw n Oht (,n Grzegorzews(8,00b. Grzegrozews (000b,00 lso consere smlng ln by vrbles Proceengs of the th Jont Conference on Informton Scences (008 Publshe by tlnts Press the uthors

2 wth fuzzy requrements. Smlng ln by ttrbutes for vgue t were consere by Hrnewcz (, 4. We rove some efnton n relmnres of fuzzy sets theory n fuzzy robblty n secton. In secton the fuzzy robblty of ccetnce of the lot, ws consere broly, n ts vlues n secl cse ws comute. In secton 4, we el wth oc bn of such ln, wth exmle. Fnlly fuzzy verge smle number s ntrouce n secton 5.. Prelmnres Let X x,..., xn} be fnte set P be robblty functon efne on ll subsets of X wth P ({ x }, n,0 ll n n. X together wth P s screte (fnte robblty functon. If be subset of X, we hve P ( P({ x}. x In rctce ll the vlue must be nown exctly. Mny tmes these vlues re estmte, or they re rove by exerts. We now ssume tht some of these vlues re uncertn n we wll moel ths uncertnty usng fuzzy number. Defnton: the fuzzy subset N of rel lne IR, wth the membersh functon : IR [0, s fuzzy number f N n only f ( N s norml (b N s fuzzy convex (c s uer sem contnuous ( su ( N s boune. Defnton: trngulr fuzzy number N s fuzzy number tht membersh functon efne by three number N where the bse of the trngle s the ntervl [, n vertex s t x=. Defnton: Due to the uncertnty n the s vlues we substtute, fuzzy number, for ech n ssume tht 0< < ll. Then X together wth the vlue s screte fuzzy robblty strbuton. We wrte P for fuzzy P n we hve P({ x }. Let x,..., xl} be subset of X. Then efne: l P ( [ { ( For 0, where stns for the n sttement [, n, ths s our restrcte fuzzy rthmetc. The fuzzy men s efne by ts cuts: [ { x ( where, S s s before. Defnton4: In m neenent Bernoll exerment let us ssume tht, robblty of success n ech exerment s not nown recsely n nees to be estmte, or obtne from exert onon. So tht vlue s uncertn n we substtute for n q for q so tht there s [ n q q[ wth q. Now let P ( r be the fuzzy robblty of r successes n m neenent trls of the exerment. Uner our restrcte fuzzy lgebr we obtn r r mr P( r[ { C q ( m Proceengs of the th Jont Conference on Informton Scences (008 Publshe by tlnts Press the uthors

3 For 0, where now S s the sttement, [, q q [, q. If P( r[ [ P r (, P ( then P ( mn{ C r r r r mr m q r r mr Cm q s} n Pr ( mx{ s} n f P[, b be the fuzzy robblty of x successes so tht fuzzy x b, then b x x mx P([, b[ { C q (4 x f P([, b[ [ P ([, b[, P ([, b[ then: b x x mx P ([, b[ mn Cm q s x n b x x mx P ([, b[ mx Cm q s } x Where S s the sme wth st cse. Exmle. Suose tht m= n n q re uncertn we use (0.,0.4,0.5 for n q (0.5,0.6,0.7 for q. Now we wll clculte the fuzzy robbltes P( n P ( tht ={0,}. If [ then q q [. Equtons of efnton 4 become: ( mn{ ( n P r P r m ( mx{ ( (( 0 obtn: on [0 snce we P([ [ ( ( ( ( ( ( where [ [ (, ( [0. 0., uner 0 we gn P ([0 [0.75,0.44 n P( [ [ P (, P ( tht P ( mn{( n P ( mx{( snce (( ( we get, ( ( 0 on [0 P( [ [( ( ( ( P (, ( ( ( ( ( uner 0 we wll hve P ( [ [0.5, ccetnce ouble smlng ln wth fuzzy rmeter In ths secton, frst we ntrouce the ouble smlng ln for clsscl ttrbutes chrcterstcs. Suose tht we wnt to nsect lot wth sze of N. Frst we choose n nsect rnom smle of sze n, n then the number of nonconformng tems ( wll be count own. If the number of observe nonconformng tems s less or equl to ccetnce number of frst smle, tht s c, then the lot wll be ccete. n f s greter thn to ccetnce number of two smles, tht s c, then the lot wll be rejecte, otherwse we choose secon rnom smle of sze n, then the number of nonconformng tems ( wll be count own. In ths stge from Proceengs of the th Jont Conference on Informton Scences (008 Publshe by tlnts Press the uthors

4 the efectve tems n ech two smles, ( +, we use for resultng bout lot. If + s less or equls to ccetnce number of two smles, then the lot wll be ccete, otherwse the lot rejecte. If the sze of lot ws very lrge, the rnom vrbles n hve bnoml strbuton wth rmeters (n, n (n,, n whch nctes the frcton of the lot s nonconformng tems. So f we reresent robblty of ccetnce on combne smles wth, n lso the robblty of the lot s ccetnce n frst n secon smles resectvely I, then I (5 I Tht nctes the robblty of observton of c efectve tems n frst rnom smle. thus c I 0 n n C ( (6 n ccorng to the neenence of two rnom vrbles n ther strbuton wll clculte wth ths formul: P c, c (7 ( c Suose tht we wnt to nsect lot wth the sze of N, whch the roorton of mge tems or the robblty of efectveness of ech of ts number s not nown recsely n s uncertn vlue. So we reresent ths rmeter wth fuzzy number whch s: (,,, [, qq [, q ouble smlng ln wth fuzzy rmeter s efne by the frst smle sze n, ccetnce number of frst smle c, secon smle sze n, n ccetnce number of two smles c. If the number of observe nonconformng tems on frst smle ( s less or equl to ccetnce number of frst smle, then the lot wll be ccete, n f s greter thn to ccetnce number of two smle, then the lot wll be rejecte, otherwse we choose secon rnom smle of sze n, then the number of nonconformng tems ( wll be count own. In ths cse from the totl efectve tems n ech two smle ( +, we for resultng bout lot. If + s less or equl to ccetnce number of two smles, then the lot be ccete, otherwse the lot rejecte, If the sze of lot ws very gret, the rnom vrbles n hve fuzzy bnoml robblty strbuton wth rmeters ( n,,(, n, n whch nctes the fuzzy roorton of the efectve tems. ccorng to ths cse f we show the fuzzy robblty of the lot s ccetnce n combne smles wth P n lso the fuzzy robblty of the lot s ccetnce n frst n secon I smles resectvely, then (8 I Exmle. Suose tht (0.0,0.0,0.0 n q (0.7,0.8,0. n n 0, n 0, c 0, c, then fuzzy robblty of ccetnce of ths lot s s follows: [ (0[ {( 0 s} I 4 Proceengs of the th Jont Conference on Informton Scences (008 Publshe by tlnts Press the uthors

5 0 (( Snce 0 on [ [ ,0.00.0,0, weobtn I 0 0 [ [(0.70.0,(0.0.0 we gn I [0 [0.774,0.044 [ P(, [ uner 0 {0 ( 4 4 snce (0( 0 on [ therefor 4 [ [0( (0.0.0, 0( ( uner 0 we obtn [0 [0.086,0.58 lh Fnlly wehve [ I [ [( ( ( ,( ( ( then [0 [0.,0. frst smle fuzzy robblty of ccetnce combne smle Fg: fuzzy robblty of ccetnce wth (0.0,0.0,0.0, 0 4 [ 4. oc -bn wth fuzzy rmeter In ssessng smlng ln, one of the mortnt crter s ts oertng chrcterstc (oc curve. Ths curve nctes the robblty of lot ccetnce n terms of fferent vlues of the frcton of efectve tems. ouble smlng ln hs mn oc curve tht show the robblty of ccetnce on combne smles. Ths ln lso hs nother oc curve tht nctes the robblty of ccetnce bse on the frst smle. In ouble smlng ln wth fuzzy rmeter, mn oc curve s s bn wth u n own boun. Ths bn nctes the fuzzy robblty of lot s ccetnce n terms of fferent vlues of the fuzzy roorton of nonconformng tems. The uncertnty egree of roorton rmeter s one of the fctors tht bnwth eens on tht. The less n uncertnty vlue results n less bnwth, n f roorton rmeter gets crs vlue, lower n uer bouns wll become equl, whch tht oc curve s n clssc stte. Knowng the uncertnty egree of roorton rmeter (gven,, n vrton of ts oston on horzontl xs, we hve fferent fuzzy number ( n hence we wll hve fferent roorton ( whch the oc bns re lotte n terms of t. To cheve ths m we conser the structure of s follows: = (, +, +, [, q q [, q Whch wth vrton of n the omn of [0,-, the mn oc bn, s lotte ccorng to the clculton of follow fuzzy robblty: [ [ (, ( [, ( 5 Proceengs of the th Jont Conference on Informton Scences (008 Publshe by tlnts Press the uthors

6 I P ( [ [ P (, P, I, P, ( mn{ P( n P ( mx{ P(, I, ( Tht the event s the event of ccetnce of lot n terms smle th, n the oc bn n term frst smle s lotte ccorng to the clculton of follow fuzzy robblty: I P ( [ [ P (, P ( P P ( mn{ ( mx{ C n C n Exmle. suose tht n 0, c, n, c, 0.0, then we hve q n q n 5 n 0.0 Then wth stuyng f ( functon, we wll hve the cut n the followng fshon: [45 (,45( 0.0 (0.8.7,0 5 (0 [45 (,0.45.7, [45( 0.0 (0.8,0.45, [45( 0.0 (0.8,45 (, Fgure reresents tht when the rocess qulty ecrese from very goo stte to moerte stte, then the oc bn wll be wer. [ ( 0.0, , fuzzy robblty ccetnc on fuzzy combne robblty smles of ccetnce on combne n f 0 then fuzzy robblty of ccetnce on frst smle s : I 0 P [0,[0 {( 0( f ( ( 0 0( ccorng to tht the f( ecresng, then: 0 [(0.8 0(0.0 (0.8 I,( 0 0( n fuzzy robblty of ccetnce on secon smle s : P (, 0 {45 (, f ( 45 ( 0 ( fuzzy robblty of ccetnce fuzzy fuzzy robblty robblty of of ccetnc on frst smle ccetnce on frst smle Fg : oc bn for ouble smlng ln wth fuzzy rmeter of n 0, c, n 5, c Usng fuzzy rthmetc, n smlfcton, n tble wll clculte. 6 Proceengs of the th Jont Conference on Informton Scences (008 Publshe by tlnts Press the uthors

7 Tble: fuzzy robblty of ccetnce wth c =, n =0, c =, n =5 I [0,0.0 [0.88, [0,0.08 [0.76, [0.0,0.0 [0.655,0.57 [0.00,0.07 [0.8,0.87 [0.0,0.04 [0.48,0.88 [0.08,0.04 [0.84,0.76 [0.0,0.05 [0.,0.655 [0.07, [0.76,0.8 [0.04,0.06 [0.884,0.48 [0.04,0.075 [0.54,0.84 [0.05,0.07 [0.848,0. [0.0577, [0.4,0.76 [0.06,0.08 [0.8,0.884 [0.075,0.074 [0.05, Fuzzy verge smle number (FSN The mn vntge of ouble smlng ln s the reuce verge smle sze, requre to rrve to goo ecson. In ths ln smle number (SN cn be n or n +n. P I s the fuzzy robblty of rwng '' frst '' smle only, whch occurs when rrvng t ecson t the frst smle (wth fuzzy robblty P( c or c. the fuzzy robblty P of hvng rw secon smle, totlng sze of n +n, occurs when we h n nconclusve outcome from the frst smle (wth fuzzy robblty P( c c. The FSN s clculte followng the efntons fuzzy men: FSN SN [ { n I ( n n ( Where, s before, S enotes the sttement [, I,, I. Hence we get FSN { n n Exmle4. Let tht c 0, c, n 0, n 0n 0.0, 0.0then FSN s sfollows: [ [ 0.0, FSN = {0+00 (- then - cut s: Tble: FSN n terms of fferent n [0,.67 [0.,.8 [.67, [0,.4 [.,.4 [.8, [0.48,. [.,. [., [0.7,. [.4,.8 [.4,.4 [0.,0. [.67,.67 [.8,.8 7 Proceengs of the th Jont Conference on Informton Scences (008 Publshe by tlnts Press the uthors

8 [FSN*,FSN**, [FSN*,.874, FSN[ [FSN**,.874, [FSN**,FSN*, where FSN* 000( 0.0 ( 0.0 nfsn** (000( ( Fgure shows FSN bn for ouble smlng ln. If the rocess qulty s very goo or very b, then the lot wll be ccete or rejecte resectvely by the frst smle. s result, the sze of the ln smle wll be of lower vlues, n f the rocess qulty s moerte, then n most cse, s to the ecson on ccetng or rejectng the lot, secon smle shoul be selecte whch wll le to n ncrese n the sze of the ln smle Concluson In the resent er we hve roose metho for esgnng ccetnce ouble smlng lns wth fuzzy qulty chrcterstc. These lns re well efne snce f the frcton of efectve tems s crs they reuce to clsscl lns. s t ws shown tht oc curves of the ln s le bn hvng hgh n low bouns. In ths ln f the rocess qulty s very goo or very b, then the fuzzy verge smle number wll be of lower vlues. References: [ J. J. Bucley (00 fuzzy robblty: new roch n lcton, hysc-velge, Helberg,Germny. [ J.J. Bucley (006 fuzzy robblty n sttstcs, srnger-verlge Berln Helberg. [ D. Dubs, H. Pre (78 Oertons of fuzzy number, Int. J. syst. [4 P. Grzegorzews(88 soft esgn of ccetnce smlng by ttrbutes, n: roceengs of the VIth nterntonl worsho on ntellgent sttstcl qulty control Wurzburg, Setember 4-6, PP.-8. fuzzy verge smle number Fg : fuzzy verge smle number for ouble smlng ln wth rmeter fuzzy of, c 0, c, n n 0 [5 P. Grzegorzews (00 b ccetnce smlng lns by ttrbutes wth fuzzy rss n qulty levels, n: Fronters n fronters n sttstcl qulty control. Vol. 6, Es. Wlrch P. Th. Lenz H. J. Srnger, Heelberg, PP [6 P. Grzegorzews (00 soft esgn of ccetnce smlng lns by vrbles, n: technologes for 8 Proceengs of the th Jont Conference on Informton Scences (008 Publshe by tlnts Press the uthors

9 contruetng ntellgent systems, Es, sernger, vol [7 O. Hrynewsz ( sttstcl ccetnce smlng wth uncertn nformton from smle n fuzzy qulty crter worng er of SRI PS, Wrsow, (n olsh. [8. Kngw,H. Oht (0, esgn for sngle smlng ttrbute ln bse on fuzzy set theory, fuzzy sets n systems, [ D. C. Montgomery (, ntroucton to sttstcl qulty control, wley New yor. [0 H. Oht,H. Ichhsh (8, Determnton of sngle-smlng ttrbute lns bse on membersh functon, Int. J. Pro, Res 6, [ J. L. Romeu, Unerstnng bnoml sequentl testng, Rc strt, Volume, Number. [ E.G. Schlng (8, ccetnce smlng qulty control, Deer, New yor. [ F. Tm,. Kngw, Oht H. (, fuzzy esgn of smlng nsecton lns by ttrbutes, Jnese journl of fuzzy theory n systems,, 5-7. Proceengs of the th Jont Conference on Informton Scences (008 Publshe by tlnts Press the uthors