Worl Amy of Sin, Enginring n Thnology 49 9 Aptn Singl Smpling ln with fuzzy prmtr with Th sing of oisson Distriution Ezztllh Bloui Jmhnh,*, Bhrm Sghpour-Gilh, Gholmhossin Yri 3 h.d Stunt, Sin n Rsrh Brnh, Islmi Az nivrsity, Thrn, Irn. _ loui8 @ yhoo.om. Dprtmnt of Sttistis, Fulty of Bsi Sin, nivrsity of Mznrn Blosr, Irn Sghpour@umz..ir 3 Irn nivrsity of Sin n Thnology, Thrn, Irn, yri @iust.. ir Astrt--This purpos of this ppr is to prsnt th ptn singl smpling pln whn th frtion of nononforming itms is fuzzy numr n ing mol s on th fuzzy oisson istriution. W hv shown tht th oprting hrtristi (o) urvs of th pln is li n hving high n low ouns whos with pns on th miguity proportion prmtr in th lot whn tht smpl siz n ptn numrs is fi. Finlly w omplt isuss opinion y numril mpl. An thn w ompr th o ns of using of inomil with th o ns of using of oisson istriution. Kywors--Sttistil qulity ontrol, ptn singl smpling, fuzzy numr. S I. ITRODCTIO TATISTICA qulity ontrol (SQC) is n ffiint mtho of improving firm s pross qulity of proution. Smpling for ptn or rjtion lot is n importnt fil in SQC. Aptn singl smpling is on of th smpling mthos for ptn or rjtion whih is long with lssil ttriut qulity hrtristi. In iffrnt ptn smpling plns th frtion of ftiv itms, is onsir s risp vlu, ut in prti th frtion of ftiv itms vlu must now tly. Mny tims ths vlus r stimt or it is provi y primnt. Th vgunss prsnt in th vlu of p from prsonl jugmnt, primnt or stimtion my trt formlly with th hlp of fuzzy st thory. As nown, fuzzy st thory is powrful mthmtil tool for moling unrtin rsulting. In this sis fining th impris proportion prmtr is s fuzzy numr. With this finition, th numr of nononforming itms in th smpl hs inomil istriution with fuzzy prmtr. Howvr if fuzzy numr p is smll w n us th fuzzy poison istriution to pproimt vlus of th fuzzy inomil. Clssil ptn smpling plns hv n stui y mny rsrhrs. Thy r thoroughly lort y Shilling (98). Singl smpling y ttriuts with rl rquirmnts wr isuss y Oht n Ihihshi (988) ngw n Oht (99), Tmi, Kngw n Oht (99),n Grzgorzwsi(998,). Grzgrozwsi (,) lso onsir smpling pln y vrils with fuzzy rquirmnts. Smpling pln y ttriuts for vgu t wr onsir y Hrniwiz (99, 994). W provi som finition n prliminris of fuzzy sts thory n fuzzy proility in stion. In stion 3 th fuzzy proility of ptn of th lot, ws onsir roly, n its vlus in spil s ws omput. In stion 4, w l with o n of suh pln, with mpl. REIMIARIES AD DEFIITIOS rmtr p (proility of suss in h primnt) of th risp inomil istriution is nown tly, ut somtims w r not l to otin t som unrtinty in th vlu p n is to stimt from rnom smpl or from prt opinion. Th risp poison istriution hs on prmtr, whih w lso ssum is not nown tly. Dfinition: th fuzzy sust of rl lin IR, with th mmrship funtion : IR [,] is fuzzy numr if n only if () is norml () is fuzzy onv () is uppr smi ontinuous () supp ( ) is oun [3]. Dfinition: A tringulr fuzzy numr is fuzzy numr tht mmrship funtion fin y thr numr 3 whr th s of th tringl is th intrvl [, 3 ] n vrt is t = [3]. Dfinitoin3: Th -ut of fuzzy numr is non-fuzzy st fin s IR; ( ). Hn w, whr hv * Corrsponing uthor 7
Worl Amy of Sin, Enginring n Thnology 49 9 inf IR; ( ) sup IR; ( ) Dfinition4: Du to th unrtinty in th i s vlus w sustitut i, fuzzy numr, for h i n ssum tht < i < ll i. Thn X togthr with th i vlu is isrt fuzzy proility istriution. W writ for fuzzy n w hv ({ i }) i. t A,..., l} sust of X. Thn fin: l ( A)[ ] { i s} () i For, whr stns for th sttmnt n i i[ ], in, i this is our rstrit fuzzy i rithmti[]. Dfinition5: In m inpnnt Brnoulli primnt lt us ssum tht p, proility of suss in h primnt is not nown prisly n ns to stimt, or otin from prt opinion. So tht p vlu is unrtin n w sustitut p for p n q for q so tht thr is p p[] n q q[] with p q. ow lt ( r ) th fuzzy proility of r susss in m inpnnt trils of th primnt. nr our rstrit fuzzy lgr w otin r r mr ( r)[ ] { Cm p q s} () For, whr now S is th sttmnt, p p[ ], q q [ ], p q. If ( r)[ ] [ r ( ), ( )] thn r r r mr r ( ) min{ C p q s} m r r mr r ( ) m{ C p q s m n } n if [, ] proility of susss so tht fuzzy, thn m ([, ] { C p q s} (3) if ([, ] [ ([, ], ([, ]] thn: m ([, ] min Cm p q s n m ([, ] m Cm p q s m th fuzzy Whr S is th sm with pst s[]. Dfinition 6: lt rnom vril hving th poison mss funtion. If () stns for th proility tht X,thn ( ) (4) For,,,... n prmtr. ow sustitut fuzzy numr for to prou th fuzzy poison proility mss funtion. t ( ) to th fuzzy proility tht X. Thn w fin -ut of this fuzzy numr s ( ) For ll, (5). t X rnom vril hving th fuzzy inomil istriution n p in th finition 4 smll. Whih mns tht ll p p r suffiintly smll. Thn, using th fuzzy poison pproimtion[]., np!. ACCETACE SIGE SAMIG A WITH FZZY ARAMETER Suppos tht w wnt to inspt lot with lrg siz of. First t rnomiz smpl of siz n from th lot, thn inspt ll itms in th smpl, n th numr of ftiv itms ( ) will ount own. If th numr of osrvtion ftiv itms is lss thn or qul to ptn numr, thn th lot will pt, othrwis th lot rjtion []. If th siz of lot lrg, th rnom vril hs inomil istriution with prmtr n n p in whih p inits th lot s ftiv itms. Howvr if th siz of smpl lrg n p is smll thn th rnom vril hs oisson pproimtion istriution with np. So, th proility for th numr of ftiv itms to tly qul to is: np ( np ) ( ) (7)! n th proility for ptn of th lot ( p ) is: (6) 8
Worl Amy of Sin, Enginring n Thnology 49 9 p np ( np) ( ) (8)! Suppos tht w wnt to inspt lot with th lrg siz of, suh tht th proportion of mg itms is not nown prisly. So w rprsnt this prmtr with fuzzy numr p s follows: p (,, ), [], 3 p p q q [], p q. A singl smpling pln with fuzzy prmtr if fin y th smpl siz n,n ptn numr,n if th numr of osrvtion ftiv prout is lss thn or qul to,th lot will ptn. If is lrg numr, thn th numr of ftiv itms in this smpl () hs fuzzy inomil istriution, n if p is smll, thn rnom vril hs fuzzy oisson istriution with prmtr n p []. So th fuzzy proility for th numr of ftiv itms in smpl siz tht is tly qul to is: ( ftiv ), (9) min np,! m! np n fuzzy ptn proility is s follows: p,! min m!! (), Empl: Th prin of Mznrn Ksht V Snt Shoml Compl mngmnt shows tht hlf prnt r ill-p. Mjor ustomrs hoos n inspt 6 itms of this prout vill in lrg stor to uy thm. If th numr of nononforming itms in this smpl quls zro or on, th ustomr will uy ll prouts in th stor. If th numr of nononforming inrss, th ustomr will not uy thm. Bus of th proportion of ftiv prouts hs plin linguistilly, w n onsir tht s fuzzy numr p (,.5,. ). Thrfor, th proility purhsing will sri in th following: 6,, p,.5,.,,.3,.6,.3,.6. ( ) n 3 p Aoring to tht th ( ) rsing, thn: (.6.3 ).3 (.6.3 ),(.3 ) p w otin p.878, unr tht is, it is t tht for vry lots in suh pross, 88 to lots will pt. lph.9.8.7.6.5.4.3...86.88.9.9.94.96.98 fuzzy proility of ptn Fig. fuzzy proility of ptn with p (,.5,.). OC-BAD WITH FZZY ARAMETER Oprting hrtristi urv is on of th importnt ritri in th smpling pln. By this urv, on oul trmin th proility of ptn or rjtion of lot hving som spifi ftiv itms []. th o urv rprsnts th prformn of th ptn smpling plns y plotting th proility of ptn lot vrsus its proution qulity, whih is prss y th proportion of nononforming itms in th lot [4]. O urv is in sltion of plns tht r fftiv in ruing ris n inits isriminting powr of th pln. Suppos tht th vnt A is th vnt of ptn of lot. Thn th fuzzy proility of ptn lot in trms of fuzzy frtion of ftiv itms woul s n with uppr n lowr ouns. Th unrtinty gr of proportion prmtr is on of th ftors tht nwith pns on tht. Th lss unrtinty vlu rsults in lss nwith, n if proportion prmtr gts risp vlu, lowr n uppr ouns will om qul, whih tht o urv is in lssi stt. Knowing th unrtinty gr of proportion prmtr (givn,, 3 ) n vrition of its position on horizontl is, w hv iffrnt fuzzy numr 9
Worl Amy of Sin, Enginring n Thnology 49 9 ( p ) n hn w will hv iffrnt proportion ( p ) whih th o ns r plott in trms of it. To hiv this im w onsir th strutur of p s follows: p = (, +, 3 + ), p p[], qq [], p q n n p ( n, n n, n3 n) Figur shows th o n of th mpl. This figur rprsnts tht whn th pross qulity rs from vry goo stt to mort stt, thn th o n will wir..9.8 Whih with vrition of in th omin of [,- 3 ], th o n, is plott oring to th lultion of follow fuzzy proility: p [ ] [ p( ), p( )] [, 3 ( 3 ) ], n n, n n3 n( 3 ) p ( A), () fuzzy proility of ptn.7.6.5.4.3.. =, n=6...3.4.5.6.7.8.9. Fig. o n for singl smpling pln with fuzzy prmtr of, n 6 min m A! A! In Ksht V Snt Compl ompny rlt to mpl w h,.5,.thn w hv p 3 n, n.n,. 99 () ( ),( ( n.n) ( n.n) Tl: fuzzy proility of ptn p p,.,( n) n.878,..,..666,.878..,.3.468,.666.3.3,.4.384,.468.4.4,.5.99,.384.5.5,.6.57,.99, n 6 Empl : suppos tht, n in mpl.5,., p,. n thn w hv 3,.,. 99 thrfor o urv in trms fuzzy Binomil istriution n fuzzy oisson istriution is s follows: p ( p) p p (.99 ),( ) (. ), pp fuzzy proility ptn.9.8.7.6.5.4.3.. inomil inomil oisson oissn..4.6.8...4.6.8. Fig.3 o n for singl smpling pln with fuzzy prmtr of, n Figur 3 n tl show tht thr two ns pproimt. Finlly it n si tht o n with using from fuzzy oisson istriution optiml pproimnt for o n with using from fuzzy inomil istriution. With rgr this, suh pln n sign s on o fuzzy oisson istriution.
Worl Amy of Sin, Enginring n Thnology 49 9 Tl: fuzzy proility of ptn, n p pp.879,.887,..6676,.879.673,.887..5437,.6676.5488,.673.3.44,.5437.4493,.5488 Figur 4 shows two o ns for n=, n=4. Initing tht o ns r onv with zro ptn numr n this ls to qui rution of fuzzy proility of ptn for proportion of ftiv itms with smll fuzzy numrs, n it will mor th inrs of n. [4] B..M. Dut,.M. Sriv, An optimiztion s pproh for signing ttriut ptn smpling plns, Int. journl of qulity & rliility mngmnt.vol. 5 no. 8,8. [5]. Grzgorzwsi(988) A soft sign of ptn smpling y ttriuts, in: proings of th VIth intrntionl worshop on intllignt sttistil qulity ontrol Wurzurg, Sptmr 4-6,.9-38. [6]. Grzgorzwsi ( ) Aptn smpling plns y ttriuts with fuzzy riss n qulity lvls, in: Frontirs in frontirs in sttistil qulity ontrol. Vol. 6, Es. Wilrih. Th. nz H. J. Springr, Hilrg,. 36-46. [7]. Grzgorzwsi () A soft sign of ptn smpling plns y vrils, in: thnologis for ontruting intllignt systms, Es, springr, vol.. pp. 75-86. [8] O. Hryniwisz (99) sttistil ptn smpling with unrtin informtion from smpl n fuzzy qulity ritri woring ppr of SRI AS, Wrsow, (in polish). [9] A. Kngw,H. Oht (99), A sign for singl smpling ttriut pln s on fuzzy st thory, fuzzy sts n systms, 37. 73-8. [] D. C. Montgomry (99), introution to sttistil qulity ontrol, wily w yor. [] H. Oht,H. Ihihshi (998), Dtrmintion of singl-smpling ttriut plns s on mmrship funtion, Int. J. ro, Rs 6, 477-485. [] J.. Romu, nrstning inomil squntil tsting, R strt, Volum, umr. [3] E.G. Shiling (98), ptn smpling qulity ontrol, Dr, w yor. [4] F. Tmi, A. Kngw, Oht H. (99), A fuzzy sign of smpling insption plns y ttriuts, Jpns journl of fuzzy thory n systms, 3, 35-37..9 fuzzy proility ptn.8.7.6.5.4.3 n=,=.. n=4, =..4.6.8...4.6 Fig.4 o n for singl smpling pln with fuzzy prmtr of n, ; n 4, V. COCSIO In th prsnt ppr w hv propos mtho for signing ptn singl smpling plns with fuzzy qulity hrtristi with using fuzzy oisson istriution. Ths plns r wll fin sin if th frtion of ftiv itms is risp thy ru to lssil plns. As it ws shown tht o urvs of th pln is li n hving high n low ouns. W h shown tht in this pln o ns r onv with zro ptn numr. REFERECES [] J. J. Buly (3) fuzzy proility: nw pproh n pplition, physi-vlg, Hilrg,Grmny. [] J.J. Buly (6) fuzzy proility n sttistis, springr-vrlg Brlin Hilrg. [3] D. Duis, H. r (978) Oprtions of fuzzy numr, Int. J. syst.