Pocdings, XVII IMKO Wold Congss, Jun 7, 003, Dubovnik, Coatia Pocdings, XVII IMKO Wold Congss, Jun 7, 003, Dubovnik, Coatia 4 XVII IMKO Wold Congss Mtology in th 3d Millnnium Jun 7, 003, Dubovnik, Coatia STTING TH PROCSS AIM: TH FFCT OF MASURMNT UNCRTAINTY Danil Hambug-Pika, Gustavo Danil Donatlli, and Calos Albto Schnid Quality Assuanc and Mtology Gou Mchanical ngining Faculty, COMAHU NATIONAL UNIVRSITY, Nuquén, Agntina Labmto Laboatoy of Mtology and Automation, FDRAL UNIVRSITY OF SANTA CATARINA, Floianóolis, Santa Cataina, Bazil Abstact In this a, a task-scific masuing caability cition is dscibd, alicabl to slct o validat masumnt tms that ovid data to st th ocss aim, whn th tchniqus known as squnc of valus o diffnc chat a usd. Th cition is basd on th stimation of th unctainty of th ocss, which chaactizs th dission of th valus that could asonably b attibutd to th ocss aft th stting ocdu. Th oosd cition is comad with th discimination atio and with th unctainty tolanc atio, showing that th last on fails to dict th masuing caability fo aim-stting oations. Kywods: stting th ocss aim, masumnt unctainty, masuing caability. INTRODUCTION Th just-in-tim manufactuing statgy is on of th most succssful answs to th cunt makt conditions: it imovs adatability to oduct and makt changs whil ducs costs by liminating stocks. To b liabl, just-intim oduction nds a suly chain intgatd by dictabl ocsss, oating on tagt with minimum vaianc. In this nvionmnt, stting th ocss aim bcoms a citical task, aticulaly whn shot oduction uns a th ul. Ral ocsss a nv on tagt. Du to th statistical otis of th mthods usd to st th ocss aim, th valu of th tu dviation fom tagt mains unknown, so lading to th conct of unctainty of ocss. This unctainty dnds on th majo factos: th ocss itslf, th ocdu usd to st th ocss aim and th unctainty of masumnt (Fig. ). A ocss has to b in statistical contol bfo stting th ocss aim. No ffots should b wastd to gulat ocsss which and/o standad dviation vay in an undictabl mann. Aft achiving th stat of contol by th us of contol chats, sval statistical tchniqus can b alid to idntify wath o not th ocss is clos nough to th tagt. Som of ths tchniqus us also contol chats to tst th hyothss that th ocss is on tagt. Th tchniqu of th squnc of valus uss an individual valu contol chat with th cntal lin st to th tagt valu. Th ocss is considd ady fo oduction whn a givn numb of succssiv masumnts, usually tn, fail to indicat any out of contol signal []. Th tchniqu of th diffnc chat is a vaiant of th fom, to b usd whn th sam ocss oducs sval at modls of diffnt nominal sizs []. ach oint in th chat snts th diffnc btwn a masud valu and th cosonding tagt valu. Individual and diffnc chats can b also usd fo ocss monitoing, simly changing th oating mod onc th ocss is on tagt. This maks ossibl using a singl tool fo th comlt quimnts of ocss oation. Pocss Dg of statistical contol Masumnt ocdu Oato nvionmnt Standad dviation Unctainty of masumnt Non-fomal dcisions Stting ocdu Standad Instumnt Wokic Saml siz Unctainty of ocss Fig.. Factos affcting th unctainty of ocss Anoth ossibility is to us -contol chats to st th ocss aim and, aftwads, to suvis oduction []. Nvthlss, -contol chats limits a basd on oduct scification and not on ocss, common-caus, vaiation. Bcaus of this, -contol is siously handicad fo th assssmnt of statistical contol, aticulaly whn ocss caability is ath high [3]. Und such limitation, on may qustion about th ability of -contol chats to st th ocss aim in an ffctiv mann.
Pocdings, XVII IMKO Wold Congss, Jun 7, 003, Dubovnik, Coatia Pocdings, XVII IMKO Wold Congss, Jun 7, 003, Dubovnik, Coatia 4 Oth tchniqus to st th ocss aim can b found in th statistical ocss contol litatu: it is not intsting to snt thm h in dtail. A common chaactistic of all th tchniqus is that th unctainty of ocss is affctd by th ocdu itslf and by th numb of masumnts usd to stimat th ocss and th ocss standad dviation. Mo accuat sttings can b obtaind incasing th numb of saml units, but this sults in high oational costs and dlays th oduction launch. Non-fomal dcisions, i.. dcisions not tiggd by th statistical ocdu, also affct th unctainty of ocss, usually nlaging it. Unctainty of masumnt is xctd to affct advsly th unctainty of ocss. Th a no quantitativ studis to hl dfining whth o not a masumnt tm can b usd to st th aim of a givn manufactuing ocss. Bcaus of this, gnal-sco caability citia a usd, lik th unctainty tolanc atio, th gag R&R% [4], th discimination atio [5] and oths, accting th hyothsis that a masumnt tm that satisfis ths citia will b accuat nough fo any quality contol activity. This a ooss a task-scific masuing caability cition, to b alid whn masumnts a usd to st th ocss aim. Th studis hav bn focusd on th alication of th tchniqu of th squnc of valus, but th sults aly also to th diffnc chat tchniqu. Sval indics to th aim-stting fomanc hav bn studid: th numb of ats (masus) ndd to st th ocss aim satisfactoily, th numb of ocss adjustmnts ndd and th standad dviation of th ossibl ocss s that can b obtaind aft concluding th aim-stting ocdu. It has bn shown that only th standad dviation of th ossibl s is snsitiv to th snc of masumnt os. This standad dviation has bn usd to quantify th unctainty of ocss, that can b viwd as an indx to th caability of masumnt tms fo ocss aimstting tasks.. TH AIM-STTING TCHNIQU This sach has bn cntd on th alication of th tchniqu known as squnc of valus (dtails about this tchniqu can b found in f. []). Th tchniqu consids two ossibl cass. Th fist on is whn th ocss standad dviation is known, in such a way that contol limits a alady availabl whn th fist unit is oducd. Th scond cas, slctd fo this simulation, is whn th standad dviation is not known and has to b stimatd fom th ocss outcoms. To do this accoding to th Shwhat uls fo contol chats, th standad dviation shall b comutd using th of a dission statistic. In this cas, th avag of th moving angs of od two is usd: mr () = y y i i i n = mr n i i= mr () wh y a th masud valus comosing th saml and n is th saml siz (n=0, as commndd by th ocdu). Thn, th stimatd standad dviation is: mr y = d ˆ (3) bing d fo moving angs of od two. =.3 Th limits of th individuals chat a symmtically ositiond aound th ocss tagt T and comutd by th following quations: bing LCL y = T A mr UCL y = T + A mr A fo moving angs of od two. =.66 Onc th contol limits a availabl, th fist tn valus can b analyzd tosctivly. To intt th chat, th fou dcision uls known as Wstn lctic Ruls a alid (Fig. ). Rul Rul Rul 4 Rul 3 UCL +/3 +/3 Tagt -/3 -/3 LCL Fig.. Wstn lctic dcision uls fo out of contol signals If any out of contol signal is dtctd, th availabl infomation is usd to stimat th ocss and comut th valu of th coction. Aft th adjustmnt, th ocss is oatd and fsh saml units a obtaind and masud, looking fo out of contol signals. If any, a nw adjustmnt is don and mo units masud. Th ocdu nds whn tn succssiv units fail to show an out of contol signal. Lik any oth statistical tool, th squnc of valus tchniqu oducs outcoms subjctd to unctainty. Masud valus and contol limits a affctd by samling vaiation, oducing diffnt ocss s whn stting th aim of th sam ocss und atability conditions. Th numb of saml units usd to st th ocss aim and th numb of ocss adjustmnts a also subjctd to havy samling vaiation. Fom this bif xlanation, som intsting facts can b highlightd gading th influnc of masumnt os on ocss st u: A constant tmatic o, affcting to th sam xtnt all masud valus, dos not modify th contol limits. In such situation, th inttation of th chat lads to inaccuat adjustmnts, sulting finally in a ocss that (4) (5)
Pocdings, XVII IMKO Wold Congss, Jun 7, 003, Dubovnik, Coatia Pocdings, XVII IMKO Wold Congss, Jun 7, 003, Dubovnik, Coatia 4 oats dviatd fom tagt. Th xctd valu of such dviation is th valu of th tmatic o. Random (.g. atability) os inflat th valu of th stimatd standad dviation, sulting in contol limits that a fath fom th tagt. Sinc th masud valus will show also mo dission, th ffct of this kind of os sults damnd. Puly lina tmatic os, with valu qual to zo at th ocss tagt, affct th stimatd standad dviation. os with ositiv slo incas th standad dviation of th masud valus, inflating th contol limits. os with ngativ slo oduc th oosit ffct. Th ffct of lina tmatic os sms to b also damnd by th colatd bhavio of contol limits and masud valus. Th modl alid in th sach consids all ths tys of os. It is dscibd in th following sction. 3. SIMULATING MASURMNT Masumnt unctainty is dfind as a aamt, associatd with th sult of a masumnt, that chaactizs th dission of th valus that could asonably b attibutd to th masuand [6]. Lt us assum that th contibutions to masumnt unctainty can b saatd into two mutually xclusiv gous of hysical quantitis, on including all andom ffcts and th oth, all tmatic ffcts. Rsnting th unctainty du to andom ffcts by a nomal andom vaiabl: ( 0; ) ~ nomal (6) th unctainty du to tmatic ffcts by a ctangula andom vaiabl: ( ) s ~ ctangula ; (7) and assuming that both vaiabls a statistically indndnt, th standad masumnt unctainty can b xssd as: u u ( y) u ( ) + u ( ) = (8) ( y) + 3 s = (9) quation (9) stats that, fo any valu of th masuand within th ocss dission limits, th masumnt sult will b affctd by a andom o of standad dviation and by an unknown tmatic o, which valu is within th intval [ ] ;. Th mathmatical modl of th masumnt ocss usd in th simulation algoithm is consistnt with th unctainty statmnt in q. (9): y = x + + (0) and wh y is th masud valu, x th valu of th manufactud chaactistic, an vnt of a andom o and andom vaiabls: and th valu of th tmatic o. In tms of Y = X + + () and Nomal andom vaiabls hav bn usd to modl th manufactud chaactistic and th andom masumnt o: X ~ nomal µ ; () and ( 0; ) ~ nomal (3) In al masumnt ocsss, tmatic os a a function of th valu of th manufactud chaactistic. It is widly acctd that, within th limits of ocss dission, most masumnt tms snt tmatic o valus that can b intolatd by a staight lin. This condition is modld by th following quation: ( x T ) = α + β (4) wh α is a constant tmatic o and β is a facto dtmining th valu of a linaly dndnt o. To fulfill th condition imosd by th unctainty statmnt, th valus of α and β a chosn at andom, in such a way that: T 4 x T + 4 (5) Th valus of α and β a maintaind constant within ach simulation un. This way, all th valus of x gnatd fo a comlt aim-stting oation a affctd by tmatic os obtaind fom th sam attn. To simulatd th ffct of th lack of knowldg, th oation is atd, choosing nw valus fo α and β. Figu 3 shows th diffnt simulation outcoms of th modl dscibd by quations (0) to (5). Th gahic in th to shows 00 o valus obtaind with µ = 00, = and << fo th sam ocss, a cas with. Th gahic in th middl shows, >> and α 0.Th gahic in th bottom shows th outcoms whn >> and β is small. Not that th two last gahics hav bn obtaind with th sam valus of and. Thus, both o chaactistics a consistnt with th sam unctainty statmnt.
Pocdings, XVII IMKO Wold Congss, Jun 7, 003, Dubovnik, Coatia Pocdings, XVII IMKO Wold Congss, Jun 7, 003, Dubovnik, Coatia 4 ; / 0 0 (6) Fo ach oint in th unctainty comonnt domain, a gou of 00 cass has bn simulatd. ach cas uss a diffnt st of andom o valus and a diffnt tmatic o function, but fulfills th sam masumnt unctainty. This way, it mak sns comuting an avag fomanc within ach gou of 00 cass and linking such fomanc with th ffct of masumnt unctainty. Th fomanc indics hav bn valuatd within ach gou, fo all th oints in th unctainty comonnt domain: Th avag numb of ats (masus) ndd to st th ocss aim; Th avag numb of ocss adjustmnts ndd to st th ocss aim; Th standad dviation of all th tu ocss s, as obtaind aft th last ocss adjustmnt, whn th aimstting ocdu has dclad th ocss on tagt : = m i= m ( µ µ ) si s (7) Fig. 3. Th sts of outcoms of th o simulation modl 4. RSULTS Th simulation algoithm imlmnting th aim-stting tchniqu dscibd in sction and th o modl dscibd in sction 3 has bn un to study th ffct of masumnt unctainty. To gt ocss-indndnt sults, th manufactuing ocss has bn st to = 0 and all th oth µ aamts hav bn dividd by. Th domain of th unctainty contibutions has bn dfind by th following inquality: wh m is th numb of cass in th gou (m=00), µ si is th of th ocss fo ach cas, aft nding th aim-stting ocdu, and µ th avag of th m s s within th gou. Th analysis of th simulation outcoms mad vidnt th nd to filt th sults to liminat statistical xtms. A small numb of cass quid an atyically high numb of units and adjustmnts to st th ocss aim. Ths cass a causd by th undstimation of th ocss standad dviation duing th initial un of 0 units, which itslf causs a gowth in th fals alam at. Such condition will b suly avoidd by an xincd oato, basd on miical infomation on how th ocdu should function. Thn, to vnt distotion of th avags, all cass snting numb of units that can b qualifid as xtms o outlis, hav bn liminatd. Aft filting, it has bn shown that th avag numb of units and th avag numb of adjustmnts a not affctd by th snc of masumnt os. This maks thm unliabl indics to masumnt ocss fomanc, bing so discadd fo th uoss of this sach. It has bn also shown that, gadlss th masumnt condition, th distibution of µ within ach gou is si clos to nomal. This bings µ s T in quation (7), in such a way that th sum can b now inttd as th sum of th squa dviations fom tagt. Th nomality mits also xanding th covag obability by th us of th z facto. This way, a 95% xandd unctainty has bn dfind fo th ocss :
Pocdings, XVII IMKO Wold Congss, Jun 7, 003, Dubovnik, Coatia Pocdings, XVII IMKO Wold Congss, Jun 7, 003, Dubovnik, Coatia 4 U =.96* (8) Th bhavio of this unctainty within th domain of masumnts unctainty comonnts can b viwd in th following gahic (Fig. 4): U Fig. 4. Th bhavio of th unctainty of ocss fo diffnt combinations of masumnt unctainty contibutions. It can b notd that, whn masumnt unctainty comonnts a bought to zo, 95% of th ocss s obtaind by th alication of th squnc of valu tchniqu would li within th intval T ±. As unctainty contibutions gow, th unctainty of ocss also gows. Fo th xtm cas, dfind by = and =, th unctainty of ocss gows u to 80%, in such a way that 95% of th tu ocss s would li within th intval T ±. 8. Th sufac dictd in Fig. 4 hav bn intolatd by olynomial gssion, using a comlt olynomial of od. Th quation obtaind follows: U = 0.936 0.5 + 0.65 + 0.395 + 0.606 + (9) This gssion modl allows xlaining 75% of th vaianc snt in th data ( R = 0.75 ), so it can b usd fo most actical uoss. This way, th unctainty of ocss bcoms an indx to th caability of a masumnt tm fo stting th ocss aim. Unlik oth xisting caability citia (.g. unctainty tolanc atio, R&R% [4], D [5]), it dos not sm ncssay dfining som miical limit to distinguish caabl and non-caabl masumnt tms. Th infomation givn by th indx ovids insight on th consquncs that masumnt unctainty has on th quality of th manufactud chaactistics. Knowing th culiaitis of th ocss at hands and th quality quimnts, an ngin o tchnician could asily dcid whth a masumnt tm can b usd fo th task o has to b imovd. 5. DISCUSSION In this sction, comaisons will b mad btwn th unctainty of ocss cition and oth masuing caability citia. Fo th sak of simlicity, only two masuing caability citia hav bn includd, on basd on th comaison with th standad dviation of th ocss (D) and oth on th comaison with th tolanc ( U Tol ). Accoding to Whl [5], th discimination atio D can b comutd as: m D = + (0) wh is th standad dviation of at masumnts m and is th standad dviation of a atability o, valuatd accoding to [5]. A tm lading to D = 4 (just caabl accoding to Whl) will hav: =. 365 0 () Considing and assuming that = =, to allow fo som tmatic sidual os that would not aa in th atability study (lik calibation siduals, long-tm dift in th nvionmntal conditions, tc.), th unctainty of ocss will sult, accoding to q. (9), U =.9. This s that a masumnt tm considd caabl accoding to th D cition would obably oduc lss than 0% nlagmnt of th unctainty of ocss. Nvthlss, ca should b takn with th influnc of unknown and sidual tmatic os. If thy a much bigg than th atability o, th unctainty of ocss could gow dastically. Lt suos now a masumnt tm snting an unctainty tolanc atio U Tol = 0., highly common in industial quality contol. Suos also that th unctainty is comosd by andom and tmatic contibutions accoding to q. (9), in such a way that =. If th caability of th manufactuing ocss C., w 67 = = 0. 884. Rlacing ths valus in q. (9), th unctainty of ocss sults U =.65. In this cas, a masumnt tm that is found caabl gading th unctainty tolanc atio, could affct havily th fomanc of th squnc of valus tchniqu. A btt unctainty tolanc atio would b ncssay (.g. btt than U Tol = 0. ), to maintain low th unctainty of ocss.
Pocdings, XVII IMKO Wold Congss, Jun 7, 003, Dubovnik, Coatia Pocdings, XVII IMKO Wold Congss, Jun 7, 003, Dubovnik, Coatia 4 Th diffnc btwn th two cass analyzd abov can b xlaind with gad to th mchanism by which th masumnt os affct th osition of th ocss. As it has bn advancd in sction, masumnt os mak th masud valus dviat fom th cosonding valus of th quality chaactistic. Th contol limits, comutd fom statistics lating thos masud valus, sult also affctd. Thn, th comaison of th dviatd valus with distotd contol limits oducs mistakn dcisions and inadquat ocss adjustmnts. Th intnsity of th caus and ffct lationshis within this chain dnds on th lvancy of masumnt os, whn comad with th standad dviation of th ocss. No fnc is mad to th oduct tolanc. This is th ason why th unctainty tolanc atio snts a oo colation with th unctainty of ocss. A simila bhavio should b xctd fom oth indics basd on oduct tolanc, lik R&R(%Tol) [4], Cg and Cgk [7]. 6. CONCLUSION In this a, a task-scific masuing caability cition has bn dscibd. It can b alid to slct o validat masumnt tms that ovid data to st th ocss aim, whn th tchniqus known as squnc of valus o diffnc chat a usd. Th cition is basd on th stimation of th unctainty of th ocss, which chaactizs th dission of th valus that could asonably b attibutd to th ocss aft th stting ocdu. This dission includs th samling vaiation, chaactistic of th statistical tchniqu, and th incmntal unctainty du to masumnt. Unlik oth citia, it dos not nd miical limit valus to judg th masuing caability, bcaus th unctainty of th ocss is dictly latd to oduct and ocss quality. This way, highly scific infomation is bought to th ngin o tchnician, who would b abl to judg considing th culiaitis of th ocss in hands. It has bn also shown that th unctainty tolanc atio can fail whn usd to assss th masuing caability of tms usd to st th ocss aim. This oblm, associatd with th us of th tolanc instad of th ocss standad dviation as fnc valu, could b also common to oth indics lik Cg, Cgk and R&R(%Tol) (not studid in this a). In th oinion of th authos, mo ffots a ncssay to lat th mtological chaactistics of masumnt tms with thi ffct on oduct quality and ocss conomy. Th sult of ths ffots should b a comlt st of siml tools to dcid th adquacy of masumnt tms fo scific sho-floo alications. This will mak mtology mo valuabl fo th oduction sonnl, making it ossibl justifying th xnditus in btt masumnt tms wh thy a tuly ndd. RFRNCS [] D.J. Whl, Shot Run SPC, SPC Pss, Knoxvill, T, 99. [] A. Wcknmann, U. Gbau, " Statistical Pocss Contol in Futu Poduction", Jounal of Mchanical ngining. vol. 44, no. -,. 4-8, 998. [3] D.J. Whl, Advancd Toics in SPC, SPC Pss, Knoxvill, T, 995. [4] AIAG, ASQ, Masumnt Systm Analysis Rfnc Manual 3 d d., Th Atomotiv Industis Action Gou, Toy, MI, 00. [5] D.J. Whl, valuating th Masumnt Pocss, SPC Pss, Knoxvill, T, 989. [6] BIPM, IC, IFCC, ISO, IUPAC, IUPAP, OIML, Guid to th xssion of Unctainty in Masumnt, Intnational Oganization fo Standadization, Gnva, 993. [7]. Ditich, A Schulz, Statistical Pocdus fo Machin and Pocss Vification, ASQ Pss, Milwauk, WI, 999. Authos: Danil Hambug-Pika, Guo d Mtología y Asguaminto d la Calidad, Fac. d Ingniía, Univsidad Nacional dl Comahu, Bunos Ais 400, (8300) Nuquén, Agntina, hon/fax +54 99 449 0357, gmac@uncoma.du.a Pof. Gustavo Danil Donatlli, LabMto, Univsidad Fdal d Santa Cataina, Caixa Postal 5053, 88040-970 Floianóolis, SC-Basil, hon +55 48 39 037, fax +55 48 39 009, donatlli@labmto.ufsc.b Pof. Calos Albto Schnid, LabMto, Univsidad Fdal d Santa Cataina, Caixa Postal 5053, 88040-970 Floianóolis, SC-Basil, hon +55 48 39 0, fax +55 48 39 009, cas@cti.og.b