Influence of Preventive Maintenance Policy on Manufacturing Systems Performances

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1 rocdings of h World Congrss on Enginring 9 Vol I WCE 9, July 1-3, 9, London, U.K. Influnc of rvniv Mainnanc olicy on Manufacuring Sysms rformancs Anonio C. Capuo, aolo Salini Absrac rvniv mainnanc (M) can bnfi sysms having an incrasing failur ra. Howvr, M is ofn schduld in ordr o minimiz h mainnanc cos or o comply wih producion planning rquirmns or insrucion from h quipmn supplirs. Givn ha mainnanc acions affc procss im variabiliy and rsourc uilizaion, such mainnanc planning criria may caus ohr advrs ffcs on h opraional prformanc of h manufacuring sysm such as an incras in Work In rocss (WI) and cycl im. In his papr som approxima quuing modls ar uiliz o asss h impac of M inrval on WI. I is shown ha WI valu is srongly influncd by h M inrval and ha mainnanc inrvals corrsponding o a minimum mainnanc cos or minimum WI can b qui diffrn. This kind of analysis can hlp in making mor informd dcisions involving WI and cos rad-offs. Indx Trms Mainnanc opimizaion, manufacuring sysms prformancs, prvniv mainnanc, quuing modls. I. INTRODUCTION Manufacuring sysms ar subjc o drioraion wih usag and ag. In cas of rpairabl sysms, mainnanc can rsor h opraional saus of manufacuring quipmn afr failurs or can prsrv i by rducing h occurrnc of brakdowns. Howvr, mainnanc downims incras rsourcs uilizaion and sysm variabiliy, ngaivly affcing som rlvan prformanc masurs of manufacuring sysms, such as work in procss (WI) and cycl im. Whil a vas body of liraur abou mainnanc planning and opimizaion xiss [1]-[5], h inracions bwn mainnanc planning and manufacuring sysms prformancs has bn scarcly invsigad. In paricular, som criria hav bn proposd o dynamically drmin mainnanc acions basd on h sysm saus [6]-[7] and o conrol manufacuring sysms aking ino accoun machins brakdowns [8]-[9]. Howvr, mos of hs approachs ar qui complx and difficul o apply in ral lif condiions. In his papr, insad, som asy o us approxima quuing modls ar uilizd o assss h impac ha prvniv mainnanc inrval has on manufacuring sysms prformanc, mainly focusing on WI. This is mad o show how h arbirary slcion of prvniv mainnanc inrval or mainnanc schduling Manuscrip rcivd April 16, 9. A. C. Capuo is wih h Univrsiy of L Aquila, L Aquila, Ialy (phon: ; fax: ; -mail: anoniocasimiro.capuo@univaq.i).. Salini is wih h Univrsiy of L Aquila, L Aquila, Ialy (-mail: paolo.salini@univaq.i). basd on minimum mainnanc cos approachs can b drimnal o WI and cycl im, so ha rad-off dcision may b rquird. Th papr is organizd as follows. A firs a brif rviw of prvniv mainnanc opimizaion approachs basd on cos minimizaion is carrid ou. Thn, availabl quuing modls for unrliabl srvrs ar survyd o slc suiabl analyical ools o sima h prformanc of manufacuring sysms subjc o corrciv and prvniv mainnanc. Subsqunly, h chosn quuing modls ar uilizd in a comparaiv mannr o xplor h bhaviour of a singl machin sysm. Som analyical rsuls ar prsnd and discussd in ordr o poin ou h rlvan issus o mainnanc plannrs. This allows mor informd dcisions. II. ECONOMIC OTIMIZATION OF REVENTIVE MAINTENANCE SCHEDULE In sysms having an incrasing failur ra (IFR), prvniv mainnanc bringing h sysm ino is original sa (as good as nw) is bnficial as i rducs h avrag failur ra. Th problm hn ariss of drmining h prvniv mainnanc inrval T. This is usually slcd by mainnanc plannrs in ordr o rspc som xrnal rquirmn (such as insrucions from h quipmn manufacurrs or dircivs from h producion planning dparmn) or, oo ofn, is dcidd arbirarily. In cas on wishs o opimiz T, a mainnanc cos minimizaion approach is usually pursud. In his rspc many policis xis [1]-[5] which mosly fall ino wo classs, namly ag rplacmn policis and priodic rplacmn policis. In ag rplacmn policy a uni is always rplacd a failur or im T if i has no faild up o im T. In boh cass an as good as nw inrvnion is assumd which mans ha h failur ra is rsord o is iniial valu. In priodic rplacmn policy, insad, a uni is rplacd priodically a plannd ims kt (k = 1,,... ). Only minimal rpair afr ach failur is mad so ha h failur ra rmains undisurbd by any rpair of failurs bwn succssiv rplacmns prformd a ims kt. Th prvniv mainnanc inrval T is ofn chosn o minimiz h ovrall mainnanc cos including boh h cos of corrciv rpairs afr a brakdown and plannd prvniv rplacmns. Ovr an infini im horizon on usually rfrs o h xpcd mainnanc cos pr uni im of a mainnanc cycl. In cas of ag rplacmn policy h avrag cos pr uni im C AU (T ) C AU ( T [ 1 T )] ( T )[ 1 T )] C T ) + CB ) = (1) T T ) + MTTF ISBN: WCE 9

2 rocdings of h World Congrss on Enginring 9 Vol I WCE 9, July 1-3, 9, London, U.K. is compud as h raio of h avrag mainnanc cos ovr a cycl o h avrag cycl duraion. In (1) C is h cos of a prvniv mainnanc inrvnion and C B is h cos of a corrciv inrvnion following a brakdown, T ) is h rliabiliy compud ovr h prvniv mainnanc inrval. MTTF is h avrag im o failur compud whn a prvniv mainnanc wih as good as nw rpair policy is adopd [1] T ) d MTTF = () 1 T ) Assuming a Wibull disribud im o failur wih paramrs β an η, h rliabiliy ) is [1] R ) = β η (. (3) In cas of priodic rplacmn, T is h fixd cycl lngh and h avrag numbr of corrciv rpairs N(T ) xpcd ovr h cycl lngh is givn by rnwal hory T = N( T ) λ ( ) d (4) B β 1 β λb ( ) = (5) η η whr λ B () is h failur ra funcion [1]. Thrfor, h mainnanc cos pr uni im is C AU ( T ) C + C N( T ) B =. (6) T Sinc wih shor T on has high prvniv mainnanc coss bu low corrciv mainnanc coss, whil for long T h opposi occurs, an opimal valu of T which minimizs C AU (T ) may xis and numrical mhods can b usd o drmin i. Howvr, his widly pracicd approach only focuss on coss bu nglcs ohr prformanc masurs of a manufacuring sysm which may b worsnd by an unsuiabl choic of T. In fac, h prvniv mainnanc inrval bsids affcing h avrag failur ra and mainnanc cos, also influncs rsourc availabiliy and uilizaion, as wll as h variabiliy of ffciv procssing ims, hus drmining congsion problms which may incras WI accumulaion and cycl im a h worksaions. III. QUEUEING MODELS FOR UNRELIABLE MACHINES Quuing hory provids a convnin mannr o quickly sima h main prformanc masurs of dynamic sysms in which discr vns alr h sa of h sysm. In quuing sysms cusomrs (i.. jobs) arriv by som arrival procss and wai in quu for h nx availabl srvr (i.. a machin). Whn h srvr bcoms availabl a cusomr is slcd from h quu and srvicd according o som disciplin bfor laving h sysm. Analyical quuing modls provid gnralizabl rsuls and xplicily show h rol of h influncing paramrs, whil his is gnrally no possibl in discr vns compur simulaion modls. Th lar, on h ohr hand, ar much mor flxibl and powrful, bu ar vry im consuming o cra and valida. Quuing hory [11] is ofn uilizd o sudy manufacuring sysms and a numbr of xbook ar availabl on his subjc [1]-[14]. Howvr, vn if from a long im a numbr of quuing modls hav bn dvlopd for unrliabl srvrs [15]-[17], mos of hm ar basd on h assumpion of xponnially disribud im o failur (i.. consan failur ra) and only addrss prmpiv inrrupions, such as brakdowns, or non-prmpiv such as prvniv mainnanc. This prvns from using simpl quuing modls o opimiz mainnanc policis as on nd o modl boh kind of inrrupions. Morovr h assumpion of consan failur ra maks h modls unsuiabl o h cas of prvniv mainnanc which is only usful whn h sysm shows an incrasing failur ra causd by progrssiv war and drioraion. Furhrmor, du o h complx naur of inrrupions in manufacuring, i is ofn difficul o proprly slc h appropria modl. To his nd Wu al. [18] propos a usful classificaion. Thy a firs disinguish bwn prmpiv and non-prmpiv inrrupions. rmpiv inrrupions ar unschduld and can occur during h procssing of a job, hus inflaing h avrag procss im rspc h valu of h naural procss im. Non-prmpiv inrrupions, insad, ar usually schduld and, in any cas, can b pospond unil h job procssing is rminad. Thn hy disinguish bwn run-basd and im-basd inrrupions. Run-basd inrrupions can occur only if WI xis in h sysm or ar indircly causd by h prsnc of WI. For insanc h brakag of a ool can occur only if h machin is procssing a par. Tim-basd inrrupions insad can occur vn in absnc of WI. Exampls of run-basd and im-basd prmpiv inrrupions ar, for insanc, brakdowns or ou of spc inpus, and powr ouags rspcivly. Cass of run-basd and im-basd non-prmpiv inrrupions insad ar, for insanc, sups and prvniv mainnanc rspcivly. Finally, hy furhr considr sa-inducd or produc-inducd vns as sub-cass of run-basd non-prmpiv vns (i.. a sa-inducd vn is an inrrupion driving from a chang of sa of h machin such as a warm up priod whn h machin passs from sand-by o working condiions). According o his classificaion, in his work w ar inrsd in run-basd prmpiv vns (i. brakdowns) and im-basd non-prmpiv vns (i.. prvniv mainnanc.). Whil h radr can consul h papr of Wu al. [18] for a mor compl classificaion of M/M/1, M/G/1 and G/G/1 quuing modls rfrring o run-basd or im-basd inrrupions whn h upim is xponnially disribud, hr w considr only modls for quus wih oisson arrivals and gnral srvic procsss in singl srvrs applicaions (M/G/1), which br fi h scop of his papr. By using h havy raffic approximaion of Whi [19], valid for quus wih gnral arrival and gnral srvic ISBN: WCE 9

3 rocdings of h World Congrss on Enginring 9 Vol I WCE 9, July 1-3, 9, London, U.K. procsss in singl srvrs applicaions (G/G/1), h quuing im (QT) can b simad as, ca + c QT ) = ρ 1 ρ whr c a is h cofficin of variaion of inrarrival im, c is h cofficin of variaion of ffciv procss im, ρ= λ/μa is h rsourc uilizaion, A h brakdown inducd srvr availabiliy, is h xpcd valu of h Effciv rocss Tim (ET). ET rprsns h avrag ffciv procss im as modifid rspc h avrag naural procss im o accoun for h inrrupions. las no ha his dfiniion of dos no includ sa-inducd, run-basd non-prmpiv vns. In cas of oisson arrivals obviously c a = 1. Hopp and Sparman [] provid h following quaions o compu h paramrs of (7) in cas of ihr prmpiv or non-prmpiv run-basd vns. c a a= (8) σ a rmpiv inrrupions = (9) A σ ( MTTR + σ r ) (1 A) σ = + (1) A A MTTR MTTR c = (11) (1 + cr ) A(1 A) c + Non-prmpiv inrrupions + = (1) N N σ (13) σ 1 = σ + + N N (7) prmpiv inrrupions hrough (9) - (1) and hn o us such valus as acual saring valus σ o compu h final and σ valus from (1)-(13). Unforunaly, (11) is valid only in cas of consan failur ra, and an xplici analyical xprssion for h cofficin of variaion of ffciv procss im wih incrasing failur ra is difficul o obain. Morovr (13) could b usd only o accoun for sups bu no for prvniv mainnanc as h lar is uslss in cas of consan failur ra whil whn h failur ra is incrasing h mainnanc inrval N influncs h acual val of h MTTF so ha h abov dscribd wo-sp procdur can no b applid. Finally, Wu al [18] poin ou ha his modl dos no accoun for im-basd and sa-inducd vns. A modl such as h abov on could b usd as an approximaion for cass including boh prmpiv and non-prmpiv inrrupions, providd ha on includs in h ffciv procss im h inflaion ffc of boh brakdowns and prvniv mainnanc inrrupions. Howvr, h problm rmains h simaion of c in cas of incrasing failur ra and non-prmpiv inrrupions. A modl for M/G/1 non-prmpiv prioriy quus wih wo prioriis, by Adan and Rsing [1], can b insad uilizd as an approximaion for im-basd non prmpiv inrrupions cass, providd ha avrag valu of h procss im is corrcd o accoun for h ffcs of prmpiv inrrupions. According o his modl λ 1 and λ ar h arrival ras of high and low prioriy jobs and μ 1 and μ h srvic ras of high and low prioriy jobs rspcivly. Hr h low prioriy jobs ar h prvniv mainnanc inrrupions. This modl compus h xpcd cycl im CT as λ CT ) = QT) + S1) + S) (15) + λ + λ λ QT) = QT1 ) + QT ) + λ + λ ρ1 R1 ) + ρ R ) (16) QT1 ) = 1 ρ1 ρ1 R1 ) + ρ R) QT ) = (1 ρ ρ )(1 ρ ) 1 1 S 1 and S ar h ffciv procss ims, ρ 1 = λ 1 /μ 1, ρ = λ /μ, whil σ c = (14) Si ) Ri ) = i = 1, S ) i (17) In (8) o (14) a is h avrag inrarrival im and σ a is sandard dviaion. σ is h sandard dviaion of naural procss im, MTTR and σ r h man valu and sandard dviaion of im o rpair, c is h cofficin of variaion of naural procss im, c r is h cofficin of variaion of rpair im, is h avrag duraion of non-prmpiv inrrupion, σ is sandard dviaion, and N is h avrag numbr of jobs procssd bwn non-prmpiv inrrupions. In cas on dals wih boh prmpiv and non-prmpiv inrrupions Hopp and Sparman [] suggs a firs o compu h and σ valus including only This modl xplicily accouns for im-basd non-prmpiv inrrupions (i.. prvniv mainnanc) whil prmpiv inrrupions should b includd in h compuaion of ffciv procss im S 1. IV. IMACT OF REVENTIVE MAINTENANCE INTERVAL ON MANUFACTURING SYSTEM ERFORMANCES In h following, h abov approxima quuing modls will b usd as a mans o dscrib h impac of prvniv mainnanc inrval on h opraional prformanc of a ISBN: WCE 9

4 rocdings of h World Congrss on Enginring 9 Vol I WCE 9, July 1-3, 9, London, U.K. manufacuring sysm. For sak of simpliciy h analysis will b limid o an lmnary manufacuring sysm composd by a singl machin, considring work in procss as h prformanc masur of inrs. WI is a rlvan prformanc masur, in fac, apar from h carrying coss i involvs, i also implis spac occupaion on h shop floor and conribus o h incras of manufacuring lad im. For a saionary sysm, in fac, h rlaion bwn WI, hroughpu λ and lad im, hr rfrrd as cycl im CT, is givn by h wll known Lil s law [] WI = λ CT. (18) In h rs of h papr h following assumpions ar mad. A consan hroughpu is assumd, according o an imposd valu of h avrag inrarrival im of jobs o b procssd. Inrarrival im of jobs has an xponnial disribuion (oisson arrivals), whil h procssing im is assumd o hav a gnral disribuion. Th adopd quuing modls will hn blong o h M/G/1 class. Th machin is assumd o hav an incrasing failur ra wih im o failur modld rsoring o a Wibull disribuion, whil prvniv mainnanc is carrid ou a consan im inrval according o a priodic rplacmn policy. Mainnanc is carrid ou undr h as good as nw assumpion, i.. afr rplacmn h quipmn failur ra is rsord o h iniial valu, whil minimal rpair is don a brakdowns. Considring ha no simpl quuing modls ar availabl which includ xplicily boh prmpiv and non-prmpiv inrrupions for unrliabl srvrs wih Wibull disribud incrasing failur ra, hr approxima quuing modls will b uilizd for sak of comparison, o sima h avrag WI a h worksaion whn h prvniv mainnanc inrval T is changd. This is mad, for insanc, o compar h T valu corrsponding o a minimum WI, if any, o h valu corrsponding o h minimum mainnanc cos pr uni im, and o assss h ffc of changing T on sysm WI. I should b poind ou ha his papr has no h goal of dvloping a nw or xac quuing modl for unrliabl srvrs subjc o boh prmpiv and non-prmpiv inrrupions, bu rahr o us som xising approxima quuing modl o poin ou h following issus ofn nglcd by mainnanc plannrs: a) WI and cycl im can b qui snsiiv o h frquncy of prvniv mainnanc acions; b) o choos a prvniv mainnanc inrval basd only on mainnanc cos minimizaion or ohr criria (producion planning rquirmns, insrucions from quipmn manufacurrs c.) may hav a ngaiv impac on ohr opraional prformancs of h manufacuring sysm such as WI and cycl im, which can also bring an advrs impac on ovrall coss; c) h valu of prvniv mainnanc inrval which minimizs h mainnanc cos can b qui diffrn from h valu which minimizs WI, hus asking for a rad-off dcision. In his scion, o provid an vidnc of h abov issus, som numrical rsuls will b shown using h following approxima quuing modls. W do no xpc ha any of h adopd modl will provid prcis numrical rsuls, owing o h larg numbr of approximaions involvd, bu hir combind uilizaion can giv a masur of h ffcs of improprly slcing h prvniv mainnanc inrvals, and can giv a qualiaiv guidanc o mainnanc plannrs. Improvd quuing modls or simulaion sudis will b rquird o obain prcis rsuls. MODEL I) This is h basic Whi modl (7) whr c is changd in a paramric mannr assuming valus.5,.75, 1. This avoids h nd o xplicily compu is valu basd on h rliabiliy characrisics of h machin and h iming of mainnanc acions. Th xpcd ffciv procss im has a valu which includs h naural procss im, and h downims occurring during h procssing of h job owing o brakdowns and prvniv mainnanc. Th compuaion is prformd uilizing (9) and (1) in squnc, considring ha in h prsn applicaion h avrag numbr of unis procssd during h im inrval T bwn wo conscuiv prvniv mainnanc acions is T A N= (19) hus obaining = + A N = 1 +. () A T A Th brakdown inducd availabiliy is MTTF A= (1) MTTF + MTTR whr MTTF is compud as shown in ()-(3) assuming a Wibull disribud im o failur. Finally, h rsourc uilizaion is compud as ρ = / a. MODEL II) This is h Whi modl (7) whr is compud as shown for modl I, whil brakdown-rlad c is compud according o Hopp and Sparman (11) bu using h MTTF valu from () compud wih IFR and prvniv mainnanc. Sinc in cas of Wibull lifim disribuion h acual failur ra is incrasing, o us a consan valu, qual o h avrag valu of h failur ra ovr inrval T is an approximaion. Howvr, givn ha sysms wih incrasing failur ra ar mor prdicabl in hir failur im, i is wll known ha h cofficin of variaion of upim is lowr han 1 [3] so ha i is xpcd ha (11) ovrsimas rahr han undrsimas h valu of c. Morovr, w can xpc h variabiliy ffc o b lss rlvan han h rsourc sauraion ffc. Th final valu of c rsuling from h inclusion of non-prmpiv inrrupions is again obaind from (1)-(14) according o h abov cid wo-sp procdur. MODEL III) This is h modl wih wo prioriis (15)-(17), whr h prvniv mainnanc acions ar xplicily modld wih h sram of arrivals having prioriy. Th sram of jobs o ISBN: WCE 9

5 rocdings of h World Congrss on Enginring 9 Vol I WCE 9, July 1-3, 9, London, U.K. b procssd is h on having prioriy 1. Th paramrs uilizd in h modl ar λ 1 = 1/ a, λ = 1/T, μ 1 = 1/, μ = 1/, S 1 ) =, S ) =, whr, includs only h ffcs of brakdowns, and is compud according o (9) wih availabiliy givn by () and (1). In all modls, xcp whr ohrwis spcifid, h cycl im is compud as CT) = QT) +, and h avrag valu of WI is compud from Lil s law (18). In h following a sampl numrical applicaion is considrd o assss h occurring phnomna and draw som conclusions. Only on numrical cas is shown hr owing o spac limiaions bu a numbr of ohr numrical xprimns confirmd ha i is rprsnaiv of a ypical sysm bhavior. Compuaions wr mad assuming h paramrs valus shown in Tabl I. Tabl I. aramrs valus for h numrical applicaion. Wibull paramr, η 16 Wibull paramr, β 4.5 Naural avrag procss im, (min) 85 Sandard dviaion of naural procss im, σ (min) WI holding cos, h ( /uni hr) 1 Man prvniv mainnanc Tim, (h) Sandard dviaion of prvniv mainnanc.5 im, σ (h) Man inrarrival im, a (min) 1 MTTR (h) (corrciv mainnanc) 5 Sandard dviaion of MTTR, σ r (h) 1 Figur 1 shows h WI rnds compud rsoring o h hr adopd modls whn T changs from o 5 hr. Curvs rfrring o modl I ar shown for c valus of.5,.75, and 1. Whil significan diffrncs in h compud valus occur, du o h diffrn approximaions involvd, all modls show h sam rnd. In paricular i is worh noing ha wo radically diffrn modls, namly Modl III and Modl I wih c =.5 show qui similar valus. Modl II show much lowr WI owing o h vry low c valu rsuling from h applicaion of Hopp and Sparman modl o h cas of IFR, which raiss doubs on h accpabiliy of his approximaion. Ovrall, as T is gradually incrasd, h WI rnd involvs a firs a rapid rducion unil a minimum is rachd, hn an incras, followd by a final sabilizaion o an asympoic valu. This may b xplaind obsrving ha, apar from h variabiliy rm, WI dircly dpnds from h rsourc sauraion which, in urn, is dircly dpndn on h valu of h ffciv procss im. Figur hn shows h variaion of ffciv procss im. In Figur a), which rfrs o modls I and II, w obsrv hr disinc zons. In h firs zon, for small valus of T, h valu of is high, bu rapidly dcrasing, bcaus of h significan impac of prvniv mainnanc downim. This downim is disribud ovr a comparaivly small numbr of pics givn h high frquncy of prvniv mainnanc acions. Conribuion of corrciv mainnanc downim is insad ngligibl as h frqun prvniv mainnanc kps h sysm failur ra low and fw brakdowns ar xpcd givn ha MTTF > T. In h hird zon, for high valus of T, h conribuion of prvniv downim bcoms ngligibl, givn ha a high numbr of pics ar procssd bwn prvniv soppags, and h corrciv downim conribuion sabilizs o h valu corrsponding o disribuing h corrciv downim ovr h avrag numbr of pics procssd bwn wo conscuiv brakdowns, givn ha T bcoms grar han MTTF and ha for high T valus h frquncy of prvniv mainnanc has only a ngligibl ffc on h failur ra which sabilizs owards an asympoic valu. In h inrmdia zon a minimum of, and hus WI, may occur or no dpnding on h rad-off bwn h comping ffcs of rducing prvniv soppags and incrasing corrciv soppags, whn T and MTTF bcom comparabl and changs in T srongly affc MTTF variaions and rsourc availabiliy. Figur b) insad shows compud according o Modl III whr only h ffc of corrciv soppags is includd, bing h prvniv soppags rad sparaly. In ha figur h asympoically sabilizing conribuion of brakdowns is vidn. Finally, Figur c) shows h rnd of compud wih modls I and II bu in a diffrn cas rspc ha of Tabl I, showing ha h abov dscribd rad-off bwn corrciv and prvniv downim ffcs on ffciv procss im dos no ncssarily giv ris o a minimum valu of. Figur 3, shows insad h WI curv of modl III suprimposd o h oal mainnanc cos pr uni im C AU (T ) curvs compud according o (6). Thr curvs ar dpicd for diffrn coupls of prvniv and corrciv inrvnion coss. This shows ha whil h acual coss involvd in mainnanc acions dfin a diffrn opimal prvniv mainnanc inrval, his inrval is obviously no rlad o h inrval minimizing WI, and significan prcn incrass of WI may occur whn T is s a h minimum mainnanc cos rspc h minimum WI valu. Thrfor, if w sum o h ovrall mainnanc cos C AU (T ), h WI holding cos, C WI (T ) = h WI, whr h ( /uni hr) is h uni WI carrying cos, w can compu a oal cos as shown in Figur 4. Th corrsponding T valu minimizing his oal cos can b anohr opion for planning a mainnanc policy. Figur 1. WI vs prvniv mainnanc inrval. ISBN: WCE 9

6 rocdings of h World Congrss on Enginring 9 Vol I WCE 9, July 1-3, 9, London, U.K. a) b) c) Figur. Trnds of ffciv procss im vs prvniv mainnanc inrval inrval or is choic mad o minimiz only h mainnanc cos can hav a drimnal ffc on h sysm prformancs. rsnd modls allow, insad, o drmin h mainnanc inrval giving h minimum oal cos or, anyway, can hlp mainnanc plannrs o mak mor informd dcision rlad o coss-prformancs rad offs. Figur 3. Trnds of WI and mainnanc coss (lgnd: Cmp = C, Cmg = C B ). Figur 4. Trnds of WI, mainnanc and oal coss. I. CONCLUSIONS In his papr som quuing modls hav bn usd o assss h ffcs ha h prvniv mainnanc inrval has on h opraional prformanc of singl machin manufacuring sysms characrizd by an incrasd failur ra. Whil h sudy poind ou ha sill som rsarch work is rquird o dvlop prcis quuing modls for unrliabl srvrs undrgoing boh prmpiv an non prmpiv inrrupions wih non xponnially disribud upims, i was also shown ha such approxima modls can hlp o quickly rprsn h ordr of magniud and h ffcs of h involvd rad offs. In paricular, i was shown ha WI can b vry snsiiv o h prvniv mainnanc inrval, and ha h minimum cos mainnanc inrval is no rlad o a minimum WI. Arbirary slcion of his REFERENCES [1] M.Bn-Daya, S.O. Duffuaa, A. Raouf, Mainnanc, Modling and Opimizaion, Springr,. [] M.A. Crspo, Th Mainnanc Managmn Framwork. Modls and Mhods for Complx Sysms Mainnanc, Springr, 7. [3] S.O. Duffuaa, A. Raouf, J.D. Campbll, lanning and Conrol of Mainnanc Sysms: Modling and Analysis, Wily, [4] T. Nakagawa, Mainnanc Thory of Rliabiliy, Springr, 5. [5] H. Wang, A survy of mainnanc policis of drioraing sysms, Europan Journal of Opraional Rsarch, Vol. 139, pp ,. [6] D. Gupa, Y. Günalay, M.M. Srinivasan, Th rlaionship bwn prvniv mainnanc and manufacuring sysms prformanc, Europan Journal of Opraional Rsarch, Vol. 13, pp , 1. [7] S.M.R. Iravani, I. Dunyas, Ingrad mainnanc and producion conrol of a drioraing producion sysm, IIE Transacions, Vol. 34, pp ,. [8] J.C. K, Th opimal conrol of an M/G/1 quuing sysm wih srvr vacaions, sarup and brakdowns, Compurs & Indusrial Enginring, Vol. 44, pp , 3. [9] D. rry, M.J.M. osnr, A corrlad M/G/1-yp quu wih randomizd srvr rpair and mainnanc mods, Opraions Rsarch Lrs, Vol. 6, pp ,. [1] C.E. Ebling, An Inroducion o Rliabiliy and Mainainabiliy Enginring, McGraw Hill, [11] D. Gross, J.F. Shorl, J.M. Thompson, C.M. Harris, Quuing Thory, Wily, 8. [1] T. Aliok, rformanc Analysis of Manufacuring Sysms, Springr, [13] J.A. Buzaco, G. Shanikumar, Sochasic Modls of Manufacuring Sysms, rnic Hall, [14] R. Suri, J.L. Sandrs, M. Kamah, rformanc valuaion of producion nworks, in Gravs, S.C. al. (Ed.), Handbook of Opraions rsarch & Managmn Scinc, Vol. 4, pp , [15] B. Avi-Izhak,. Naor, Som quuing problms wih srvic saion subjc o srvr brakdown, Opraions Rsarch, Vol. 1, pp. 33-3, 196. [16] A. Fdrgrun, L. Grn, Quuing sysms wih srvic inrrupions, Opraions Rsarch, Vol. 34, No. 5, pp , [17] H.C. Whi, L.S. Chrisi, Quuing wih prmpiv prioriis or wih brakdowns, Opraions Rsarch, Vol. 6, pp , [18] K. Wu, L.F. McGinnis, B. Zwar, Quuing modls for singl machin manufacuring sysms wih inrrupions, roc. 8 Winr Simulaion Confrnc, IEEE, pp [19] W.Whi, Approximaions for h GI/G/m quu, roducion and Opraions Managmn, Vol., pp , 1993 [] W.J. Hopp, M.L. Sparman, Facory hysics, McGraw Hill,. [1] I. Adan, J. Rsing, Quuing Thory, Lcur Nos. Availabl: hp:// [] J.D.C. Lil, A roof of h quuing formula: L = λw, Opraions Rsarch, Vol. 9, pp , [3] J. Li, S.M. Mrkov, On h cofficins of variaion of upim and downim in manufacuring quipmn, Mahmaical roblms in Enginring, Vol. 1, pp. 1-6, 5. ISBN: WCE 9

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