Maintenance Models. Prof. Robert C. Leachman IEOR 130, Methods of Manufacturing Improvement Spring, 2011

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1 Mainenance Models Prof Rober C Leachman IEOR 3, Mehods of Manufacuring Improvemen Spring, Inroducion The mainenance of complex equipmen ofen accouns for a large porion of he coss associaed wih ha equipmen I has been esimaed, for example, ha he mainenance coss in he miliary comprise almos one-hird of all of he operaing coss incurred Alernaive philosophies of equipmen mainenance are as follows: Breadown mainenance The equipmen is pu in service and operaed unil i fails Mainenance forces hen repair he equipmen and aemp o resore i as closely as possible o a lie-new condiion, whereupon he equipmen is pu bac in operaion Mainenance is confined o repair following failures Prevenaive mainenance The equipmen is periodically aen ou of service for scheduled mainenance including replacemen of worn componens, inspecion and cleaning, ec The frequency of machine mainenance may be based on hours of usage, number of machine cycles, calendar ime, ec Repairs following failures also are performed as required on an on-call basis (Hopefully, he prevenaive mainenance maes failures less liely) 3 Predicive mainenance The equipmen is coninually moniored or frequenly inspeced by manual or auomaed means Required mainenance is idenified and performed upon inspecion (This philosophy presumes a level of nowledge abou he equipmen sufficien o be able o deec flaws or wear in he equipmen upon inspecion) As before, repairs following failures are performed as required on an on-call basis (Hopefully, he frequen inspecion or coninuous monioring maes failures unliely) Generally, mainenance evolves from breadown mainenance o prevenaive mainenance o predicive mainenance as he manufacuring organizaion and sysems become more sophisicaed and nowledgeable abou he equipmen The famous TPM (oal producive mainenance) paradigm sresses he imporance of compleing he evoluion o predicive mainenance in order o aain he highes produciviy Sandard mainenance erminology is as follows: MBTF Mean ime beween failures This is he expeced ime from compleion of repair following failure unil he nex failure

2 MTTR Mean ime o repair This is he expeced ime o repair he equipmen afer failure, including ime waiing for pars and experise, ime o effec he repairs, and ime o es and re-qualify he equipmen before reurning o producion mode Availabiliy The fracion of ime he equipmen is available for producion operaion For he case of breadown mainenance, he machine follows a renewal process, alernaing beween operaing and failure saes In ha case he availabiliy may be calculaed as A MTBF MTBF MTTR Scheduling Prevenaive Mainenance The purpose of prevenaive mainenance is o decrease he lielihood ha a machine will fail when in operaion A he hear of such a policy is he assumpion ha i coss more o correc a failed machine han i does o perform scheduled mainenance For example, if failure means ha he producion line mus be sopped, whereas prevenaive mainenance could be accomplished a a convenien ime when he producion line is no operaing, hen he cos of planned mainenance would be less han he cos of unplanned mainenance As anoher example, failure in operaion migh resul in desrucion of wor-in-process, whereas planned mainenance migh be performed wihou consuming any produc The absrac problem of planning mainenance inervals is mahemaically equivalen o he absrac problem of planning replacemen inervals for he iem We can hin of mainenance in erms of resoring an iem o lie-new condiion, in effec a replacemen of he iem As before, he case of ineres is where i coss more o replace a failed machine han i does o replace a woring one Because of he memoryless propery of he exponenial disribuion, if an iem or a group of iems obeys an exponenial failure law (ie, is expeced lifeime unil failure does no decrease as i ges older), hen here is no advanage o replacing prior o failure In he exponenial case, he lielihood ha failure will occur in a ime is he same jus afer a planned replacemen as i is for an iem ha has been operaing for an arbirary amoun of ime Hence planned mainenance sraegies have value only if he iem exhibis aging, ha is, i has an increasing failure rae funcion Planned Replacemen of a Single Iem Consider a single piece of coninuously operaing equipmen whose lifeime unil failure is a random variable T wih nown cumulaive disribuion funcion F() We assume ha T is a coninuous random variable Suppose ha i coss c o replace he iem when i fails and c < c o replace he iem prior o failure We assume ha planned replacemens are made exacly ime unis afer he las replacemen The goal is o find he opimal

3 value of o minimize he average cos per uni ime of boh planned and unplanned replacemens (or equivalenly, boh planned and unplanned mainenance) A cycle is he ime beween successive replacemens (or mainenance procedures) Because he process resars iself afer each replacemen, irrespecive of wheher he replacemen was planned or unplanned, we may apply he renewal mehod o obain an expression for he expeced cos per uni ime Tha is, E cos per uni ime E Ecos per cycle lengh of a cycle Now cos per cycle c replacemen is he resul of a failure E P c P replacemen is planned Noice ha P{replacemen is he resul of a failure} = P{ T } = F(), and P{replacemen is planned} = P{ T } = F(), where T is he lifeime of he iem placed ino service a he end of he previous cycle I follows ha per cycle c F( ) c ( F( )) E cos Le T be he ime of failure of he iem placed ino service a he end of he previous cycle Clearly, he nex replacemen will occur a min(t, ) Hence E lengh of a cycle EminT, minx, f ( x) xf ( x) f ( x) xf ( x) [ F( )] The expeced cos per uni ime, say ), is hus expressed as ) c F( ) c [ F( )] xf ( x) [ F( )] 3

4 The disribuion f(x) is ypically no nown in a pracical case However, i is ofen pracical o collec daa on lifeimes in erms of he probabiliy of failure in discree ime periods of life =,, 3, For example, p 3 represens he fracion of iems ha fail in he hird period of service afer replacemen For ineger lifeimes, we can approximae he parial lifeime expecaion erm in erms of he discree daa as xf ( x) p Our final expression for he objecive funcion is hen cf ( ) c[ F( )] ) p [ F( )] The soluion sraegy is o compue ) for =,, 3, and idenify which value of provides he lowes expeced cos per uni ime and herefore he bes planned mainenance inerval We remar ha in a pracical case i may be difficul o esimae he coss c and c involved However, i is almos always pracical o collec and analyze daa concerning he ime required for prevenaive mainenance procedures and for unplanned mainenance (MTTR, including ime waiing for repairs and ime o re-qualify equipmen) If c is se o he ime required o perform unplanned mainenance and c is se o be he ime required o perform planned mainenance, hen ) expresses he expeced down ime per uni ime, ie, availabiliy In ha case, minimizing ) is equivalen o maximizing availabiliy Exponenial Lifeimes I was assered above ha here is no advanage o planned replacemen when he lifeime has an exponenial disribuion We now mahemaically prove his Suppose ha F( ) e Then he expeced lengh of each cycle is e ( e x xe e ) I follows ha e ( ) 4

5 c ( e ) ce ) ( e ) c ( c c ) e e As, he erm e, so ha ) c Furhermore, G () I can be shown using calculus ha ) is monoonically decreasing for all Hence he opimal soluion is, ie, planned replacemen should never be made We have shown ha if he lifeime disribuion is exponenial (ie, i has a consan failure rae), hen here is no economy in replacing an iem prior o he ime i fails This also holds if he failure rae is decreasing 3 Bloc Replacemen for a Group of Iems In cerain circumsances i is more economical o replace groups of iems a he same ime raher han one by one For example, i could be more economical o replace all he ires on a ruc when a replacemen is made The coss of ransporing he ruc o a service area, placing he ruc on a lif, and paying a echnician o moun and balance he ires could be comparable o he cos of a ire iself If all he ires were replaced simulaneously, his cos would be incurred less ofen han if he ires were individually replaced This secion develops a model o deermine he opimal ime o replace an enire group of iems To eep he mahemaics racable, we will assume ha he lifeime of each operaing uni is a discree random variable wih a nown disribuion Tha is, suppose p is he probabiliy ha an iem fails in period assuming i was placed in service a period These probabiliies may be esimaed direcly from hisorical daa or compued from a coninuous disribuion Assume ha n iems are placed ino service a ime Suppose here is no bloc replacemen and all iems ha fail in a period are replaced a he end of ha period We will also assume for simpliciy ha p is he acual proporion of unis periods old ha fail Then he number of failures occurring in period is n = n p In period he proporion of he original group of iems ha fail is n p, and he proporion of iems placed ino service in period ha fail is n p Hence, he expeced number of failures in period is n = n p + n p Coninuing wih his argumen, we obain n n p n p n p Now suppose ha individual replacemens cos a each and he enire bloc of n componens can be replaced for a If all n iems were replaced a he end of each period, he cos each period would be a + a n If all n iems were replaced a he end of every oher period, he cos incurred every wo periods would be a + a (n + n ) or an average 5

6 per period cos of [a + a (n + n )] / Similarly, he average per period cos of replacing all n iems afer periods is ) a a j n j The opimal number of periods o replace all n iems is he value of ha minimizes ) The minimum value of ) should be compared o he expeced cos per period assuming ha iems are replaced as hey fail Le E ( T ) p represen he expeced lifeime of a single componen Then he cos of maing replacemens on a one-a-a-ime basis is a / E(T) or n a / E(T) for he enire bloc of iems This should be compared o he opimal value of ) o deermine if a bloc replacemen sraegy is economical Acnowledgemen These lecure noes are parially adaped from Nahmias () Bibliography Nahmias, Seven () Producion and Operaions Analysis, fourh ediion, McGraw Hill Irwin, New Yor 6